text: 3 Pauli observables (excluding Z) whose eigenbases form C ⊂ C ⊂ C C where . . . but the sets no longer mutually unbiased bases. In the qudit case, diﬀerent or- 1 2 3 k>3 form a group. The operations in the ﬁrst two levels arise derings and phases of the measurement operators can naturally in the context of fault-tolerant quantum com- have an eﬀect on the quantumly achievable expectation putation, whereas an operation from third level is typi- value. B cally also required to enable universal quantum compu- The expectation value of from Eq. (13) can be related C ≤ ≤ tation. Properties of for multiple qubits are discussed to the quantum value, 0 ν 1, of a nonlocal game 3 C in [30] whereas the diagonal single-qudit subset of , as [5, 6], which(cid:88)takes the general form 3 discussed in [27], is given by M above. It is in this | (cid:105)a,b,c | (cid:105) sense that we say states fa,b,c and Ja,b,c reﬂect the | structure of (the third level of) the Cliﬀord hierarchy. ν = P(x, y)V (a, b, x, y)P(a, b x, y) (15) ∈Z B. Generalizing the CHSH game a,b,x,y p The familiar two-qubit CHSH inequality takes the form for measurement settings (x, y), outcomes (a, b) and (cid:104)B(cid:105) ≤ ∈ { } LHV 2 with a Bell operator a predicate or pay-oﬀ function V 0, 1 . By choos- max (cid:88) ing a ﬂat probability distribution over all measure- ment settings, P(x, y) = 1/p2, and a winning condition B = XX + XY + Y X − Y Y = 1 ωxyA B (9) V (a, b, x, y) = δa+b+xy,0 we ar(cid:88)rive at x y x,y=0 { } { } { } where A , A = B , B = X, Y (10) 1 (cid:104) |B| (cid:105) 1 | 0 1 0 1 ν = Ψ Ψ = P(a, b x, y), (16) p2 p2 a+b+xy=0 − and ω = 1 is a p-th root of unity. The maximum √ where the right hand side of the above expression is expectation value using local hidden variables (LHV) is a more general form for CHSH-type games, amenable to 2 whereas quantum mechanics can achieve 2 2 which is B POVMs as well as projective measurements (see also a the maximum eigenvalue, λ ( ), of the Bell operator. max discussion in [10]). The optimal shared state that Alice and Bob can measure (cid:104)B(cid:105) | (cid:105) The maximum classical value LHV is given by ﬁnd- in this case is given by J from Eq. (7) i.e., max 1,0,0 ing local hidden variable assignments √ (cid:104)B(cid:105)QM = (cid:104)J |B|J (cid:105) = 2 2. (11) (cid:126)a = (a0, . . . , ax, . . . , ap−1) ∈ Zpp max 1,0,0 1,0,0 (cid:126)b = (b , . . . , b , . . . , b − ) ∈ Zp 0 y p 1 p Generalizing the above CHSH Bell operator to qudits that tell Alice and Bob which element of the spectrum of [9], we ﬁnd it convenient to deﬁne two closely related their observable they should output for a given measure- operators (cid:88) (cid:88) ment setting i.e., (cid:55)→ (cid:55)→ A ωax , B ωby. B∗ 1 x y = (cid:88) ωnxyAnBn, (12) p n∈Z∗p x,y∈Zp x y The best choices of (cid:126)a an(cid:88)d (cid:126)b maximize the quantity B 1 − = p ∈Z ωnxyAnxByn. (13) (cid:104)B(cid:105)LHV = 1 (cid:88)p 1 ωnxy(ωax)n(ωby)n n,x,y p max p n,x,y=0 B B∗ I quwenhtelyreλω =(Be2)π=i/pλso t(hBa∗t) + =p. Th+e tprapc2elaesnsdvceorsnisoen- = ∈Z δax+by+xy,0 B∗ max max a,b,x,y p is more convenient to work with but the maximal B value of is a more useful quantity. i.e., they give the best classical strategy for participants To aid our analysis we use the Pauli measurement op- trying to maximize the number of instances where a + erators adopted in [9] b + xy = 0 mod p. Liang et al. [10] provide an explicit classical strategy achieving (cid:104)B(cid:105) ≥ − A = ωx(2x+1)/2D | , B = ωy(y+1)/4D | (14) LHV 3p 2. x (1 x) y (1 y/2) max Turning to quantum strategies, Bavarian and Shor’s ∈ Z where x, y , so that A = B . The operators result [11] (restricted to q = p) says that the quantum { ∈ Z } p y/2 y D | , j are a generalization of the qubit mea- value (16) of a generalized CHSH game is bounded above (1 j) { p } surements X, Y used above insofar as they are the (via Information Causality [12]) by ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Pauli

text: observables

text: (excluding

text: Z)

text: whose

text: eigenbases

text: form

text: C

text: C

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text: but

text: the

text: sets

text: no

text: longer

text: mutually

text: unbiased

text: bases.

text: In

text: the

text: qudit

text: case,

text: diﬀerent

text: or-

text: form

text: a

text: group.

text: The

text: operations

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text: the

text: ﬁrst

text: two

text: levels

text: arise

text: derings

text: and

text: phases

text: of

text: the

text: measurement

text: operators

text: can

text: naturally

text: in

text: the

text: context

text: of

text: fault-tolerant

text: quantum

text: com-

text: have

text: an

text: eﬀect

text: on

text: the

text: quantumly

text: achievable

text: expectation

text: putation,

text: whereas

text: an

text: operation

text: from

text: third

text: level

text: is

text: typi-

text: value.

text: B

text: cally

text: also

text: required

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text: universal

text: quantum

text: compu-

text: The

text: expectation

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text: of

text: from

text: Eq.

text: can

text: be

text: related

text: C

text: tation.

text: Properties

text: of

text: for

text: multiple

text: qubits

text: are

text: discussed

text: to

text: the

text: quantum

text: value,

text: ν

text: of

text: a

text: nonlocal

text: game

text: C

text: in

text: whereas

text: the

text: diagonal

text: single-qudit

text: subset

text: of

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text: the

text: general

text: form

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text: is

text: given

text: by

text: M

text: above.

text: It

text: is

text: in

text: this

text: sense

text: that

text: we

text: say

text: states

text: fa,b,c

text: and

text: Ja,b,c

text: reﬂect

text: the

text: structure

text: of

text: (the

text: third

text: level

text: of)

text: the

text: Cliﬀord

text: hierarchy.

text: ν

text: P(x,

text: y)V

text: (a,

text: b,

text: x,

text: y)P(a,

text: b

text: x,

text: y)

text: ∈Z

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text: the

text: CHSH

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text: a,b,x,y

text: p

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text: CHSH

text: inequality

text: takes

text: the

text: form

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text: (x,

text: y),

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text: b)

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text: measure-

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text: P(x,

text: y)

text: and

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text: B

text: XX

text: XY

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text: Y

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text: ωxyA

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text: b,

text: x,

text: y)

text: we

text: at

text: x

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text: where

text: A

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text: B

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text: X,

text: Y

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text: Ψ

text: Ψ

text: P(a,

text: b

text: x,

text: y),

text: and

text: ω

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text: a

text: p-th

text: root

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text: unity.

text: The

text: maximum

text: where

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text: right

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text: of

text: the

text: above

text: expression

text: is

text: expectation

text: value

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text: local

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text: (LHV)

text: is

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text: general

text: form

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text: CHSH-type

text: games,

text: amenable

text: to

text: whereas

text: quantum

text: mechanics

text: can

text: achieve

text: which

text: is

text: B

text: POVMs

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text: projective

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text: maximum

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text: λ

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text: Bell

text: operator.

text: max

text: discussion

text: in

text: The

text: optimal

text: shared

text: state

text: that

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text: and

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text: measure

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text: maximum

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text: given

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text: Eq.

text: i.e.,

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text: |B|J

text: ax,

text: Zpp

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text: (b

text: b

text: b

text: Zp

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text: the

text: above

text: CHSH

text: Bell

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text: to

text: qudits

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text: tell

text: Alice

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text: spectrum

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text: ﬁnd

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text: deﬁne

text: two

text: closely

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text: observable

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text: should

text: output

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text: a

text: given

text: measure-

text: operators

text: ment

text: setting

text: i.e.,

text: A

text: ωax

text: B

text: ωby.

text: B∗

text: x

text: y

text: ωnxyAnBn,

text: p

text: n∈Z∗p

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text: x

text: y

text: The

text: best

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text: maximize

text: the

text: quantity

text: B

text: p

text: ∈Z

text: ωnxyAnxByn.

text: ωnxy(ωax)n(ωby)n

text: n,x,y

text: p

text: max

text: p

text: B

text: B∗

text: I

text: quwenhtelyreλω

text: t(hBa∗t)

text: =p.

text: Th+e

text: ∈Z

text: B∗

text: max

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text: p

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text: aid

text: our

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text: we

text: use

text: the

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text: measurement

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text: the

text: number

text: of

text: instances

text: where

text: a

text: erators

text: adopted

text: in

text: b

text: xy

text: mod

text: p.

text: Liang

text: et

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text: provide

text: an

text: explicit

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text: A

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text: x

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text: Bavarian

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text: Z

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text: x,

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text: quantum

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text: j)

text: p

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text: X,

text: Y

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text: (via

text: Information

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text: ##LTLine##

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text: ##LTLine##


text: 6 − (cid:104)B(cid:105) (cid:104)B(cid:105) provided p does not divide n. Applied to Eq. (30) the p p(1 + p√p1) 4−p QmMax LmHaxV above bound implies 3 6.4641 6.4115 6 5 13.9443 20. 13.0902 12 (cid:104)B(cid:105) S ≤ QM = λ ( ) 4p. (33) 7 22.8745 28. 19.4112 19 max max 11 44.1662 44. 34.6464 37 13 56.2666 52. 48.3481 47 ≥ (cid:16)For p >(cid:17) 7 this bound is more restrictive than the 17 82.9697 68. 55.1022 66 ≥ Information Causality Sbound(cid:104)B. (cid:105) In Table I we compare 19 97.4602 76. 72.6084 79 this bound with λ ( ) = QM and with the bound 23 128.508 92. 74.8954 ≥≥ 99 p 1 + p√−1 imposmedaxby Informamtaixon Causality. 29 179.785 116. 104.819 135 p The qutrit case must be handled individually – terms − like 1/12 are ill-deﬁned and the Weil bound is not ap- TABLE I. Comparison of the bounds imposed by Information plicable. One can explicitly check that the optimal quan- Causality (ﬁrst column) and Weil (second column) with the tum state for Alice and Bob to measure is what we have optimal quantum and local hidden variable strategies (third come to expect i.e. a maximally entangled state that is and fourth columns). The LHV lower bounds are reproduced (cid:104)B(cid:105) isomorphic to a magic state, from [10] whereas the operator norm QM is attained by ma|x (cid:105) measuring the maximally entangled states Ja,b,c . Because √ we have restricted ourselves to Pauli measurements, the quan- (cid:104)B(cid:105) (cid:104) |B| (cid:105) tity (cid:104)B(cid:105)QmMax is less than what is achievable in the unrestricted QmMax = J1,1,0 J1,1,0 = 3 3 cos(π/18) = 6.4115. case. (34) Liang et al. [10] have shown that this is truly the quantum maximum; even allowing for POVMs this value Our proof provides an explicit expression showing ex- |(cid:104) | (cid:105)| cannot be surpassed. actly how probabilities ψV f 2 are permuted be- B a,b,c | (cid:105) B. Minimum uncertainty equatorial states tween diﬀerent bases. For the magic state f− − 1/12, 1/8,c under consideration, the overlap with vector V in basis B Deﬁne a generalized equatorial state in terms of a vec- B is constant whenever (cid:18) (cid:19) tor of phases φ(cid:126) = (φ := 0, φ , . . . , φ − ), 0 1 p 1 (cid:88) − 1 VB = B B + 2 . (28) | (cid:105) √1 p−1 | (cid:105) ψ (φ(cid:126)) = eiφk k . (35) eq p k=0 Combining thi(cid:88)s with Eq. (25) we ﬁnd − This state has p 1 free real parameters, which is ex- λ (S) = |(cid:104)ψ−B(B+12)|f− − (cid:105)|2 (29) actly half the number of parameters that specify an arbi- max B 1/12, 1/8,c trary pure state. This is already a very important class B|(cid:104) | (cid:105)| of states for cryptographic protocols e.g., optimal cloning = p + f−1/12,−1/8,c 2 (30) procedures are known for states of this form [43, 44]. The total collision-entropic uncertainty across p non- computational Pauli bases is bounded fro(cid:32)m below(cid:33) by All overlaps between (non-computational) MUB vec- (cid:88) (cid:88) tors and magic states take the same form viz. (cid:88) − − |(cid:104) | (cid:105)| ≥ − 2 1 log ψ (φ(cid:126)) ψV 4 p log p (36) |(cid:104)ψBV |fa,b,c(cid:105)|2 = p1| ωak3+(b−2−1B)k2+(c−V )k|2 (31) B∈Zp (cid:80) V eq B p k |(cid:104) | (cid:105)| since ψ (φ(cid:126)) V 4 = 1/p and we already know V eq The values of these overlaps are constrained by the from Eq. (18) that the sum over all p + 1 bases is iden- (cid:80) Weil bound [41, 42], which says that a polynomial f of tically 2 for any pure state. Because of the balanced- Z | (cid:105) degree n over a p(cid:88)rime ﬁeld satisﬁes ness property of f proven in Sec. we have that p |(cid:104) | (cid:105)| a,b,c f ψV 4 is independent of B and therefore the √ V a,b,c B | | ≤ − above inequality is saturated. In summary, of all equa- ωf(x) (n 1) p (32) torial states (35) it turns out that magic states minimize x the total collision entropy and saturate the above en- tropic uncertainty relation. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: provided

text: p

text: does

text: not

text: divide

text: n.

text: Applied

text: to

text: Eq.

text: the

text: p

text: QmMax

text: LmHaxV

text: above

text: bound

text: implies

label: equation

text: S

text: QM

text: λ

text: max

text: max

text: p

text: this

text: bound

text: is

text: more

text: restrictive

text: than

text: the

text: Information

text: Causality

text: In

text: Table

text: I

text: we

text: compare

text: this

text: bound

text: with

text: λ

text: QM

text: and

text: with

text: the

text: bound

text: p

text: p√−1

text: imposmedaxby

text: Informamtaixon

text: Causality.

text: p

text: The

text: qutrit

text: case

text: must

text: be

text: handled

text: individually

text: terms

text: like

text: are

text: ill-deﬁned

text: and

text: the

text: Weil

text: bound

text: is

text: not

text: ap-

text: TABLE

text: I.

text: Comparison

text: of

text: the

text: bounds

text: imposed

text: by

text: Information

text: plicable.

text: One

text: can

text: explicitly

text: check

text: that

text: the

text: optimal

text: quan-

text: Causality

text: (ﬁrst

text: column)

text: and

text: Weil

text: (second

text: column)

text: with

text: the

text: tum

text: state

text: for

text: Alice

text: and

text: Bob

text: to

text: measure

text: is

text: what

text: we

text: have

text: optimal

text: quantum

text: and

text: local

text: hidden

text: variable

text: strategies

text: (third

text: come

text: to

text: expect

text: i.e.

text: a

text: maximally

text: entangled

text: state

text: that

text: is

text: and

text: fourth

text: columns).

text: The

text: LHV

text: lower

text: bounds

text: are

text: reproduced

text: isomorphic

text: to

text: a

text: magic

text: state,

text: from

text: whereas

text: the

text: operator

text: norm

text: QM

text: is

text: attained

text: by

text: ma|x

text: measuring

text: the

text: maximally

text: entangled

text: states

text: Ja,b,c

text: Because

text: we

text: have

text: restricted

text: ourselves

text: to

text: Pauli

text: measurements,

text: the

text: quan-

text: |B|

text: tity

text: is

text: less

text: than

text: what

text: is

text: achievable

text: in

text: the

text: unrestricted

text: QmMax

text: case.

text: Liang

text: et

text: al.

text: have

text: shown

text: that

text: this

text: is

text: truly

text: the

text: quantum

text: maximum;

text: even

text: allowing

text: for

text: POVMs

text: this

text: value

text: Our

text: proof

text: provides

text: an

text: explicit

text: expression

text: showing

text: ex-

text: cannot

text: be

text: surpassed.

text: actly

text: how

text: probabilities

text: ψV

text: f

text: are

text: permuted

text: be-

text: B

text: a,b,c

text: B.

text: Minimum

text: uncertainty

text: equatorial

text: states

text: tween

text: diﬀerent

text: bases.

text: For

text: the

text: magic

text: state

text: f−

text: under

text: consideration,

text: the

text: overlap

text: with

text: vector

text: V

text: in

text: basis

text: B

text: Deﬁne

text: a

text: generalized

text: equatorial

text: state

text: in

text: terms

text: of

text: a

text: vec-

text: B

text: is

text: constant

text: whenever

text: tor

text: of

text: phases

text: (φ

text: φ

text: φ

text: p

text: VB

text: B

text: B

text: √1

text: ψ

text: eiφk

text: k

text: eq

text: p

text: Combining

text: with

text: Eq.

text: we

text: ﬁnd

text: This

text: state

text: has

text: p

text: free

text: real

text: parameters,

text: which

text: is

text: ex-

text: λ

text: (S)

text: actly

text: half

text: the

text: number

text: of

text: parameters

text: that

text: specify

text: an

text: arbi-

text: max

text: B

text: trary

text: pure

text: state.

text: This

text: is

text: already

text: a

text: very

text: important

text: class

text: of

text: states

text: for

text: cryptographic

text: protocols

text: e.g.,

text: optimal

text: cloning

text: p

text: procedures

text: are

text: known

text: for

text: states

text: of

text: this

text: form

text: The

text: total

text: collision-entropic

text: uncertainty

text: across

text: p

text: non-

text: computational

text: Pauli

text: bases

text: is

text: bounded

text: by

text: All

text: overlaps

text: between

text: (non-computational)

text: MUB

text: vec-

text: tors

text: and

text: magic

text: states

text: take

text: the

text: same

text: form

text: viz.

text: log

text: ψ

text: ψV

text: p

text: log

text: p

text: B∈Zp

text: V

text: eq

text: B

text: p

text: k

text: since

text: ψ

text: V

text: and

text: we

text: already

text: know

text: V

text: eq

text: The

text: values

text: of

text: these

text: overlaps

text: are

text: constrained

text: by

text: the

text: from

text: Eq.

text: that

text: the

text: sum

text: over

text: all

text: p

text: bases

text: is

text: iden-

text: Weil

text: bound

text: which

text: says

text: that

text: a

text: polynomial

text: f

text: of

text: tically

text: for

text: any

text: pure

text: state.

text: Because

text: of

text: the

text: balanced-

text: Z

text: degree

text: n

text: over

text: a

text: ﬁeld

text: satisﬁes

text: ness

text: property

text: of

text: f

text: proven

text: in

text: Sec.

text: we

text: have

text: that

text: p

text: a,b,c

text: f

text: ψV

text: is

text: independent

text: of

text: B

text: and

text: therefore

text: the

text: V

text: a,b,c

text: B

text: above

text: inequality

text: is

text: saturated.

text: In

text: summary,

text: of

text: all

text: equa-

text: ωf(x)

text: (n

text: p

text: torial

text: states

text: it

text: turns

text: out

text: that

text: magic

text: states

text: minimize

text: x

text: the

text: total

text: collision

text: entropy

text: and

text: saturate

text: the

text: above

text: en-

text: tropic

text: uncertainty

text: relation.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ∼ Figure 3. Dynamic spectrum of a 7-hr long observation of PSR J1614–2230, after interference signals have been removed from the data (showing up as dark lines in the plot). Lighter color indicates higher intensity of the pulsar signal, with separate bright islands referred to as scintles. Analyses of the scintles in the dynamic spectrum can be used to estimate scintillation parameters for a pulsar. 2π∆ν τ = C (5) d d 1 where C is a constant that depends on the geometry and spectrum of the electron density, 1 − and with expected value ranging from 0.6 1.5 [19]. By creating dynamic spectra of each observation for each of the PTA pulsars, the scintillation parameters can be monitored, and in particular the variation of the scattering delays can be analyzed. This is important, since a constant delay would not cause any additional errors in timing precision, no matter its magnitude, but a highly variable delay would contribute largely to the timing noise. Therefore, the maximum variation of the delays need to be monitored, and can be compared to the individual TOA error to see if the scattering is one of the dominating sources ∼ of noise in the data. At the commonly used observing frequency band centered at 1.4 GHz, for most PTA pulsars, the scattering delay variations are in general much smaller than the current TOA uncertainty [7, 20], and hence are currently not limiting the timing precision. With longer data spans and new developments in instrumentation, the timing precision will improve, and hence a detailed scattering analysis may become increasingly important in the future. The latest developments in ISM mitigation include a technique to estimate pulse broadening times and scattering delays using the signal-processing tool Cyclic Spectroscopy (CS; [21, 22]). This technique allows determination of the ISM impulse response function and extraction of the intrinsic pulse proﬁle for highly scattered pulsars with a large ﬂux density. CS backends have ##LTFigure## ##LTFigure##

label: caption

text: Figure

text: Dynamic

text: spectrum

text: of

text: a

text: long

text: observation

text: of

text: PSR

text: after

text: interference

text: signals

text: have

text: been

text: removed

text: from

text: the

text: data

text: (showing

text: up

text: as

text: dark

text: lines

text: in

text: the

text: plot).

text: Lighter

text: color

text: indicates

text: higher

text: intensity

text: of

text: the

text: pulsar

text: signal,

text: with

text: separate

text: bright

text: islands

text: referred

text: to

text: as

text: scintles.

text: Analyses

text: of

text: the

text: scintles

text: in

text: the

text: dynamic

text: spectrum

text: can

text: be

text: used

text: to

text: estimate

text: scintillation

text: parameters

text: for

text: a

text: pulsar.

text: 2π∆ν

text: τ

text: C

text: d

text: d

text: where

text: C

text: is

text: a

text: constant

text: that

text: depends

text: on

text: the

text: geometry

text: and

text: spectrum

text: of

text: the

text: electron

text: density,

text: and

text: with

text: expected

text: value

text: ranging

text: from

text: By

text: creating

text: dynamic

text: spectra

text: of

text: each

text: observation

text: for

text: each

text: of

text: the

text: PTA

text: pulsars,

text: the

text: scintillation

text: parameters

text: can

text: be

text: monitored,

text: and

text: in

text: particular

text: the

text: variation

text: of

text: the

text: scattering

text: delays

text: can

text: be

text: analyzed.

text: This

text: is

text: important,

text: since

text: a

text: constant

text: delay

text: would

text: not

text: cause

text: any

text: additional

text: errors

text: in

text: timing

text: precision,

text: no

text: matter

text: its

text: magnitude,

text: but

text: a

text: highly

text: variable

text: delay

text: would

text: contribute

text: largely

text: to

text: the

text: timing

text: noise.

text: Therefore,

text: the

text: maximum

text: variation

text: of

text: the

text: delays

text: need

text: to

text: be

text: monitored,

text: and

text: can

text: be

text: compared

text: to

text: the

text: individual

text: TOA

text: error

text: to

text: see

text: if

text: the

text: scattering

text: is

text: one

text: of

text: the

text: dominating

text: sources

text: of

text: noise

text: in

text: the

text: data.

text: At

text: the

text: commonly

text: used

text: observing

text: frequency

text: band

text: centered

text: at

text: GHz,

text: for

text: most

text: PTA

text: pulsars,

text: the

text: scattering

text: delay

text: variations

text: are

text: in

text: general

text: much

text: smaller

text: than

text: the

text: current

text: TOA

text: uncertainty

text: and

text: hence

text: are

text: currently

text: not

text: limiting

text: the

text: timing

text: precision.

text: With

text: longer

text: data

text: spans

text: and

text: new

text: developments

text: in

text: instrumentation,

text: the

text: timing

text: precision

text: will

text: improve,

text: and

text: hence

text: a

text: detailed

text: scattering

text: analysis

text: may

text: become

text: increasingly

text: important

text: in

text: the

text: future.

text: The

text: latest

text: developments

text: in

text: ISM

text: mitigation

text: include

text: a

text: technique

text: to

text: estimate

text: pulse

text: broadening

text: times

text: and

text: scattering

text: delays

text: using

text: the

text: signal-processing

text: tool

text: Cyclic

text: Spectroscopy

text: (CS;

text: This

text: technique

text: allows

text: determination

text: of

text: the

text: ISM

text: impulse

text: response

text: function

text: and

text: extraction

text: of

text: the

text: intrinsic

text: pulse

text: proﬁle

text: for

text: highly

text: scattered

text: pulsars

text: with

text: a

text: large

text: ﬂux

text: density.

text: CS

text: backends

text: have

text: ##LTFigure##

text: ##LTFigure##


text: 5 is a signature of the PG of the system, while the energy split- The interplay between mass and coupling allows us to re- ting of the two levels is associated to the superconducting T . late global trends in the phase diagram for cuprate supercon- c We see that Fig. 3a) mimics the generic phase diagram of the ductors with the strength of the SU(2) symmetry breaking. ∗ cuprates where the PG line T abruptly plunges inside the SC In the electron doped compounds the coupling between AF dome at some value of oxygen doping. modes and conduction electrons is believed to be weaker than Although it is very encouraging to see that the SU(2) regime for the hole doped case. From the above analysis we ﬁnd that has a non-zero probability to exist, one can wonder whether the SU(2) symmetry is less respected in that case leading to a the paramagnon mass in hole doped cuprate superconductors smaller PG dome and overall smaller SC, see Fig. 7. For the is small, since typically the AF correlation length is of few same reasons La compounds where the coupling is also be- lattice constants40. The issue is addressed in Fig. 4a), where lieved to be small behave similarly. On the other hand, hole- the values of χ and ∆ are shown for a ﬁxed mass as a func- doped cuprates like YBCO live in the large mass and large tion of the coupling constant λ. Here again a generic pattern coupling regime. This results in broad gapped regions in the emerges. For small λ the SU(2) symmetry is broken, but sur- BZ where the symmetry is well respected so that both the PG prisingly, above a certain threshold of λ, the SU(2) symmetry and SC dome are large, as shown in Fig. 6. is almost completely restored. As seen in Fig. 4a), the bigger Finally, let us mention that a major effect of a magnetic the mass is, the stronger the coupling constant needs to be for ﬁeld is to invert the order of the level splitting between the the symmetry to be restored. Figure 4b) shows that this be- CDW and the SC components10,11. This will favor the charge havior is quite general among the different compounds. Note order compared to the SC pairing. We believe that Umklapp however that the electron -doped compound is less sensitive scattering can have a similar effect to invert the level ordering, to the effect of increasing the coupling constant, compared to but leave detailed investigation for further studies. the other hole-doped ones, for which the SU(2) symmetry is restored for large enough λ. B. Spacial dependence of the splitting The SU(2) symmetry is not only broken due to the curva- ture of the Fermi surface at the hot spots, but it is typically broken in the BZ away from the eight hot spots. Figure 5 | | | | shows the typical shape of χ and ∆ for four compounds under investigation for the most unfavorable case for the sym- metry, that is for small values of the mass and coupling con- stant. The level splitting is shown as a density plot in the bottom. It is rather small almost everywhere in the BZ and 0.1 at the hotspot positions with maxima of the order of 20-40% around the “shadow” Fermi surface. The main learning from Π0 0.05 these plots is that the variations of the Fermi surface geometry gives a rather small departing from the SU(2)-degeneracy for 0 a various range of compounds. In all cases, the SU(2) sym- 0Π metry is well respected at the hotspot positions. | | FIG. 6. (Color online) Generic picture of the gap functions χ , In Fig. 6 we place ourselves in the strong coupling and | | | − | | | strong mass regime and plot the variation of |χ| , |∆| and ∆ and χ ∆ / ∆max in the ﬁrst BZ for hole-doped cuprates, here | − | explicitly shown for YBCO. The ﬁgures correspond to the large mass χ ∆ /∆. The level splitting is also shown in Fig. 6 and and large coupling limit (λ = 160 and mass m = 0.5, so that E0 (cid:39) found to be much smaller than the previous case in Fig. 5, and 160K and Eeff = 226K) where the SU(2) symmetry is well respected. has dropped to an order of 5-10% 2. Interestingly, the typical | | | | shape of χ and ∆ in the BZ has changed compared to Fig. 5, with maxima now around the zone edge. This is the justi- ﬁcation that “hot regions” instead of “hot spots” is the correct description of hole doped cuprate superconductors within the IV. CONCLUSION | | spin-fermion model. Note that since the maximum of χ is now at the zone edge, the wave vector corresponding to the In conclusion, this paper gives ﬁrm ground to the intuition associated charge order is now parallel to the x/y axes of the that the charge sector is a key player in the physics of cuprate system, in similarity with the ﬁndings of Ref. [12]. superconductors. While the main instability is still the AF C. Global trends in cuprate superconductors ordering, the d-wave bond order relates to the d-wave pair- ing through an SU(2) symmetry. We have shown that there exists a wide range of parameters where the SU(2) degen- eracy is fulﬁlled, which gives a natural explanation for the large PG regime observed in certain compounds. We argue 2 The gap functions and level splitting of the other compounds are very sim- that compounds like electron doped cuprates or the La com- ilar and therefore not shown in Fig. 6 pounds are outside the regime of SU(2) degeneracy, and the ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: is

text: a

text: signature

text: of

text: the

text: PG

text: of

text: the

text: system,

text: while

text: the

text: energy

text: split-

text: The

text: interplay

text: between

text: mass

text: and

text: coupling

text: allows

text: us

text: to

text: re-

text: ting

text: of

text: the

text: two

text: levels

text: is

text: associated

text: to

text: the

text: superconducting

text: T

text: late

text: global

text: trends

text: in

text: the

text: phase

text: diagram

text: for

text: cuprate

text: supercon-

text: c

text: We

text: see

text: that

text: Fig.

text: mimics

text: the

text: generic

text: phase

text: diagram

text: of

text: the

text: ductors

text: with

text: the

text: strength

text: of

text: the

text: symmetry

text: breaking.

text: cuprates

text: where

text: the

text: PG

text: line

text: T

text: abruptly

text: plunges

text: inside

text: the

text: SC

text: In

text: the

text: electron

text: doped

text: compounds

text: the

text: coupling

text: between

text: AF

text: dome

text: at

text: some

text: value

text: of

text: oxygen

text: doping.

text: modes

text: and

text: conduction

text: electrons

text: is

text: believed

text: to

text: be

text: weaker

text: than

text: Although

text: it

text: is

text: very

text: encouraging

text: to

text: see

text: that

text: the

text: regime

text: for

text: the

text: hole

text: doped

text: case.

text: From

text: the

text: above

text: analysis

text: we

text: ﬁnd

text: that

text: has

text: a

text: non-zero

text: probability

text: to

text: exist,

text: one

text: can

text: wonder

text: whether

text: the

text: symmetry

text: is

text: less

text: respected

text: in

text: that

text: case

text: leading

text: to

text: a

text: the

text: paramagnon

text: mass

text: in

text: hole

text: doped

text: cuprate

text: superconductors

text: smaller

text: PG

text: dome

text: and

text: overall

text: smaller

text: SC,

text: see

text: Fig.

text: For

text: the

text: is

text: small,

text: since

text: typically

text: the

text: AF

text: correlation

text: length

text: is

text: of

text: few

text: same

text: reasons

text: La

text: compounds

text: where

text: the

text: coupling

text: is

text: also

text: be-

text: lattice

text: The

text: issue

text: is

text: addressed

text: in

text: Fig.

text: where

text: lieved

text: to

text: be

text: small

text: behave

text: similarly.

text: On

text: the

text: other

text: hand,

text: hole-

text: the

text: values

text: of

text: χ

text: and

text: are

text: shown

text: for

text: a

text: ﬁxed

text: mass

text: as

text: a

text: func-

text: doped

text: cuprates

text: like

text: YBCO

text: live

text: in

text: the

text: large

text: mass

text: and

text: large

text: tion

text: of

text: the

text: coupling

text: constant

text: λ.

text: Here

text: again

text: a

text: generic

text: pattern

text: coupling

text: regime.

text: This

text: results

text: in

text: broad

text: gapped

text: regions

text: in

text: the

text: emerges.

text: For

text: small

text: λ

text: the

text: symmetry

text: is

text: broken,

text: but

text: sur-

text: BZ

text: where

text: the

text: symmetry

text: is

text: well

text: respected

text: so

text: that

text: both

text: the

text: PG

text: prisingly,

text: above

text: a

text: certain

text: threshold

text: of

text: λ,

text: the

text: symmetry

text: and

text: SC

text: dome

text: are

text: large,

text: as

text: shown

text: in

text: Fig.

text: is

text: almost

text: completely

text: restored.

text: As

text: seen

text: in

text: Fig.

text: the

text: bigger

text: Finally,

text: let

text: us

text: mention

text: that

text: a

text: major

text: effect

text: of

text: a

text: magnetic

text: the

text: mass

text: is,

text: the

text: stronger

text: the

text: coupling

text: constant

text: needs

text: to

text: be

text: for

text: ﬁeld

text: is

text: to

text: invert

text: the

text: order

text: of

text: the

text: level

text: splitting

text: between

text: the

text: the

text: symmetry

text: to

text: be

text: restored.

text: Figure

text: shows

text: that

text: this

text: be-

text: CDW

text: and

text: the

text: SC

text: This

text: will

text: favor

text: the

text: charge

text: havior

text: is

text: quite

text: general

text: among

text: the

text: different

text: compounds.

text: Note

text: order

text: compared

text: to

text: the

text: SC

text: pairing.

text: We

text: believe

text: that

text: Umklapp

text: however

text: that

text: the

text: electron

text: -doped

text: compound

text: is

text: less

text: sensitive

text: scattering

text: can

text: have

text: a

text: similar

text: effect

text: to

text: invert

text: the

text: level

text: ordering,

text: to

text: the

text: effect

text: of

text: increasing

text: the

text: coupling

text: constant,

text: compared

text: to

text: but

text: leave

text: detailed

text: investigation

text: for

text: further

text: studies.

text: the

text: other

text: hole-doped

text: ones,

text: for

text: which

text: the

text: symmetry

text: is

text: restored

text: for

text: large

text: enough

text: λ.

text: B.

text: Spacial

text: dependence

text: of

text: the

text: splitting

text: The

text: symmetry

text: is

text: not

text: only

text: broken

text: due

text: to

text: the

text: curva-

text: ture

text: of

text: the

text: Fermi

text: surface

text: at

text: the

text: hot

text: spots,

text: but

text: it

text: is

text: typically

text: broken

text: in

text: the

text: BZ

text: away

text: from

text: the

text: eight

text: hot

text: spots.

text: Figure

text: shows

text: the

text: typical

text: shape

text: of

text: χ

text: and

text: for

text: four

text: compounds

text: under

text: investigation

text: for

text: the

text: most

text: unfavorable

text: case

text: for

text: the

text: sym-

text: metry,

text: that

text: is

text: for

text: small

text: values

text: of

text: the

text: mass

text: and

text: coupling

text: con-

text: stant.

text: The

text: level

text: splitting

text: is

text: shown

text: as

text: a

text: density

text: plot

text: in

text: the

text: bottom.

text: It

text: is

text: rather

text: small

text: almost

text: everywhere

text: in

text: the

text: BZ

text: and

text: at

text: the

text: hotspot

text: positions

text: with

text: maxima

text: of

text: the

text: order

text: of

text: around

text: the

text: “shadow”

text: Fermi

text: surface.

text: The

text: main

text: learning

text: from

text: Π0

text: these

text: plots

text: is

text: that

text: the

text: variations

text: of

text: the

text: Fermi

text: surface

text: geometry

text: gives

text: a

text: rather

text: small

text: departing

text: from

text: the

text: for

text: a

text: various

text: range

text: of

text: compounds.

text: In

text: all

text: cases,

text: the

text: sym-

text: 0Π

text: metry

text: is

text: well

text: respected

text: at

text: the

text: hotspot

text: positions.

text: FIG.

text: (Color

text: online)

text: Generic

text: picture

text: of

text: the

text: gap

text: functions

text: χ

text: In

text: Fig.

text: we

text: place

text: ourselves

text: in

text: the

text: strong

text: coupling

text: and

text: strong

text: mass

text: regime

text: and

text: plot

text: the

text: variation

text: of

text: |χ|

text: and

text: and

text: χ

text: ∆max

text: in

text: the

text: ﬁrst

text: BZ

text: for

text: hole-doped

text: cuprates,

text: here

text: explicitly

text: shown

text: for

text: YBCO.

text: The

text: ﬁgures

text: correspond

text: to

text: the

text: large

text: mass

text: χ

text: The

text: level

text: splitting

text: is

text: also

text: shown

text: in

text: Fig.

text: and

text: and

text: large

text: coupling

text: limit

text: (λ

text: and

text: mass

text: m

text: so

text: that

text: found

text: to

text: be

text: much

text: smaller

text: than

text: the

text: previous

text: case

text: in

text: Fig.

text: and

text: and

text: Eeff

text: where

text: the

text: symmetry

text: is

text: well

text: respected.

text: has

text: dropped

text: to

text: an

text: order

text: of

text: Interestingly,

text: the

text: typical

text: shape

text: of

text: χ

text: and

text: in

text: the

text: BZ

text: has

text: changed

text: compared

text: to

text: Fig.

text: with

text: maxima

text: now

text: around

text: the

text: zone

text: edge.

text: This

text: is

text: the

text: justi-

text: ﬁcation

text: that

text: “hot

text: regions”

text: instead

text: of

text: “hot

text: spots”

text: is

text: the

text: correct

text: description

text: of

text: hole

text: doped

text: cuprate

text: superconductors

text: within

text: the

text: IV.

text: CONCLUSION

text: spin-fermion

text: model.

text: Note

text: that

text: since

text: the

text: maximum

text: of

text: χ

text: is

text: now

text: at

text: the

text: zone

text: edge,

text: the

text: wave

text: vector

text: corresponding

text: to

text: the

text: In

text: conclusion,

text: this

text: paper

text: gives

text: ﬁrm

text: ground

text: to

text: the

text: intuition

text: associated

text: charge

text: order

text: is

text: now

text: parallel

text: to

text: the

text: x/y

text: axes

text: of

text: the

text: that

text: the

text: charge

text: sector

text: is

text: a

text: key

text: player

text: in

text: the

text: physics

text: of

text: cuprate

text: system,

text: in

text: similarity

text: with

text: the

text: ﬁndings

text: of

text: Ref.

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text: 7 T. Schmitt, G. Ghiringhelli, N. Barišic´, M. K. Chan, C. J. Dorow, G. Yu, X. Zhao, B. Keimer, and M. Greven, Nat. Commun. 5, 5875 1 P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006). (2014). 2 E. Gull, O. Parcollet, and A. J. Millis, Phys. Rev. Lett. 110, 216405 (220313R)..-H. He, M. Hashimoto, H. Karapetyan, J. D. Koralek, J. P. Hinton, J. P. Testaud, V. Nathan, Y. Yoshida, H. Yao, K. Tanaka, W. Meevasana, 3 S. Sorella, G. B. Martins, F. Becca, C. Gazza, L. Capriotti, A. Parola, andRE. .GD. aMgoototore,,PDhy. sH..RLeuv., LS.e-tKt..8M8,o1,1M70.0I2sh(i2k0a0d2o),.H. Eisaki, Z. Hussain, T. P. Devereaux, S. A. Kivelson, J. Orenstein, A. Kapitulnik, and 4 M. R. Norman and C. Pépin, Rep. Prog. Phys. 66, 1547 (2003). Z.-X. Shen, Science 331, 1579 (2011). 5 A. Abanov, A. V. Chubukov, and J. Schmalian, Adv. Phys. 52, 119 (202043I).. M. Vishik, M. Hashimoto, R.-H. He, W.-S. Lee, F. Schmitt, D. Lu, R. G. Moore, C. Zhang, W. Meevasana, T. Sasagawa, S. Uchida, 6 A. Chubukov, D. Pines, and J. Schmalian, in Superconductivity, edited bKy. FKu.jBitae,nSn.emIshanidna,aMnd. JI.shKikeattdeors,oYn. Y(Sopsrhiindgae,rHB.eErliisnakHi,eZid.eHlbuesrsga,in, T. P. Devereaux, and Z.-X. Shen, PNAS 109, 18332 (2012). 2008). 25 Y. Wang and A. Chubukov, Phys. Rev. B 90, 035149 (2014). 7 Close to a magnetic instability, the Spin-Fermion model should capture2t6heDr.eFle.vAangttelrobwe-regn,eDrg. yS.pMhyeslicchseorft,thaenHduMb.bKar.dKmaosdheylawp,itPhhayrsb.iRtreavry. B 91, 054502 (2015). interaction strength, see e.g. Ref. [6]. 27 P. A. Lee, Phys. Rev. X 4, 031017 (2014). 8 M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075128 (2010). 28 E. Fradkin, S. A. Kivelson, and J. M. Tranquada, arXiv:1407.4480 [cond-mat.supr-con] (2014). 9 K. B. Efetov, H. Meier, and C. Pépin, Nat. Phys. 9, 442 (2013). 29 X. Montiel, T. Kloss, C. Pépin, S. Benhabib, Y. Gallais, and A. Sacuto, arXiv:1504.03951 [cond-mat.supr-con] (2015). 10 H. Meier, M. Einenkel, C. Pépin, and K. B. 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Chatterjee, T. Kondo, T. Takeuchi, H. Ikuta,TY..TWakaenugc,hai,nHd.EI.kWuta. ,HPu.dMsoisnt,aNrka,tR. P. hSy.sM. 4a,rk6i9e6w(ic2z0,0A8)..Bansil, S. Sachdev, E. W. Hudson, and J. E. Hoffman, Science 344, 608 (2014). 16 S. E. Sebastian, N. Harrison, R. Liang, D. A. Bonn, W. N. Hardy, C. H.35MKie.lkFeu,jiatan,dCG. .KG. .KLimon,zIa.rLicehe,,PJ.hLyse.eR, Mev.. HLeatmt.i1d0ia8n,,1I9.6A4.0F3ir(m20o1,2S)..Mukhopadhyay, H. Eisaki, S. Uchida, M. J. Lawler, E. A. Kim, and J. C. 17 G. Ghiringhelli, M. Le Tacon, M. Minola, S. Blanco-Canosa, C. Mazzoli,DNa.vBis.,BSrcoioeknecse,3G4.4M, 6.1D2e(2L0u1c4a),.A. Frano, D. G. Hawthorn, F. He, T. Loew, M. M. Sala, D. C. Peets, M. Salluzzo, E. Schierle, R. Sutarto,3G6. AT..DSaaws,aPtzhkyys,.ER.eWv.eBsc8h6k,e0,5B4.5K1e8im(2e0r1,2a)n.d L. Braicovich, Science 337, 821 (2012). 37 D. Sénéchal and A.-M. S. Tremblay, Phys. Rev. Lett. 92, 126401 (2004). 18 T. Wu, H. Mayaffre, S. Krämer, M. Horvatic´, C. Berthier, P. L. Kuhns,3A8 .WP.eRheayvees,chRe.cLkieadngth,aWt t.hNis.aHpaprrdoyx,iDm.aAtio.nBiosnrne,liaabnldeMto.-cHal.cJuulaliteent,he spacial distribution of the gap functions. Nat. Commun. 4, 2113 (2013). 39 R. Comin, R. Sutarto, F. He, E. da Silva Neto, L. Chauviere, A. Frano, R. Liang, W. N. Hardy, D. Bonn, Y. Yoshida, H. Eisaki, J. E. 19 D. LeBoeuf, S. Kramer, W. N. Hardy, R. Liang, D. A. Bonn, and C. ProuHsto, fNfmata.nP,hBys..K9e,i7m9e(r2, 0G1.3A).. Sawatzky, and A. Damascelli, arXiv:1402.5415 [cond-mat.supr-con] (2014). 20 M. Le Tacon, A. Bosak, S. M. Souliou, G. Dellea, T. Loew, R. Heid, K4.0-PV. B. Hohinnkeonv,,GS..GPhaiirlihnegsh,ePl.liB, Mou.rgKersi,scYh.,Saidnids,BA. .KIeviamneorv,,NAa.t.KPuhlyask.ov, C. T. Lin, D. P. Chen, C. Bernhard, and B. Keimer, Nature 430, 650 10, 52 (2014). (2004). 21 L. E. Hayward, D. G. Hawthorn, R. G. Melko, and S. Sachdev, Scienc4e13T4h3e, 1g3a3p6fu(2n0c1ti4o)n.s and level splitting of the other compounds are very similar and therefore not shown in Fig. 6. 22 W. Tabis, Y. Li, M. Le Tacon, L. Braicovich, A. Kreyssig, M. Minola, G. Dellea, E. Weschke, M. J. Veit, M. Ramazanoglu, A. I. Goldman,

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text: Devereaux,

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text: FKu.jBitae,nSn.emIshanidna,aMnd.

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text: THE NONLINEAR STEEPEST DESCENT METHOD 13 The inverse scattering transform formalism [1, 2, 6] implies that the solution u(x, t)f (1.2) with initial data u(x, 0) = u (x) is given by 0 ∠ u(x, t) = 2i lim (kM(x, t, k)) , (3.4) →∞ 12 k where M(x, t, k(cid:40)) is the unique solution of the L2-RH problem · ∈ C \ R M(x, t, ) I + E˙ 2( ), ∈ R (3.5) M (x, t, k) = M−(x, t, k)J(x, t, k) for a.e. k , + (cid:32) (cid:33) with − | | − − − 1 r(k) 2 r(k¯)e 2ikx 8ik3t ∈ R J(x, t, k) = , k . (3.6) r(k)e2ikx+8ik3t 1 The function r(k) satisﬁes r(k) = r(−k¯), k ∈ R, (3.7a) and | | sup r(k) < 1. (3.7b) ∈R k (cid:40) An elaborate analysis of (3.1) shows that if ∈ R u Cm+1( ), 0 (1 + x)nu(i)(x) ∈ L1(R), i = 0, 1, . . . , m + 1, (3.8) 0 ≥ ∈ R for some integers n, m 1, then r Cn( ) and − − | | → ∞ ∈ R r(j)(k) = O(k m 1), k , k , j = 0, 1, . . . , n. (3.9) If the initial data u (x) atisfy (3.8) for n = 1 and m = 4, then the limit in (3.4) exists ∈ R × 0 ∞ for each (x, t) [0, ) and deﬁnes a classical solution u(x, t) of (1.2) with initial data u(x, 0) = u (x), but lower regularity requirements are also possible. 0 ∈ R Since the matrix Re J = 1(J + J¯T ) is positive deﬁnite for k , the homogeneous 2R RH problem determined by ( , J) has only the zero solution (see Theorem 9.3 of [15]). Hence Lemma A.2 implies that the L2-RH problem (3.5) has a unique solution whenever ∈ R ∩ ∞ R ∩ R r C( ) L ( ) L2( ). 4. Asymptotics in the similarity sector We use Theorem 2.1 to derive the asymptotics of the solution of the mKdV equation in the similarity sector. Let ζ = x/t and deﬁne the variables k = k (ζ) and τ = τ(x, t) by 0 0 ##LTLine## ##LTLine## ##LTLine##

text: THE

text: NONLINEAR

text: STEEPEST

text: DESCENT

text: METHOD

text: The

text: inverse

text: scattering

text: transform

text: formalism

text: implies

text: that

text: the

text: solution

text: u(x,

text: t)f

text: with

text: initial

text: data

text: u(x,

text: u

text: (x)

text: is

text: given

text: by

text: u(x,

text: t)

text: lim

text: (kM(x,

text: t,

text: k))

text: k

text: where

text: M(x,

text: t,

text: is

text: the

text: unique

text: solution

text: of

text: the

text: problem

text: C

text: R

text: M(x,

text: t,

text: I

text: E˙

text: R

text: M

text: (x,

text: t,

text: k)

text: M−(x,

text: t,

text: k)J(x,

text: t,

text: k)

text: for

text: a.e.

text: k

text: with

text: r(k)

text: r(k¯)e

text: R

text: J(x,

text: t,

text: k)

text: k

text: The

text: function

text: r(k)

text: satisﬁes

text: r(k)

text: r(−k¯),

text: k

text: R,

text: and

text: sup

text: r(k)

text: ∈R

text: k

text: An

text: elaborate

text: analysis

text: of

text: shows

text: that

text: if

text: R

text: u

text: x)nu(i)(x)

text: i

text: m

text: R

text: for

text: some

text: integers

text: n,

text: m

text: then

text: r

text: Cn(

text: and

text: R

text: r(j)(k)

text: O(k

text: m

text: k

text: k

text: j

text: n.

text: If

text: the

text: initial

text: data

text: u

text: (x)

text: atisfy

text: for

text: n

text: and

text: m

text: then

text: the

text: limit

text: in

text: exists

text: R

text: for

text: each

text: (x,

text: t)

text: and

text: deﬁnes

text: a

text: classical

text: solution

text: u(x,

text: t)

text: of

text: with

text: initial

text: data

text: u(x,

text: u

text: (x),

text: but

text: lower

text: regularity

text: requirements

text: are

text: also

text: possible.

text: R

text: Since

text: the

text: matrix

text: Re

text: J

text: J¯T

text: is

text: positive

text: deﬁnite

text: for

text: k

text: the

text: homogeneous

text: RH

text: problem

text: determined

text: by

text: J)

text: has

text: only

text: the

text: zero

text: solution

text: (see

text: Theorem

text: of

text: Hence

text: Lemma

text: implies

text: that

text: the

text: problem

text: has

text: a

text: unique

text: solution

text: whenever

text: R

text: R

text: R

text: r

text: C(

text: L

text: Asymptotics

text: in

text: the

text: similarity

text: sector

text: We

text: use

text: Theorem

text: to

text: derive

text: the

text: asymptotics

text: of

text: the

text: solution

text: of

text: the

text: mKdV

text: equation

text: in

text: the

text: similarity

text: sector.

text: Let

text: ζ

text: x/t

text: and

text: deﬁne

text: the

text: variables

text: k

text: k

text: (ζ)

text: and

text: τ

text: τ(x,

text: t)

text: by

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 16 THE N(cid:40)ONLINEAR STEEPEST DESCENT METHOD | | δ−(ζ, k), k > k0, ∈ R δ (ζ, k) = − | | | | k . (4.7) + δ−(ζ, k)(1 r(k) 2), k < k , 0 Moreover, the symmetry (3.7a) implies that − − − δ(ζ, k) = δ(ζ, k¯) = δ(ζ, k) 1. (4.8) It follows that ∆ obeys the symmetries − − − ∆(ζ, k) = ∆(ζ, k¯) = ∆(ζ, k) 1. (4.9) × Lemma 4.4. The 2 2-matrix valued function ∆(ζ, k) satisﬁes · · − ∈ C \ R ∩ ∞ C \ R ∆(ζ, ), ∆(ζ, ) 1 I + E˙ 2( ) E ( ), ∈ I for each ζ . (cid:18) (cid:19) Proof. First note that − k k iν δ(ζ, k) = 0 eχ(ζ,k), (4.10) k + k 0 (cid:90) where ν(ζ) is given by (4.5), 1 ds χ(ζ, k) = ψ(ζ, s) − , (4.11) 2πi R s k (cid:40) (cid:0) (cid:1) and the function ψ(ζ, s) is deﬁned by −| | − 1 r(s) 2 ln −| | , k < s < k , ψ(ζ, s) = 1 r(k0) 2 0 0 0, otherwise. · ∈ R · ∈ ∞ C \ R ∈ I Since ψ(ζ, ) H1( ), we have χ(ζ, ) E ( ) for each ζ ; in fact, see Lemma 23.3 in [2], | | (cid:107) · (cid:107) ∞ sup sup χ(ζ, k) < sup ψ(ζ, ) R < . (4.12) ∈I ∈C\R ∈I H1( ) ζ k ζ · · − ∈ ∞ C \ R ∈ I Hence δ(ζ, ), δ(ζ, ) 1 E ( ) for each ζ . − → ∞ · · − ∈ Since χ(ζ, k) = O(k 1) uniformly as k , Lemma A.4 shows that δ(ζ, ), δ(ζ, ) 1 C \ R (cid:53) 1 + E˙ 2( ). By Lemma 4.4 and Lemma A.5, the function M˜ deﬁned by (cid:40) M˜ (x, t, k) = M(x, t, k)∆(ζ, k) satisﬁes the L2-RH problem · ∈ C \ R M˜ (x, t, ) I + E˙ 2( ), ∈ R M˜ (x, t, k) = M˜−(x, t, k)J˜(x, t, k) for a.e. k , + ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: THE

text: STEEPEST

text: DESCENT

text: METHOD

text: δ−(ζ,

text: k),

text: k

text: R

text: δ

text: (ζ,

text: k)

text: k

text: δ−(ζ,

text: r(k)

text: k

text: k

text: Moreover,

text: the

text: symmetry

text: implies

text: that

text: δ(ζ,

text: k)

text: δ(ζ,

text: k¯)

text: δ(ζ,

text: k)

text: It

text: follows

text: that

text: obeys

text: the

text: symmetries

text: ∆(ζ,

text: k)

text: ∆(ζ,

text: k¯)

text: ∆(ζ,

text: k)

text: Lemma

text: The

text: valued

text: function

text: ∆(ζ,

text: k)

text: satisﬁes

text: C

text: R

text: C

text: R

text: ∆(ζ,

text: ∆(ζ,

text: I

text: E˙

text: E

text: I

text: for

text: each

text: ζ

text: Proof.

text: First

text: note

text: that

text: k

text: k

text: iν

text: δ(ζ,

text: k)

text: eχ(ζ,k),

text: k

text: k

text: where

text: ν(ζ)

text: is

text: given

text: by

text: ds

text: χ(ζ,

text: k)

text: ψ(ζ,

text: s)

text: R

text: s

text: k

text: and

text: the

text: function

text: ψ(ζ,

text: s)

text: is

text: deﬁned

text: by

text: r(s)

text: ln

text: k

text: s

text: k

text: ψ(ζ,

text: s)

text: otherwise.

text: R

text: C

text: R

text: I

text: Since

text: ψ(ζ,

text: we

text: have

text: χ(ζ,

text: E

text: for

text: each

text: ζ

text: in

text: fact,

text: see

text: Lemma

text: in

text: sup

text: sup

text: χ(ζ,

text: k)

text: sup

text: ψ(ζ,

text: R

text: ∈I

text: ∈C\R

text: ∈I

text: ζ

text: k

text: ζ

text: C

text: R

text: I

text: Hence

text: δ(ζ,

text: δ(ζ,

text: E

text: for

text: each

text: ζ

text: Since

text: χ(ζ,

text: k)

text: O(k

text: uniformly

text: as

text: k

text: Lemma

text: shows

text: that

text: δ(ζ,

text: δ(ζ,

text: C

text: R

text: E˙

text: By

text: Lemma

text: and

text: Lemma

text: the

text: function

text: M˜

text: deﬁned

text: by

text: M˜

text: (x,

text: t,

text: k)

text: M(x,

text: t,

text: k)∆(ζ,

text: k)

text: satisﬁes

text: the

text: problem

text: C

text: R

text: M˜

text: (x,

text: t,

text: I

text: E˙

text: R

text: M˜

text: (x,

text: t,

text: k)

text: M˜−(x,

text: t,

text: k)J˜(x,

text: t,

text: k)

text: for

text: a.e.

text: k

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: THE NONLINEAR STEEPEST DESCENT METHOD 17 where (cid:32) (cid:33) − J˜(x, t, k) = ∆−1(ζ, k)J(x, t, k)∆ (ζ, k) + = δδ−+((ζζ,,kk))(1 − |r(k)|2) −δ−(ζ, k)δ+(ζ, k)r(k¯)e−tΦ(ζ,k) , k ∈ R. (4.13) 1 r(k)etΦ(ζ,k) δ+(ζ,k) δ−(ζ,k)δ(cid:32)+(ζ,k) δ−(ζ,k) (cid:33)  ∈ R In view of the jump (4.7) of δ(ζ, k), this gives, for k ,   − | | − −  1 r(k) 2 δ(ζ, k)2r(k¯)e tΦ(ζ,k)  | | δ(ζ, k)−2r(k)etΦ(ζ,k) 1 ,  k > k0, J˜(x, t, k) =  1 −δ (ζ, k)2 −r|(k¯) | e−tΦ(ζ,k) | | δ−(ζ, k)−2 −r|(k) | etΦ(ζ,k) + 1 −1 |rr((kk))|22 , k < k0. 1 r(k) 2 The upshot of the above conjugation is that we can now factorize the jump matrix as (cid:40) follows: − | | ∈ R B 1B , k > k , k , J˜ = −u l | | 0 ∈ R (4.14) b 1b k < k , k , l u 0 (cid:18) (cid:19) (cid:18) (cid:19) where − − (cid:18) 1 0 (cid:19) 1 (cid:18) δ (ζ, k)2r (k)e tΦ(ζ,k)(cid:19) B = − , b = + 2 , l δ(ζ, k) 2r (k)etΦ(ζ,k) 1 u 0 1 1 − 1 0 1 δ(ζ, k)2r (k)e tΦ(ζ,k) b = − − , B = 4 , (4.15) l δ−(ζ, k) 2r (k)etΦ(ζ,k) 1 u 0 1 3 { } and the functions r (k) 4 are deﬁned by j 1 r(k¯) r1(k) = r(k), r2(k) = − , 1 r(k)r(k¯) r(k) r3(k) = − , r4(k) = r(k¯). 1 r(k)r(k¯) Our next goal is to deform the contour. However, we ﬁrst need to introduce analytic { } approximations of r (k) 4. j 1 Step 2: Introduce analytic approximations. The following lemma describes how to decompose r , j = 1, . . . , 4, into an analytic part r and a small remainder r . We j j,a j,r introduce open domains U = U (ζ), j = 1, . . . , 4, as in Figure 3 so that j j { | } ∪ { | } ∪ k Re Φ(ζ, k) < 0 = U U , k Re Φ(ζ, k) > 0 = U U . (4.16) 1 3 2 4 (cid:40) Lemma 4.5. There exist decompositions | | ∈ R r (x, t, k) + r (x, t, k), j = 1, 4, k > k , k , j,a j,r 0 rj(k) = | | ∈ R r (x, t, k) + r (x, t, k), j = 2, 3, k < k , k , j,a j,r 0 { } where the functions r , r 4 have the following properties: j,a j,r j=1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: THE

text: NONLINEAR

text: STEEPEST

text: DESCENT

text: METHOD

text: where

text: J˜(x,

text: t,

text: k)

text: ∆−1(ζ,

text: k)J(x,

text: t,

text: k)∆

text: (ζ,

text: k)

text: −δ−(ζ,

text: k)δ+(ζ,

text: k)r(k¯)e−tΦ(ζ,k)

text: k

text: R.

text: r(k)etΦ(ζ,k)

text: δ+(ζ,k)

text: δ−(ζ,k)

text: R

text: In

text: view

text: of

text: the

text: jump

text: of

text: δ(ζ,

text: k),

text: this

text: gives,

text: for

text: k

text: r(k)

text: δ(ζ,

text: tΦ(ζ,k)

text: δ(ζ,

text: k

text: J˜(x,

text: t,

text: k)

text: −δ

text: (ζ,

text: −r|(k¯)

text: e−tΦ(ζ,k)

text: δ−(ζ,

text: −r|(k)

text: etΦ(ζ,k)

text: −1

text: k

text: r(k)

text: The

text: upshot

text: of

text: the

text: above

text: conjugation

text: is

text: that

text: we

text: can

text: now

text: factorize

text: the

text: jump

text: matrix

text: as

text: follows:

text: R

text: B

text: k

text: k

text: k

text: J˜

text: −u

text: l

text: R

text: b

text: k

text: k

text: k

text: l

text: u

text: where

text: δ

text: (ζ,

text: (k)e

text: B

text: b

text: l

text: δ(ζ,

text: k)

text: (k)etΦ(ζ,k)

text: u

text: δ(ζ,

text: (k)e

text: tΦ(ζ,k)

text: b

text: B

text: l

text: δ−(ζ,

text: k)

text: (k)etΦ(ζ,k)

text: u

text: and

text: the

text: functions

text: r

text: (k)

text: are

text: deﬁned

text: by

text: j

text: r(k¯)

text: r(k),

text: r(k)r(k¯)

text: r(k)

text: r(k¯).

text: r(k)r(k¯)

text: Our

text: next

text: goal

text: is

text: to

text: deform

text: the

text: contour.

text: However,

text: we

text: ﬁrst

text: need

text: to

text: introduce

text: analytic

text: approximations

text: of

text: r

text: (k)

text: j

text: Step

text: Introduce

text: analytic

text: approximations.

text: The

text: following

text: lemma

text: describes

text: how

text: to

text: decompose

text: r

text: j

text: into

text: an

text: analytic

text: part

text: r

text: and

text: a

text: small

text: remainder

text: r

text: We

text: j

text: j,a

text: j,r

text: introduce

text: open

text: domains

text: U

text: U

text: (ζ),

text: j

text: as

text: in

text: Figure

text: so

text: that

text: j

text: j

text: k

text: Re

text: Φ(ζ,

text: k)

text: U

text: U

text: k

text: Re

text: Φ(ζ,

text: k)

text: U

text: U

text: Lemma

text: There

text: exist

text: decompositions

text: R

text: r

text: (x,

text: t,

text: k)

text: r

text: (x,

text: t,

text: k),

text: j

text: k

text: k

text: k

text: j,a

text: j,r

text: rj(k)

text: R

text: r

text: (x,

text: t,

text: k)

text: r

text: (x,

text: t,

text: k),

text: j

text: k

text: k

text: k

text: j,a

text: j,r

text: where

text: the

text: functions

text: r

text: r

text: have

text: the

text: following

text: properties:

text: j,a

text: j,r

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: THE NONLINEAR STEEPEST DESCENT METHOD 19 (cid:88) Let 10 a (ζ) f (ζ, k) = j− , 0 (k i)j j=4 { } where a (ζ) 10 are such that j 4 (cid:88) 6 − − → f (ζ, k) = p (ζ)(k k )j + O((k k )7), k k , (4.20) 0 j 0 0 0 j=0 ∈ I for each ζ . It is easy to verify that (4.20) imposes seven linear conditions on the ∈ I a (ζ)’s that are linearly independent for each ζ ; hence the coeﬃcients a (ζ) exist and j { } j are unique. The a (ζ)’s are polynomials in p (ζ) 6 with coeﬃcients that are polynomials j I j 0 | | ∞ in k . Thus, since the interval is bounded, we have sup ∈I a (ζ) < for each j. 0 − ζ j Let f(ζ, k) = r (k) f (ζ, k). The following properties hold: 1 0 ∈ I ∈ C (i) For each ζ , f (ζ, k) is a rational function of k with no poles in U¯+. (cid:40) 0 ∞ 1 (ii) f (ζ, k) coincides with r (k) to order six at k and to order three at ; more precisely 0 1 0 − − → ∂nf O((k k0)7 n), k k0, ∈ R ∈ I (ζ, k) = − → ∞ k , ζ , n = 0, 1, 2, (4.21) ∂kn O(k 4+2n), k , ∈ I where the error terms are uniform with respect to ζ . ∈ I The decomposition of r (k) can now be derived as follows. For each ζ , the map (cid:55)→ 1 k φ = φ(ζ, k) where − − φ = iΦ(ζ, k) = 24k2k + 8k3. (4.22) 0 ∞ → − ∞ is a bijection [k , ) (cid:40)[ 16k3, ), so we may deﬁne a function F (ζ, φ) by 0 0 − ≥ − (k−i)3 f(ζ, k), φ 16k3, ∈ I ∈ R F (ζ, φ) = k k0 − 0 ζ , φ . (4.23) 0, φ < 16k3, 0 ∈ I (cid:54) − For each ζ , the(cid:18)function F (ζ, φ)(cid:19)is(cid:20)smooth for φ =(cid:21) 16k3 and 0 − ∂nF 1 ∂ n (k i)3 ≥ − (ζ, φ) = − − f(ζ, k) , φ 16k3. (4.24) ∂φn 24(k2 k2) ∂k k k 0 0 0 Equations (4.21) and (4.24) together with the trivial inequalities − k ≤ k k ≤ ≥ 0 1 and 1 for k k , (4.25) 0 k + k k + k 0 0 (cid:12) (cid:12) (cid:12) · ∈(cid:12) R show th(cid:12)at F (ζ, ) (cid:12) C1( ) for each ζ and that (cid:12) (cid:12) ∂nF ≤ C ∈ − ∞ ∈ I ∂φn (ζ, φ) (1 + |φ|)23 , φ ( 16k03, ), ζ , n = 0, 1, 2. Hence ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: THE

text: NONLINEAR

text: STEEPEST

text: DESCENT

text: METHOD

text: Let

text: a

text: (ζ)

text: f

text: (ζ,

text: k)

text: j−

text: (k

text: i)j

text: where

text: a

text: (ζ)

text: are

text: such

text: that

text: j

text: f

text: (ζ,

text: k)

text: p

text: (ζ)(k

text: k

text: )j

text: O((k

text: k

text: k

text: k

text: j

text: I

text: for

text: each

text: ζ

text: It

text: is

text: easy

text: to

text: verify

text: that

text: imposes

text: seven

text: linear

text: conditions

text: on

text: the

text: I

text: a

text: (ζ)’s

text: that

text: are

text: linearly

text: independent

text: for

text: each

text: ζ

text: hence

text: the

text: coeﬃcients

text: a

text: (ζ)

text: exist

text: and

text: j

text: j

text: are

text: unique.

text: The

text: a

text: (ζ)’s

text: are

text: polynomials

text: in

text: p

text: (ζ)

text: with

text: coeﬃcients

text: that

text: are

text: polynomials

text: j

text: I

text: j

text: in

text: k

text: Thus,

text: since

text: the

text: interval

text: is

text: bounded,

text: we

text: have

text: sup

text: ∈I

text: a

text: (ζ)

text: for

text: each

text: j.

text: ζ

text: j

text: Let

text: f(ζ,

text: k)

text: r

text: (k)

text: f

text: (ζ,

text: k).

text: The

text: following

text: properties

text: hold:

text: I

text: C

text: (i)

text: For

text: each

text: ζ

text: f

text: (ζ,

text: k)

text: is

text: a

text: rational

text: function

text: of

text: k

text: with

text: no

text: poles

text: in

text: U¯+.

text: (ii)

text: f

text: (ζ,

text: k)

text: coincides

text: with

text: r

text: (k)

text: to

text: order

text: six

text: at

text: k

text: and

text: to

text: order

text: three

text: at

text: more

text: precisely

text: ∂nf

text: O((k

text: n),

text: k

text: R

text: I

text: (ζ,

text: k)

text: k

text: ζ

text: n

text: ∂kn

text: O(k

text: k

text: I

text: where

text: the

text: error

text: terms

text: are

text: uniform

text: with

text: respect

text: to

text: ζ

text: I

text: The

text: decomposition

text: of

text: r

text: (k)

text: can

text: now

text: be

text: derived

text: as

text: follows.

text: For

text: each

text: ζ

text: the

text: map

text: k

text: φ

text: φ(ζ,

text: k)

text: where

text: φ

text: iΦ(ζ,

text: k)

text: is

text: a

text: bijection

text: [k

text: so

text: we

text: may

text: deﬁne

text: a

text: function

text: F

text: (ζ,

text: φ)

text: by

text: f(ζ,

text: k),

text: φ

text: I

text: R

text: F

text: (ζ,

text: φ)

text: k

text: ζ

text: φ

text: φ

text: I

text: For

text: each

text: ζ

text: F

text: (ζ,

text: for

text: φ

text: and

text: ∂nF

text: n

text: (k

text: (ζ,

text: φ)

text: f(ζ,

text: k)

text: φ

text: ∂φn

text: ∂k

text: k

text: k

text: Equations

text: and

text: together

text: with

text: the

text: trivial

text: inequalities

text: k

text: k

text: k

text: and

text: for

text: k

text: k

text: k

text: k

text: k

text: k

text: R

text: show

text: F

text: (ζ,

text: for

text: each

text: ζ

text: and

text: that

text: ∂nF

text: C

text: I

text: ∂φn

text: (ζ,

text: φ)

text: φ

text: ζ

text: n

text: Hence

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: THE NONLINEAR STEEPEST DESCENT METHOD 27 where the constant C is independent of ζ and k. Proof. We ﬁrst prove that (cid:0) (cid:1) | − | ≤ | − | | | − || ∈ I ∈ χ(ζ, k) χ(ζ, k ) C k k 1 + ln k k , ζ , k k + X(cid:15). (4.44) 0 0 0 0 1 Integration by parts in the deﬁnitio(cid:90)n (4.11) of χ gives (cid:0) (cid:1) − 1 k0 − − | | χ(ζ, k) = ln(s k)d ln 1 r(s) 2 . (4.45) 2πi − (cid:12)k(cid:90)0 (cid:18) (cid:19) (cid:12) (cid:12) (cid:0) (cid:1)(cid:12) (cid:12) (cid:12) Hence (cid:12) (cid:12) − | − | 1 k0 s k − | | χ(ζ, k) χ(ζ, k ) = ln − d ln 1 r(s) 2 . 0 2π − s k k0 0 This gives (cid:90) (cid:12) (cid:12) (cid:90) (cid:12) (cid:18) (cid:19)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) − − | − | ≤ k0 s k ≤ 2k0 k k χ(ζ, k) χ(ζ, k ) C ln − ds C ln 1 + 0 du. (4.46) 0 − s k u k0 0 0 (cid:90) (cid:12) (cid:18) (cid:19)(cid:12) (cid:90) (cid:12) | − (cid:12) | (cid:12) (cid:12) The change of var(cid:12)iables v = k k(cid:12) /u yields (cid:12) (cid:12) (cid:12) (cid:12)0 − ∞ 2k0 k k0 | − | iπ dv 0 ln 1 + u du = k(cid:40) k0 |k−k0| ln(1 + ve 4 ) v2 . (cid:12) (cid:12) 2k0 (cid:12) (cid:12) Since ≤ ≤ ≤ v, 0 v 2, iπ ln(1 + ve 4 ) C ≤ ∞ (cid:90) (cid:12) (cid:18) (cid:19)(cid:12) (cid:18)ln(cid:90)v, 2 v <(cid:90) , (cid:19) (cid:12) (cid:12) we conclude t(cid:12)hat (cid:12) (cid:12) (cid:12) − ∞ 2k0 k k ≤ | − | 2 dv ln v 0 ln 1 + du C k k + dv 0 u 0 |k−k0| v 2 v2 ≤ | − | | |2k0− || ∈ C k k ( ln k k + C), k k + X(cid:15). 0 0 0 1 This proves (4.44). Using the inequality (2.23) together with (4.12) and (4.44), we estimate | − − − | ≤ | − || − − − | e 2χ(ζ,k) e 2χ(ζ,k0) e 2χ(ζ,k0) e 2[χ(ζ,k) χ(ζ,k0)] 1 ≤ | − | | − | C χ(ζ, k) χ(ζ, k ) e2 Re(χ(ζ,k) χ(ζ,k0)) 0 ≤ | − | | | − || ∈ I ∈ C k k (1 + ln k k ), ζ , k k + X(cid:15). 0 0 0 1 Th(cid:12)is proves (4.42). (cid:12) (cid:12) (cid:18) (cid:19)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) B(cid:12) y (2.23), (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) − −2iν(ζ)ln(k+k0 ) ≤ (cid:12)(cid:12) (cid:18) k + k0 (cid:19)(cid:12)(cid:12) |Re(2iν(ζ)ln(k+k0 ))| 1 e 2k0 2(cid:12)ν(ζ) ln (cid:12)e 2k0 (cid:12) 2k0 (cid:12) − ≤ k k ≤ − | − | ∈ I ∈ C ln 1 + 0 Ck 1 k k , ζ , k k + X(cid:15), 2k 0 0 0 1 0 which proves (4.43). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: THE

text: NONLINEAR

text: STEEPEST

text: DESCENT

text: METHOD

text: where

text: the

text: constant

text: C

text: is

text: independent

text: of

text: ζ

text: and

text: k.

text: Proof.

text: We

text: ﬁrst

text: prove

text: that

text: I

text: χ(ζ,

text: k)

text: χ(ζ,

text: k

text: C

text: k

text: k

text: ln

text: k

text: k

text: ζ

text: k

text: k

text: Integration

text: by

text: parts

text: in

text: the

text: of

text: χ

text: gives

text: χ(ζ,

text: k)

text: ln(s

text: k)d

text: ln

text: r(s)

text: Hence

text: s

text: k

text: χ(ζ,

text: k)

text: χ(ζ,

text: k

text: ln

text: d

text: ln

text: r(s)

text: 2π

text: s

text: k

text: This

text: gives

text: s

text: k

text: k

text: k

text: χ(ζ,

text: k)

text: χ(ζ,

text: k

text: C

text: ln

text: ds

text: C

text: ln

text: du.

text: s

text: k

text: u

text: The

text: change

text: of

text: v

text: k

text: /u

text: yields

text: k

text: iπ

text: dv

text: ln

text: u

text: du

text: ve

text: Since

text: v,

text: v

text: iπ

text: ve

text: C

text: v

text: we

text: conclude

text: k

text: k

text: dv

text: ln

text: v

text: ln

text: du

text: C

text: k

text: k

text: dv

text: u

text: v

text: C

text: k

text: k

text: ln

text: k

text: k

text: C),

text: k

text: k

text: This

text: proves

text: Using

text: the

text: inequality

text: together

text: with

text: and

text: we

text: estimate

text: e

text: e

text: e

text: e

text: C

text: χ(ζ,

text: k)

text: χ(ζ,

text: k

text: Re(χ(ζ,k)

text: I

text: C

text: k

text: k

text: ln

text: k

text: k

text: ζ

text: k

text: k

text: proves

text: y

text: k

text: e

text: ln

text: k

text: k

text: I

text: C

text: ln

text: Ck

text: k

text: k

text: ζ

text: k

text: k

text: which

text: proves

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: THE NONLINEAR STEEPEST DESCENT METHOD 29 ∞ for each j, respectively. We let E (D) denote the space of bounded analytic functions in C D. A Carleson jump contour is a connected subset Γ of ˆ such that: ∩ C (a) Γ is the union of ﬁnitely many oriented arcs3 each pair of which have at most endpoints in common. C \ (b) ˆ Γ is the union of two disjoint open sets D and D− each of which has a ﬁnite + C number of simply connected components in ˆ. (c) Γ is the positively oriented boundary of D and the negatively oriented boundary of − + D−, i.e. Γ = ∂D = ∂D−. { } { +−} ∈ J (d) If D+ n and D m are the components of D and D−, then ∂D+ for j = j 1 −j∈1J + j 1, . . . , n, and ∂D for j = 1, . . . , m. ≥ j × Let n 1 be an integer and let Γ be a Carleson jump contour. Given an n n-matrix → C valued function v : Γ GL(n, ), a solution of the Lp-RH problem determined by (Γ, v) is × ∈ C \ an n n-matrix valued function m I + E˙ p(ˆ Γ) such that the nontangential boundary values m± satisfy m = m−v a.e. on Γ. If Γ˜ denotes the Carleson jump contour Γ with + ⊂ \ the orientation reversed on a subset Γ Γ and v˜ is deﬁned by v˜ = v on Γ Γ and − ∈ 0 0 v˜ = v 1 on Γ , then we say that m I + E˙ p(D) satisﬁes the Lp-RH problem determined 0 by (Γ˜, v˜) if and only if m satisﬁes the Lp-RH problem determined by (Γ, v). We next list some facts about Smirnoﬀ classes and L2-RH problems; detailed proofs of − can be found in [12]. We assume that Γ = ∂D = ∂D− is a Carleson jump contour. + ∈ ∈ If f E˙ 2(D ) or f E˙ 2(D−), then the nontangential limits of f(z) as z approaches + ∈ the boundary exist a.e. on Γ and the boundary function belongs to L2(Γ). If h L2(Γ), C (cid:90) then the Cauchy transform h deﬁned by C 1 h(s) ∈ C \ ( h)(z) = − ds, z Γ, (A.3) 2πi s z Γ C ∈ ∪ C satisﬁes h E˙ 2(D D−). We denote the nontangential boundary values of f from + C C C C the left and right sides of Γ by f and −f respectively. Then and − are bounded C − C + + operators on L2(Γ) and − = I. ±+∈ ∩ ∞ C ∞ → Given two functions w L2(Γ) L (Γ), we deﬁne the operator : L2(Γ)+L (Γ) w L2(Γ) by C C − C (f) = (fw ) + −(fw+). (A.4) w + Then (cid:8) (cid:9) (cid:107)C (cid:107) ≤ (cid:107) (cid:107) (cid:107) −(cid:107) B C max w+ ∞ , w ∞ . (A.5) w (L2(Γ)) L (Γ) L (Γ) {(cid:107)C (cid:107) (cid:107)C (cid:107) } ∞ B where C = max B , − B < and (L2(Γ)) denotes the Banach + (L2(Γ)) → (L2(Γ)) space of bounded linear maps L2(Γ) L2(Γ). − − ± ± ± ∓ The next lemma shows that if v = (v ) 1v+ and w = v I then the L2-RH problem determined by (Γ, v) is equivalent to the following singular integral equation for ∈ µ I + L2(Γ): 3A subset Γ ⊂ C is an arc if it is homeomorphic to a connected subset I of the real line which contains at least two distinct points. ##LTLine## ##LTLine## ##LTLine##

text: THE

text: NONLINEAR

text: STEEPEST

text: DESCENT

text: METHOD

text: for

text: each

text: j,

text: respectively.

text: We

text: let

text: E

text: (D)

text: denote

text: the

text: space

text: of

text: bounded

text: analytic

text: functions

text: in

text: C

text: D.

text: A

text: Carleson

text: jump

text: contour

text: is

text: a

text: connected

text: subset

text: Γ

text: of

text: ˆ

text: such

text: that:

label: list

text: C

text: (a)

text: Γ

text: is

text: the

text: union

text: of

text: ﬁnitely

text: many

text: oriented

text: each

text: pair

text: of

text: which

text: have

text: at

text: most

text: endpoints

text: in

text: common.

text: C

text: (b)

text: ˆ

text: Γ

text: is

text: the

text: union

text: of

text: two

text: disjoint

text: open

text: sets

text: D

text: and

text: D−

text: each

text: of

text: which

text: has

text: a

text: ﬁnite

text: C

text: number

text: of

text: simply

text: connected

text: components

text: in

text: ˆ.

text: (c)

text: Γ

text: is

text: the

text: positively

text: oriented

text: boundary

text: of

text: D

text: and

text: the

text: negatively

text: oriented

text: boundary

text: of

text: D−,

text: i.e.

text: Γ

text: ∂D

text: ∂D−.

text: J

text: (d)

text: If

text: D+

text: n

text: and

text: D

text: m

text: are

text: the

text: components

text: of

text: D

text: and

text: D−,

text: then

text: ∂D+

text: for

text: j

text: j

text: j

text: n,

text: and

text: ∂D

text: for

text: j

text: m.

text: j

text: Let

text: n

text: be

text: an

text: integer

text: and

text: let

text: Γ

text: be

text: a

text: Carleson

text: jump

text: contour.

text: Given

text: an

text: n

text: n-matrix

text: C

text: valued

text: function

text: v

text: Γ

text: GL(n,

text: a

text: solution

text: of

text: the

text: Lp-RH

text: problem

text: determined

text: by

text: (Γ,

text: v)

text: is

text: C

text: an

text: n

text: n-matrix

text: valued

text: function

text: m

text: I

text: E˙

text: p(ˆ

text: Γ)

text: such

text: that

text: the

text: nontangential

text: boundary

text: values

text: m±

text: satisfy

text: m

text: m−v

text: a.e.

text: on

text: Γ.

text: If

text: Γ˜

text: denotes

text: the

text: Carleson

text: jump

text: contour

text: Γ

text: with

text: the

text: orientation

text: reversed

text: on

text: a

text: subset

text: Γ

text: Γ

text: and

text: v˜

text: is

text: deﬁned

text: by

text: v˜

text: v

text: on

text: Γ

text: Γ

text: and

text: v˜

text: v

text: on

text: Γ

text: then

text: we

text: say

text: that

text: m

text: I

text: E˙

text: p(D)

text: satisﬁes

text: the

text: Lp-RH

text: problem

text: determined

text: by

text: (Γ˜,

text: v˜)

text: if

text: and

text: only

text: if

text: m

text: satisﬁes

text: the

text: Lp-RH

text: problem

text: determined

text: by

text: (Γ,

text: v).

text: We

text: next

text: list

text: some

text: facts

text: about

text: Smirnoﬀ

text: classes

text: and

text: problems;

text: detailed

text: proofs

text: of

text: can

text: be

text: found

text: in

text: We

text: assume

text: that

text: Γ

text: ∂D

text: ∂D−

text: is

text: a

text: Carleson

text: jump

text: contour.

text: If

text: f

text: E˙

text: or

text: f

text: E˙

text: then

text: the

text: nontangential

text: limits

text: of

text: f(z)

text: as

text: z

text: approaches

text: the

text: boundary

text: exist

text: a.e.

text: on

text: Γ

text: and

text: the

text: boundary

text: function

text: belongs

text: to

text: If

text: h

text: C

text: then

text: the

text: Cauchy

text: transform

text: h

text: deﬁned

text: by

text: C

text: h(s)

text: C

text: h)(z)

text: ds,

text: z

text: Γ,

text: s

text: z

text: Γ

text: C

text: C

text: satisﬁes

text: h

text: E˙

text: D−).

text: We

text: denote

text: the

text: nontangential

text: boundary

text: values

text: of

text: f

text: from

text: C

text: C

text: C

text: C

text: the

text: left

text: and

text: right

text: sides

text: of

text: Γ

text: by

text: f

text: and

text: −f

text: respectively.

text: Then

text: and

text: are

text: bounded

text: C

text: C

text: operators

text: on

text: and

text: I.

text: C

text: Given

text: two

text: functions

text: w

text: L

text: (Γ),

text: we

text: deﬁne

text: the

text: operator

text: (Γ)

text: w

text: by

text: C

text: C

text: C

text: (f)

text: (fw

text: −(fw+).

text: w

text: Then

text: B

text: C

text: max

text: w+

text: w

text: w

text: L

text: (Γ)

text: L

text: (Γ)

text: B

text: where

text: C

text: max

text: B

text: B

text: and

text: denotes

text: the

text: Banach

text: space

text: of

text: bounded

text: linear

text: maps

text: The

text: next

text: lemma

text: shows

text: that

text: if

text: v

text: (v

text: and

text: w

text: v

text: I

text: then

text: the

text: problem

text: determined

text: by

text: (Γ,

text: v)

text: is

text: equivalent

text: to

text: the

text: following

text: singular

text: integral

text: equation

text: for

text: µ

text: I

text: subset

text: Γ

text: C

text: is

text: an

text: arc

text: if

text: it

text: is

text: homeomorphic

text: to

text: a

text: connected

text: subset

text: I

text: of

text: the

text: real

text: line

text: which

text: contains

text: at

text: least

text: two

text: distinct

text: points.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 THE NONLINEAR STEEPEST DESCENT METHOD (cid:107)S (cid:107) ∞ sup B < . (2.2) ∈I Γˆ (L2(Γˆ)) ζ Consider the following family of L2-RH problems parametrized by the two parameters ∈ I (cid:40) ζ and t > 0: · ∈ C \ m(ζ, t, ) I + E˙ 2(ˆ Γ), ∈ (2.3) m (ζ, t, k) = m−(ζ, t, k)v(ζ, t, k) for a.e. k Γ, + where the jump matrix v(ζ, t, k) satisﬁes · · − ∈ ∩ ∩ ∞ ∈ I w(ζ, t, ) := v(ζ, t, ) I L2(Γ) L2(Γ) L (Γ), ζ , t > 0, (2.4) · ∈ I det v(ζ, t, ) = 1 a.e. on Γ, ζ , t > 0, (2.5) and − ∈ I ∈ v(ζ, t, k) = v(ζ, t, k¯), ζ , t > 0, k Γ. (2.6) ± (cid:48) \ Let τ := tρ2. Let Γ be the union of the two crosses k + X(cid:15) and let Γ = Γ Γ . X 0 X (cid:40) Suppose (cid:107) · (cid:107) 1 − → ∞ ∈ I (cid:107)w(ζ, t, ·)(cid:107)Lp(Γ(cid:48)) = O((cid:15)p−τ 1), τ→ ∞ , ζ∈ I , p = 1, 2, (2.7) w(ζ, t, ) ∞ (cid:48) = O(τ 1), τ , ζ , L (Γ ) ∈ I uniformly with respect to ζ . Moreover, suppose that near k the normalized jump 0 (cid:18) (cid:19) matrix (cid:15)z ∈ v (ζ, t, z) = v ζ, t, k + , z Xρ, (2.8) 0 0 ρ (cid:32) (cid:33) has the form        (cid:32) −1 0 (cid:33), z ∈ Xρ,  R (ζ, t, z)z 2iν(ζ)etφ(ζ,z) 1 1  1 − − (cid:32)1 R (ζ, t, z)z2iν(ζ)e tφ(ζ,z)(cid:33) ∈  2 , z Xρ,  0 1 2  v0(ζ, t, z) = (cid:32)− 1− (cid:33)0 , z ∈ Xρ, (2.9)  R (ζ, t, z)z 2iν(ζ)etφ(ζ,z) 1 3  3 − 1 R (ζ, t, z)z2iν(ζ)e tφ(ζ,z) ∈ 4 , z Xρ, 0 1 4 where: ##LTLine## ##LTLine## ##LTLine##

text: THE

text: NONLINEAR

text: STEEPEST

text: DESCENT

text: METHOD

text: sup

text: B

text: ∈I

text: Γˆ

text: ζ

text: Consider

text: the

text: following

text: family

text: of

text: problems

text: parametrized

text: by

text: the

text: two

text: parameters

text: I

text: ζ

text: and

text: t

text: C

text: m(ζ,

text: t,

text: I

text: E˙

text: Γ),

text: m

text: (ζ,

text: t,

text: k)

text: m−(ζ,

text: t,

text: k)v(ζ,

text: t,

text: k)

text: for

text: a.e.

text: k

text: Γ,

text: where

text: the

text: jump

text: matrix

text: v(ζ,

text: t,

text: k)

text: satisﬁes

text: I

text: w(ζ,

text: t,

text: v(ζ,

text: t,

text: I

text: L

text: (Γ),

text: ζ

text: t

text: I

text: det

text: v(ζ,

text: t,

text: a.e.

text: on

text: Γ,

text: ζ

text: t

text: and

text: I

text: v(ζ,

text: t,

text: k)

text: v(ζ,

text: t,

text: k¯),

text: ζ

text: t

text: k

text: Γ.

text: Let

text: τ

text: Let

text: Γ

text: be

text: the

text: union

text: of

text: the

text: two

text: crosses

text: k

text: and

text: let

text: Γ

text: Γ

text: Γ

text: X

text: X

text: Suppose

text: I

text: t,

text: τ→

text: ζ∈

text: I

text: p

text: w(ζ,

text: t,

text: O(τ

text: τ

text: ζ

text: L

text: (Γ

text: I

text: uniformly

text: with

text: respect

text: to

text: ζ

text: Moreover,

text: suppose

text: that

text: near

text: k

text: the

text: normalized

text: jump

text: matrix

text: v

text: (ζ,

text: t,

text: z)

text: v

text: ζ,

text: t,

text: k

text: z

text: Xρ,

text: ρ

text: has

text: the

text: form

text: −1

text: z

text: Xρ,

text: R

text: (ζ,

text: t,

text: z)z

text: R

text: (ζ,

text: t,

text: z

text: Xρ,

text: t,

text: z)

text: (cid:32)−

text: 1−

text: z

text: Xρ,

text: R

text: (ζ,

text: t,

text: z)z

text: R

text: (ζ,

text: t,

text: tφ(ζ,z)

text: z

text: Xρ,

text: where:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 36 THE NONLINEAR STEEPEST DESCENT METHOD [11] J. Lenells, The Degasperis-Procesi equation on the half-line, Nonlinear Anal. 76 (2013), 122– 139. [12] J. Lenells, Matrix Riemann-Hilbert problems with jumps across Carleson contours, preprint, arXiv:1401.2506. [13] S. V. Manakov, Nonlinear Fraunhofer diﬀraction, Zh. Eksp. Teor. Fiz. 65 (1973), 1392–1398 (in Russian); Sov. Phys. JETP 38 (1974), 693–696 (in English). [14] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors. NIST Handbook of Mathematical Functions, Cambridge University Press, New York, NY, 2010. [15] X. Zhou, The Riemann-Hilbert problem and inverse scattering, SIAM J. Math. Anal. 20 (1989), 966–986.

text: THE

text: NONLINEAR

text: STEEPEST

text: DESCENT

text: METHOD

label: reference

text: J.

text: Lenells,

text: The

text: Degasperis-Procesi

text: equation

text: on

text: the

text: half-line,

text: Nonlinear

text: Anal.

text: J.

text: Lenells,

text: Matrix

text: Riemann-Hilbert

text: problems

text: with

text: jumps

text: across

text: Carleson

text: contours,

text: preprint,

text: S.

text: V.

text: Manakov,

text: Nonlinear

text: Fraunhofer

text: diﬀraction,

text: Zh.

text: Eksp.

text: Teor.

text: Fiz.

text: (in

text: Russian);

text: Sov.

text: Phys.

text: JETP

text: (in

text: English).

text: F.

text: W.

text: J.

text: Olver,

text: D.

text: W.

text: Lozier,

text: R.

text: F.

text: Boisvert,

text: and

text: C.

text: W.

text: Clark,

text: editors.

text: NIST

text: Handbook

text: of

text: Mathematical

text: Functions,

text: Cambridge

text: University

text: Press,

text: New

text: York,

text: NY,

text: X.

text: Zhou,

text: The

text: Riemann-Hilbert

text: problem

text: and

text: inverse

text: scattering,

text: SIAM

text: J.

text: Math.

text: Anal.