text: The proofs of Theorems 5 and 6 are given in Appendices and , respectively, and based on a generalization of the methods proposed in [9,23] however the nonlinear agent dynamics results in a diﬀerent Lyapunov function max ∈V ϕ∗(x¯ (t)). The analysis are based on estimating the i i scalar function ϕ∗(x¯ (t)) agent by agent and thus yields an estimate of the convergence rate. i They are diﬀerent from the contradiction arguments used in the proofs of Theorems 1 and 2 where the convergence rate is unclear. · Remark 4 If the scalar function ϕ∗( ) satisﬁes an additional condition, i.e., there exist 0 < ≤ c c , such that 1 2 (cid:107) (cid:107) ≤ ≤ (cid:107) (cid:107) ∀ ∈ R c η 2 ϕ∗(η) c η 2, η n, 1 2 then Theorem 5 leads to global exponential synchronization. Remark 5 For the leader-follower case, Theorems 5 and 6 show that global asymptotic syn- chronization is achieved while for the leaderless case, while global asymptotic ϕ-synchronization is achieved as shown in Theorems 1 and 2. For the leader-follower case, the uniformly jointly leader connected in Theorem 5 requires the leader to be a center node, while for the leaderless case, the uniformly jointly strongly connected in Theorem 1 requires every node to be a cen- ter node. Thus, the connectivity condition of the leaderless case is stronger than that of the leader-follower case. 4.2 Lipschitz Inherent Dynamics In this section, we extend the result for the case when the agent dynamics is globally Lipschitz to leader-follower networks. Our main result for this case is given in the following theorem whose proof can be found in Appendix . G Theorem 7 Let Assumption 2 hold. Suppose that ¯ is uniformly jointly leader connected. σ(t) The multi-agent system (32) achieves global exponential synchronization if L < ρˆ∗/2, where ρˆ∗ is a constant depending on the network parameters. 21

text: The

text: proofs

text: of

text: Theorems

text: and

text: are

text: given

text: in

text: Appendices

text: and

text: respectively,

text: and

text: based

text: on

text: a

text: generalization

text: of

text: the

text: methods

text: proposed

text: in

text: however

text: the

text: nonlinear

text: agent

text: dynamics

text: results

text: in

text: a

text: diﬀerent

text: Lyapunov

text: function

text: max

text: ∈V

text: ϕ∗(x¯

text: (t)).

text: The

text: analysis

text: are

text: based

text: on

text: estimating

text: the

text: i

text: i

text: scalar

text: function

text: ϕ∗(x¯

text: (t))

text: agent

text: by

text: agent

text: and

text: thus

text: yields

text: an

text: estimate

text: of

text: the

text: convergence

text: rate.

text: i

text: They

text: are

text: diﬀerent

text: from

text: the

text: contradiction

text: arguments

text: used

text: in

text: the

text: proofs

text: of

text: Theorems

text: and

text: where

text: the

text: convergence

text: rate

text: is

text: unclear.

text: Remark

text: If

text: the

text: scalar

text: function

text: ϕ∗(

text: satisﬁes

text: an

text: additional

text: condition,

text: i.e.,

text: there

text: exist

text: c

text: c

text: such

text: that

text: R

text: c

text: η

text: ϕ∗(η)

text: c

text: η

text: η

text: n,

text: then

text: Theorem

text: leads

text: to

text: global

text: exponential

text: synchronization.

text: Remark

text: For

text: the

text: leader-follower

text: case,

text: Theorems

text: and

text: show

text: that

text: global

text: asymptotic

text: syn-

text: chronization

text: is

text: achieved

text: while

text: for

text: the

text: leaderless

text: case,

text: while

text: global

text: asymptotic

text: ϕ-synchronization

text: is

text: achieved

text: as

text: shown

text: in

text: Theorems

text: and

text: For

text: the

text: leader-follower

text: case,

text: the

text: uniformly

text: jointly

text: leader

text: connected

text: in

text: Theorem

text: requires

text: the

text: leader

text: to

text: be

text: a

text: center

text: node,

text: while

text: for

text: the

text: leaderless

text: case,

text: the

text: uniformly

text: jointly

text: strongly

text: connected

text: in

text: Theorem

text: requires

text: every

text: node

text: to

text: be

text: a

text: cen-

text: ter

text: node.

text: Thus,

text: the

text: connectivity

text: condition

text: of

text: the

text: leaderless

text: case

text: is

text: stronger

text: than

text: that

text: of

text: the

text: leader-follower

text: case.

text: Lipschitz

text: Inherent

text: Dynamics

text: In

text: this

text: section,

text: we

text: extend

text: the

text: result

text: for

text: the

text: case

text: when

text: the

text: agent

text: dynamics

text: is

text: globally

text: Lipschitz

text: to

text: leader-follower

text: networks.

text: Our

text: main

text: result

text: for

text: this

text: case

text: is

text: given

text: in

text: the

text: following

text: theorem

text: whose

text: proof

text: can

text: be

text: found

text: in

text: Appendix

text: G

text: Theorem

text: Let

text: Assumption

text: hold.

text: Suppose

text: that

text: is

text: uniformly

text: jointly

text: leader

text: connected.

text: σ(t)

text: The

text: multi-agent

text: system

text: achieves

text: global

text: exponential

text: synchronization

text: if

text: L

text: ρˆ∗/2,

text: where

text: ρˆ∗

text: is

text: a

text: constant

text: depending

text: on

text: the

text: network

text: parameters.


text: Deﬁnition 3 (i) The multi-agent system (1) achieves global asymptotic synchronization if − ∈ V ≥ ∈ R lim →∞(x (t) x (t)) = 0 for any i, j , any t 0 and any x(t ) nN. t i j 0 0 ≥ (ii) Multi-agent system (1) achieves global exponential synchronization if there exist γ 1 and λ > 0 such that (cid:107) − (cid:107) ≤ − − (cid:107) − (cid:107) ≥ max x (t) x (t) 2 γe λ(t t0) max x (t ) x (t ) 2, t t , (2) { }∈V×V i j { }∈V×V i 0 j 0 0 i,j i,j ≥ ∈ R for any t 0 and any x(t ) nN. 0 0 Remark 1 ϕ-synchronization is a type of output synchronization where the output of agent ∈ V i is chosen to be ϕ(x ). It is related to but diﬀerent from χ-synchronization [18,19] since ϕ i is a function of an individual agent state while χ is a function of all agent states. 2.2 Non-expansive Inherent Dynamics In this section, we focus on when the nonlinear inherent agent dynamics is non-expansive as indicated by the following assumption. R → R Assumption 1 ϕ : n is a continuously diﬀerentiable positive deﬁnite convex function satisfying ∞ (i). lim(cid:107) (cid:107)→∞ ϕ(η) = ; η (cid:104)∇ (cid:105) ≤ ∈ R ≥ (ii). ϕ(η), f(t, η) 0 for any η n and any t 0. The following lemma shows how Assumption 1 enforces non-expansive dynamics. Lemma 1 Let Assumption 1 hold. Along the multi-agent dynamics (1), max ∈V ϕ(x (t)) is i i ≥ non-increasing for all t 0. We now state main results for the non-expansive case. Theorem 1 Let Assumption 1 hold. The multi-agent system (1) achieves global asymptotic G ϕ-synchronization if is uniformly jointly strongly connected. σ(t) 4

text: Deﬁnition

label: paragraph

text: (i)

text: The

text: multi-agent

text: system

text: achieves

text: global

text: asymptotic

text: synchronization

text: if

text: V

text: R

text: lim

text: →∞(x

text: (t)

text: x

text: (t))

text: for

text: any

text: i,

text: j

text: any

text: t

text: and

text: any

text: x(t

text: nN.

text: t

text: i

text: j

text: (ii)

text: Multi-agent

text: system

text: achieves

text: global

text: exponential

text: synchronization

text: if

text: there

text: exist

text: γ

text: and

text: λ

text: such

text: that

label: equation

text: max

text: x

text: (t)

text: x

text: (t)

text: γe

text: λ(t

text: max

text: x

text: (t

text: x

text: (t

text: t

text: t

text: }∈V×V

text: i

text: j

text: }∈V×V

text: i

text: j

text: i,j

text: i,j

text: R

text: for

text: any

text: t

text: and

text: any

text: x(t

text: nN.

text: Remark

text: ϕ-synchronization

text: is

text: a

text: type

text: of

text: output

text: synchronization

text: where

text: the

text: output

text: of

text: agent

text: V

text: i

text: is

text: chosen

text: to

text: be

text: ϕ(x

text: It

text: is

text: related

text: to

text: but

text: diﬀerent

text: from

text: χ-synchronization

text: since

text: ϕ

text: i

text: is

text: a

text: function

text: of

text: an

text: individual

text: agent

text: state

text: while

text: χ

text: is

text: a

text: function

text: of

text: all

text: agent

text: states.

text: Non-expansive

text: Inherent

text: Dynamics

text: In

text: this

text: section,

text: we

text: focus

text: on

text: when

text: the

text: nonlinear

text: inherent

text: agent

text: dynamics

text: is

text: non-expansive

text: as

text: indicated

text: by

text: the

text: following

text: assumption.

text: R

text: R

text: Assumption

text: ϕ

text: n

text: is

text: a

text: continuously

text: diﬀerentiable

text: positive

text: deﬁnite

text: convex

text: function

text: satisfying

text: (i).

text: ϕ(η)

text: η

text: R

text: (ii).

text: ϕ(η),

text: f(t,

text: η)

text: for

text: any

text: η

text: n

text: and

text: any

text: t

text: The

text: following

text: lemma

text: shows

text: how

text: Assumption

text: enforces

text: non-expansive

text: dynamics.

text: Lemma

text: Let

text: Assumption

text: hold.

text: Along

text: the

text: multi-agent

text: dynamics

text: max

text: ∈V

text: ϕ(x

text: (t))

text: is

text: i

text: i

text: non-increasing

text: for

text: all

text: t

text: We

text: now

text: state

text: main

text: results

text: for

text: the

text: non-expansive

text: case.

text: Theorem

text: Let

text: Assumption

text: hold.

text: The

text: multi-agent

text: system

text: achieves

text: global

text: asymptotic

text: G

text: ϕ-synchronization

text: if

text: is

text: uniformly

text: jointly

text: strongly

text: connected.

text: σ(t)


text: Accretion onto very low mass companions of young stars 5 110000..0000 110000..0000 GSC06214−00210 GSC06214−00210 Fix Parameter Fitting Free Parameter Fitting ÅÅ−1−1)) 1100..0000 ÅÅ−1−1)) 1100..0000 −2−2m m −2−2m m cc cc −1−1g s g s 11..0000 −1−1g s g s 11..0000 erer erer −17−170 0 −17−170 0 11 11 x (x ( 00..1100 x (x ( 00..1100 uu uu FlFl FlFl 00..0011 00..0011 22..55••110033 55..00••110033 11..00••110044 11..55••110044 22..55••110044 22..55••110033 55..00••110033 11..00••110044 11..55••110044 22..55••110044 WWaavveelleennggtthh ((ÅÅ)) WWaavveelleennggtthh ((ÅÅ)) GQ Lup GQ Lup Fix Parameter Fitting Free Parameter Fitting ÅÅ−1−1)) 110000 ÅÅ−1−1)) 110000 −2−2m m −2−2m m cc cc −1−1g s g s −1−1g s g s erer erer −17−170 0 −17−170 0 11 11 x (x ( x (x ( uu uu FlFl 1100 FlFl 1100 22..55••110033 55..00••110033 11..00••110044 11..55••110044 22..55••110044 22..55••110033 55..00••110033 11..00••110044 11..55••110044 22..55••110044 WWaavveelleennggtthh ((ÅÅ)) WWaavveelleennggtthh ((ÅÅ)) DH Tau DH Tau Fix Parameter Fitting Free Parameter Fitting ÅÅ−1−1)) 1100..0000 ÅÅ−1−1)) 1100..0000 −2−2m m −2−2m m cc cc −1−1g s g s 11..0000 −1−1g s g s 11..0000 erer erer −17−170 0 −17−170 0 11 11 x (x ( 00..1100 x (x ( 00..1100 uu uu FlFl FlFl 00..0011 00..0011 22..55••110033 55..00••110033 11..00••110044 11..55••110044 22..55••110044 22..55••110033 55..00••110033 11..00••110044 11..55••110044 22..55••110044 WWaavveelleennggtthh ((ÅÅ)) WWaavveelleennggtthh ((ÅÅ)) Fig. 2.— Optical and NIR SED with BT-settl grid and slab emission of GSC06214-00210 (top), GQ Lup b (middle), and DH Tau b (bottom). The left ﬁgures are the results for ﬁxed parameter ﬁtting. The right ﬁgures are for free parameter ﬁtting. Red square points are observed photometry results. Green dashed lines are emission from a pure hydrogen layer, gray lines are the ﬂuxes from BT-settl model spectra ,dark blue lines are the total ﬂux of photosphere and accretion model. The integrate ﬂux of the models are plotted in blue points. The SEDs present clear optical excess emissions. 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text: Accretion onto very low mass companions of young stars 7 −6 −6 Herczeg et al. (2008,2009) This Paper −7 IGn(glllebbr) n egt eatl .a (l2. 0(11939)8) −7 Natta et al. (2006) R gl aco et al. (2011,2012) −8 Alcala et al. (2013) −8 Th s Paper yr)) −9 GQ yr)) −9 GQ / / ⊙−10 ⊙−10 M M M/(acc −11 DH GSC M/(acc −11 DH GSC g( −12 g( −12 o o l l −13 −13 −14 −14 −15 −15 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 log(M/M⊙) log(M/M⊙) −6 This Paper −7 Fang et al. (2009) 100 y −8 sit GSC o n yr)) −9 GQ Lumi /⊙−10 b M/(Macc −11 DH GSC sity/Sla 10-1 DH log( −12 mino −13 Lu GQ α −14 H 10-2 −15 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 10-7 10-6 10-5 10-4 10-3 10-2 10-1 log(M/M⊙) Accretion Luminosity Fig. 3.— Accretion properties of the planetary mass companions compared to other young accretors. The three planetary mass companions have accretion rates that are one order of magnitude higher than expected from the correlation between object mass and UV excess accretion rates (upper left) and from accretion rates obtained from ombined measurements of the Hα, Hβ and He I λ5876 lines (low±er left), but are consistent with accretion rates measured in ρ Oph from Paβ and Brγ (upper right). The yellow shaded region shows the 1σ region from best ﬁt line between mass and UV excess accretion rates and demonstrates the oﬀsets between diﬀerent studies. Lower right: The ratio of Hα luminosity to slab luminosity increases with lower accretion luminosity. ACCRETION ONTO PLANETARY MASS COMPANIONS OF LOW-MASS YOUNG STARS Yifan Zhou1,2, Gregory J. Herczeg1,2, Adam L. Kraus3, Stanimir Metchev4,5 and Kelle L. Cruz6,7 Subject headings: stars: pre-main sequence — planetary systems: protoplanetary disks — stars: low-mass, brown dwarfs — accretion, accretion disks 4. DISCUSSION though this emission was undetected in some spectra of GQ Lup b. For GSC 06214-00210 b, the accretion lu- Accretion is detected in both the Hα line and in excess minosity measured here (when including Hα luminosity) optical continuum emission from DH Tau b, GSC 06214- − ± − ± of log L(cid:32) /L(cid:12) = 4.6 0.5 is similar to the 4.4 1.3 00210 b, and GQ Lup b. Accretion had been previ- acc measured from Paβ by Bowler et al. (2011), although our ously detected for all three objects in Paβ emission, al- direct measurements have much smaller error bars. The accretion rates calculated here demonstrate that 1 Kavli Institute for Astronomy and Astrophysics, Peking Uni- these objects have their own disks. The separations (100– versity, Yi He Yuan Lu 5, Haidian Qu, Beijing 100871, People’s 300 AU) of the companion objects to their primaries are Republic of China 2 Department of Astronomy, School of Physics, Peking Uni- too large to support such high accretion rates. Moreover, versity, Yi He Yuan Lu 5, Haidian District, Beijing 100871, P.R. there is no evidence to indicate the presence of a primor- China dial disk around the primary star GSC 06214-00210. 3 Department of Astronomy, The University of Texas at These sources are at the brown dwarf-planet mass Austin, Austin, TX 78712, USA 4 Department of Physics & Astronomy, The University of boundary, are among the lowest mass sources with ac- Western Ontario, London, ON N6A 3K7, Canada cretion measured directly from excess optical spectra, 5 Department of Physics and Astronomy, State University of and are the lowest mass companions to stellar primary New York, Stony Brook, NY 11794, USA 6 Hunter College, City University of New York, Dept. of stars with measured accretion rates. Previous detections Physics and Astronomy, New York, NY 10065 of accretion onto very low mass objects has been diag- 7 American Museum of Natural History, Astrophysics Dept, nosed through emission lines (e.g. Bowler et al. 2011; New York, NY 10025 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: Accretion

text: onto

text: very

text: low

text: mass

text: companions

text: of

text: young

text: stars

text: −6

text: −6

text: Herczeg

text: et

text: al.

text: This

text: Paper

text: −7

text: IGn(glllebbr)

text: n

text: egt

text: eatl

text: .a

text: −7

text: Natta

text: et

text: al.

text: R

text: gl

text: aco

text: et

text: al.

text: −8

text: Alcala

text: et

text: al.

text: −8

text: Th

text: s

text: Paper

text: yr))

text: −9

text: GQ

text: yr))

text: −9

text: GQ

text: ⊙−10

text: ⊙−10

text: M

text: M

text: M/(acc

text: DH

text: GSC

text: M/(acc

text: DH

text: GSC

text: g(

text: g(

text: o

text: o

text: l

text: l

text: log(M/M⊙)

text: log(M/M⊙)

text: −6

text: This

text: Paper

text: −7

text: Fang

text: et

text: al.

text: y

text: −8

text: sit

text: GSC

text: o

text: n

text: yr))

text: −9

text: GQ

text: Lumi

text: b

text: M/(Macc

text: DH

text: GSC

text: sity/Sla

text: DH

text: log(

text: mino

text: Lu

text: GQ

text: α

text: H

text: log(M/M⊙)

text: Accretion

text: Luminosity

text: Fig.

text: Accretion

text: properties

text: of

text: the

text: planetary

text: mass

text: companions

text: compared

text: to

text: other

text: young

text: accretors.

text: The

text: three

text: planetary

text: mass

text: companions

text: have

text: accretion

text: rates

text: that

text: are

text: one

text: order

text: of

text: magnitude

text: higher

text: than

text: expected

text: from

text: the

text: correlation

text: between

text: object

text: mass

text: and

text: UV

text: excess

text: accretion

text: rates

text: (upper

text: left)

text: and

text: from

text: accretion

text: rates

text: obtained

text: from

text: ombined

text: measurements

text: of

text: the

text: Hα,

text: Hβ

text: and

text: He

text: I

text: lines

text: (low±er

text: left),

text: but

text: are

text: consistent

text: with

text: accretion

text: rates

text: measured

text: in

text: ρ

text: Oph

text: from

text: Paβ

text: and

text: Brγ

text: (upper

text: right).

text: The

text: yellow

text: shaded

text: region

text: shows

text: the

text: 1σ

text: region

text: from

text: best

text: ﬁt

text: line

text: between

text: mass

text: and

text: UV

text: excess

text: accretion

text: rates

text: and

text: demonstrates

text: the

text: oﬀsets

text: between

text: diﬀerent

text: studies.

text: Lower

text: right:

text: The

text: ratio

text: of

text: Hα

text: luminosity

text: to

text: slab

text: luminosity

text: increases

text: with

text: lower

text: accretion

text: luminosity.

text: ACCRETION

text: ONTO

text: PLANETARY

text: MASS

text: COMPANIONS

text: OF

text: LOW-MASS

text: YOUNG

text: STARS

text: Yifan

text: Gregory

text: J.

text: Adam

text: L.

text: Stanimir

text: and

text: Kelle

text: L.

text: Subject

text: headings:

text: stars:

text: pre-main

text: sequence

text: planetary

text: systems:

text: protoplanetary

text: disks

text: stars:

text: low-mass,

text: brown

text: dwarfs

text: accretion,

text: accretion

text: disks

text: DISCUSSION

text: though

text: this

text: emission

text: was

text: undetected

text: in

text: some

text: spectra

text: of

text: GQ

text: Lup

text: b.

text: For

text: GSC

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text: the

text: accretion

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text: Accretion

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text: detected

text: in

text: both

text: the

text: Hα

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text: excess

text: minosity

text: measured

text: here

text: (when

text: including

text: Hα

text: luminosity)

text: optical

text: continuum

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text: DH

text: Tau

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text: GSC

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text: log

text: is

text: similar

text: to

text: the

text: b,

text: and

text: GQ

text: Lup

text: b.

text: Accretion

text: had

text: been

text: previ-

text: acc

text: measured

text: from

text: Paβ

text: by

text: Bowler

text: et

text: al.

text: although

text: our

text: ously

text: detected

text: for

text: all

text: three

text: objects

text: in

text: Paβ

text: emission,

text: al-

text: direct

text: measurements

text: have

text: much

text: smaller

text: error

text: bars.

text: The

text: accretion

text: rates

text: calculated

text: here

text: demonstrate

text: that

text: Kavli

text: Institute

text: for

text: Astronomy

text: and

text: Astrophysics,

text: Peking

text: Uni-

text: these

text: objects

text: have

text: their

text: own

text: disks.

text: The

text: separations

text: versity,

text: Yi

text: He

text: Yuan

text: Lu

text: Haidian

text: Qu,

text: Beijing

text: People’s

text: AU)

text: of

text: the

text: companion

text: objects

text: to

text: their

text: primaries

text: are

text: Republic

text: of

text: China

text: Department

text: of

text: Astronomy,

text: School

text: of

text: Physics,

text: Peking

text: Uni-

text: too

text: large

text: to

text: support

text: such

text: high

text: accretion

text: rates.

text: Moreover,

text: versity,

text: Yi

text: He

text: Yuan

text: Lu

text: Haidian

text: District,

text: Beijing

text: P.R.

text: there

text: is

text: no

text: evidence

text: to

text: indicate

text: the

text: presence

text: of

text: a

text: primor-

text: China

text: dial

text: disk

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text: Previous

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text: Astronomy,

text: New

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text: onto

text: very

text: low

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text: objects

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text: been

text: diag-

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text: of

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text: History,

text: Astrophysics

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text: nosed

text: through

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text: retain theirs identities. However, unlike in the soliton case, the collisions are not elastic. Experimental realization of the ideas presented here was suggested earlier [17]. ACKNOWLEDGMENTS The authors acknowledge support of a Polish Government Research Grant and also the Foundation for Polish Science Team Programme co-ﬁnanced by the EU European Regional Development Fund. ≈ Appendix A: Expansion in the case η η−. + (cid:113) ± Assume the diﬀerence between µ and µ− to be small. Introduce µ± = µ(1 δ) and √ + ξ = µx. We now have η± = µ f±. Equations (21) now lead to 1+γ d2f± − ± f± + f±3 + βf∓2 f± = δf±, (A1) dξ2 − where β = (1 γ)/(1 + γ). We now expand f± in δ: ≈ f± f(0) + f±(1) δ. (A2) (cid:113) Equation (A1) in zero order is: − ⇒ d2 f(0) f(0) + (1 + β)f(0)3 = 0 f(0) = 2 sech(ξ). (A3) dξ2 1+β And in the next order we ﬁnd ± Lˆf±(1) + 2βf(0)2f∓(1) = f(0), (A4) (cid:18) (cid:19) where d2 − d2 − 3 + β Lˆ = 1 + (3 + β)f(0)2 = 1 + 2 sech2ξ. (A5) dξ2 dξ2 1 + β (cid:20) (cid:21) Adding the two equations (A4) yields d2 − 1 + 3(1 + β)f(0)2 (f(1) + f−(1)) = 0 (A6) dξ2 + ∝ Solved by f(1) + f−(1) df(0)(ξ)/dξ, corresponding to a shift in position, a trivial transfor- + (cid:114) − mation. Thus, without loss of generality we may assume f(1) = f−(1). Deﬁne + − ≡ 2 f(1) f−(1) ∆f. (A7) + 1 + β 20 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: retain

text: theirs

text: identities.

text: However,

text: unlike

text: in

text: the

text: soliton

text: case,

text: the

text: collisions

text: are

text: not

text: elastic.

text: Experimental

text: realization

text: of

text: the

text: ideas

text: presented

text: here

text: was

text: suggested

text: earlier

text: ACKNOWLEDGMENTS

text: The

text: authors

text: acknowledge

text: support

text: of

text: a

text: Polish

text: Government

text: Research

text: Grant

text: and

text: also

text: the

text: Foundation

text: for

text: Polish

text: Science

text: Team

text: Programme

text: co-ﬁnanced

text: by

text: the

text: EU

text: European

text: Regional

text: Development

text: Fund.

label: section

text: Appendix

text: A:

text: Expansion

text: in

text: the

text: case

text: η

text: η−.

text: Assume

text: the

text: diﬀerence

text: between

text: µ

text: and

text: µ−

text: to

text: be

text: small.

text: Introduce

text: µ±

text: δ)

text: and

text: ξ

text: µx.

text: We

text: now

text: have

text: η±

text: µ

text: f±.

text: Equations

text: now

text: lead

text: to

text: f±

text: βf∓2

text: f±

text: δf±,

text: where

text: β

text: γ).

text: We

text: now

text: expand

text: f±

text: in

text: δ:

text: f±

text: δ.

text: Equation

text: in

text: zero

text: order

text: is:

text: sech(ξ).

text: And

text: in

text: the

text: next

text: order

text: we

text: ﬁnd

text: where

text: β

text: Lˆ

text: β

text: Adding

text: the

text: two

text: equations

text: yields

text: Solved

text: by

text: corresponding

text: to

text: a

text: shift

text: in

text: position,

text: a

text: trivial

text: transfor-

text: mation.

text: Thus,

text: without

text: loss

text: of

text: generality

text: we

text: may

text: assume

text: Deﬁne

text: ∆f.

text: β

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text: 2. The polar state (cid:104) (cid:105) Considering Eq. (3) T = 1, we recover a normal sech-type soliton: √ − − − Φ = 2 k sech [k(x x 2pt)] eıpxeı(k2 p2)tχ. (11) 0     † The constrain χ χ¯ = 1 implies that          −√1 e−ıβ sin α   0  2 U χ(polar) = eıτ cos α = eıτ ˆ(β, α, θ) 1 , (12) √1 eıβ sin α 0 2 and one can check that the local mean spin density vanishes identically. Generalization of this soliton solution is straightforward. Since it is a spinless state, the spin mixing interaction term in Eq. (2) vanishes and what remains is stratiﬁed by (11) for all γ. We will refer to   this solution as ageneralized polar soliton       √  0  − − − U Φ(polar) = 2 k sech [k(x x 2pt)] eıpxeı(k2 p2)teıτ ˆ 1 . (13) 0 0 Notice that the amplitude of the soliton is diﬀerent from that of the ferromagnetic soliton, which leads to the diﬀerent relation between the total number of atoms and parameter k N = 4k. (14) tot The energy diﬀerence between ferromagnetic and polar solitons, with the same number of atoms, is: − − 1 Eferro Epolar = N3 γ(2 + γ). (15) 48 tot 3. Split solitons (cid:104) (cid:105) If 0 < T < 1, for every moment t, each component of the local mean spin density vector (cid:82) † fα(x, t) = Φ fˆαΦ, is an anisymmetric function of x with respect to a certain point x (t). node This implies that the total mean spin of this state, f = dx f(x), is equal to 0. Careful tot (cid:104) (cid:105) examination of the density proﬁle of this kind of soliton reveals the reason for this. For T close to 0 the density splits into two disjointed peaks traveling with the same velocity. Each 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: polar

text: state

text: Considering

text: Eq.

text: T

text: we

text: recover

text: a

text: normal

text: sech-type

text: soliton:

text: Φ

text: k

text: sech

text: [k(x

text: x

text: The

text: constrain

text: χ

text: χ¯

text: implies

text: that

text: −√1

text: e−ıβ

text: sin

text: α

text: U

text: χ(polar)

text: eıτ

text: cos

text: α

text: eıτ

text: ˆ(β,

text: α,

text: θ)

text: √1

text: eıβ

text: sin

text: α

text: and

text: one

text: can

text: check

text: that

text: the

text: local

text: mean

text: spin

text: density

text: vanishes

text: identically.

text: Generalization

text: of

text: this

text: soliton

text: solution

text: is

text: straightforward.

text: Since

text: it

text: is

text: a

text: spinless

text: state,

text: the

text: spin

text: mixing

text: interaction

text: term

text: in

text: Eq.

text: vanishes

text: and

text: what

text: remains

text: is

text: stratiﬁed

text: by

text: for

text: all

text: γ.

text: We

text: will

text: refer

text: to

text: this

text: solution

text: as

text: ageneralized

text: polar

text: soliton

text: U

text: Φ(polar)

text: k

text: sech

text: [k(x

text: x

text: ˆ

text: Notice

text: that

text: the

text: amplitude

text: of

text: the

text: soliton

text: is

text: diﬀerent

text: from

text: that

text: of

text: the

text: ferromagnetic

text: soliton,

text: which

text: leads

text: to

text: the

text: diﬀerent

text: relation

text: between

text: the

text: total

text: number

text: of

text: atoms

text: and

text: parameter

text: k

text: N

text: tot

text: The

text: energy

text: diﬀerence

text: between

text: ferromagnetic

text: and

text: polar

text: solitons,

text: with

text: the

text: same

text: number

text: of

text: atoms,

text: is:

text: Eferro

text: Epolar

text: γ).

text: tot

text: Split

text: solitons

text: If

text: T

text: for

text: every

text: moment

text: t,

text: each

text: component

text: of

text: the

text: local

text: mean

text: spin

text: density

text: vector

text: fα(x,

text: t)

text: Φ

text: fˆαΦ,

text: is

text: an

text: anisymmetric

text: function

text: of

text: x

text: with

text: respect

text: to

text: a

text: certain

text: point

text: x

text: (t).

text: node

text: This

text: implies

text: that

text: the

text: total

text: mean

text: spin

text: of

text: this

text: state,

text: f

text: dx

text: f(x),

text: is

text: equal

text: to

text: Careful

text: tot

text: examination

text: of

text: the

text: density

text: proﬁle

text: of

text: this

text: kind

text: of

text: soliton

text: reveals

text: the

text: reason

text: for

text: this.

text: For

text: T

text: close

text: to

text: the

text: density

text: splits

text: into

text: two

text: disjointed

text: peaks

text: traveling

text: with

text: the

text: same

text: velocity.

text: Each

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 1 3 2 4 5 α(1) β(3) β(4) α(5) α(2) = β(2) α(2) = β(2) α(3) β(1) β(5) α(4) Figure 7: An example to illustrate the result of Proposition 4.2 where α(1) = β(4), β(3) = α(5), α(3) = β(5) and β(1) = α(4). α(1) = β(4) α(5) = β(3) α(1) = β(4) α(5) = β(3) α(2) = β(2) α(2) = β(2) α(3) = β(5) α(4) = β(1) α(3) = β(5) α(4) = β(1) H K Figure 8: The smallest unstable graphs where a triangle is taken to a two directed triangles sharing a vertex. 1 α(4) = β(1) 2 3 α(2) = β(2) α(3) = β(3) 4 α(1) = β(4) Figure 9: The smallest unstable graph which has a TF-automorphism taking a triangle to the mixed graph illustrated in Figure 4(c). 12 ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: An

label: caption

text: example

text: to

text: illustrate

text: the

text: result

text: of

text: Proposition

text: where

text: and

text: H

text: K

text: Figure

text: The

text: smallest

text: unstable

text: graphs

text: where

text: a

text: triangle

text: is

text: taken

text: to

text: a

text: two

text: directed

text: triangles

text: sharing

text: a

text: vertex.

text: Figure

text: The

text: smallest

text: unstable

text: graph

text: which

text: has

text: a

text: TF-automorphism

text: taking

text: a

text: triangle

text: to

text: the

text: mixed

text: graph

text: illustrated

text: in

text: Figure

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: − 4. Letting m = 2k + e, whenever d(v, w) = k 1, there is a vertex u adjacent to w such that d(v, u) = k. Proof. First note that if uv is an arc of G and both u, v belong to the same H , then the image of uv is an arc of H , hence of G, because (α, β) acts i i+1 like (α , β ) in H . If u, v do not belong to the same H , then they belong to i i i i consecutive graphs, say H , H , so α(u) and β(u) belong to the consecu- i i+1 tive graphs H , H , and are adjacent because all the arcs between these i+1 i+2 two graphs belong to G. This proves (1). − Concerning distance, a path from u in H to v in H , where s = (m e)/2, 0 s must pass through all graphs H , H , ..., H − or else H − , H − , ..., H . 1 2 s 1 m 1 m 2 s+1 Since such a path can be found, d(u, v) = s. For two vertices in, say, H (cid:54) i and H with i = j, the same argument shows that d(u, v) cannot exceed s. j Finally, if u, v lie in the same H , they have a common neighbour in H , i i+1 then d(u, v) is less or equal than 2 (regardless to their distance within H ). i This proves (2). ∈ What about triangles? If uv is an edge of some H then letting w H i i+1 the vertex w is adjacent to both u and v, hence uvw is a triangle. If uv is an ∈ ∈ edge of some [H , H ], say with u H and v H , take any neighbour i i+1 i i+1 w of u in H (recall the assumption about no isolated vertices) and get the i triangle uvw. This proves (3). Finally, since m = 2k + e, diam G is at least k + e. Let us remark that − ∈ ∈ if d(v, w) = k 1 and, say, v V(H ), then w V(H ) where j is either − i j i + 2 or i 2. Take u in H or H − , respectively. Then d(v, u) = k. This i+3 i 3 proves (4). Note that the structure of the single H ’s is immaterial here, − i because two vertices at distance at least k 1 from each other must come (cid:3) from a diﬀerent H . i Until now, we did not mention that the concerned TF-isomorphisms are non-trivial, so all the above would work ﬁne for the case of isomorphisms too. But adding the hypothesis that at least one of them is non-trivial, the obtained graph G has a non-trivial TF-isomorphism, namely (α, β) as de- scribed above. The statements (1)-(3) show that there are unstable graphs of arbitrarily high diameter, where each edge belongs to a triangle. Surowski [10], [11] proved various results concerning graph stability. In [10] Proposition 2.1, he claims that if G is a connected graph of diameter ≥ d 4 in which every edge lies in a triangle, then G is stable. However, by ≥ taking m 7 in Theorem 5.1 we get inﬁnitely many counterexamples to this claim by taking all the H isomorphic to the same vertex-determining i bipartite graph, because such a vertex-transitive graph is unstable, therefore 14

label: list

text: Letting

text: m

text: e,

text: whenever

text: d(v,

text: w)

text: k

text: there

text: is

text: a

text: vertex

text: u

text: adjacent

text: to

text: w

text: such

text: that

text: d(v,

text: u)

text: k.

text: Proof.

text: First

text: note

text: that

text: if

text: uv

text: is

text: an

text: arc

text: of

text: G

text: and

text: both

text: u,

text: v

text: belong

text: to

text: the

text: same

text: H

text: then

text: the

text: image

text: of

text: uv

text: is

text: an

text: arc

text: of

text: H

text: hence

text: of

text: G,

text: because

text: (α,

text: β)

text: acts

text: i

text: like

text: (α

text: β

text: in

text: H

text: If

text: u,

text: v

text: do

text: not

text: belong

text: to

text: the

text: same

text: H

text: then

text: they

text: belong

text: to

text: i

text: i

text: i

text: i

text: consecutive

text: graphs,

text: say

text: H

text: H

text: so

text: α(u)

text: and

text: β(u)

text: belong

text: to

text: the

text: consecu-

text: i

text: tive

text: graphs

text: H

text: H

text: and

text: are

text: adjacent

text: because

text: all

text: the

text: arcs

text: between

text: these

text: two

text: graphs

text: belong

text: to

text: G.

text: This

text: proves

text: Concerning

text: distance,

text: a

text: path

text: from

text: u

text: in

text: H

text: to

text: v

text: in

text: H

text: where

text: s

text: (m

text: s

text: must

text: pass

text: through

text: all

text: graphs

text: H

text: H

text: H

text: or

text: else

text: H

text: H

text: H

text: s

text: m

text: m

text: Since

text: such

text: a

text: path

text: can

text: be

text: found,

text: d(u,

text: v)

text: s.

text: For

text: two

text: vertices

text: in,

text: say,

text: H

text: i

text: and

text: H

text: with

text: i

text: j,

text: the

text: same

text: argument

text: shows

text: that

text: d(u,

text: v)

text: cannot

text: exceed

text: s.

text: j

text: Finally,

text: if

text: u,

text: v

text: lie

text: in

text: the

text: same

text: H

text: they

text: have

text: a

text: common

text: neighbour

text: in

text: H

text: i

text: then

text: d(u,

text: v)

text: is

text: less

text: or

text: equal

text: than

text: (regardless

text: to

text: their

text: distance

text: within

text: H

text: i

text: This

text: proves

text: What

text: about

text: triangles?

text: If

text: uv

text: is

text: an

text: edge

text: of

text: some

text: H

text: then

text: letting

text: w

text: H

text: i

text: the

text: vertex

text: w

text: is

text: adjacent

text: to

text: both

text: u

text: and

text: v,

text: hence

text: uvw

text: is

text: a

text: triangle.

text: If

text: uv

text: is

text: an

text: edge

text: of

text: some

text: [H

text: H

text: say

text: with

text: u

text: H

text: and

text: v

text: H

text: take

text: any

text: neighbour

text: i

text: i

text: w

text: of

text: u

text: in

text: H

text: (recall

text: the

text: assumption

text: about

text: no

text: isolated

text: vertices)

text: and

text: get

text: the

text: i

text: triangle

text: uvw.

text: This

text: proves

text: Finally,

text: since

text: m

text: e,

text: diam

text: G

text: is

text: at

text: least

text: k

text: e.

text: Let

text: us

text: remark

text: that

text: if

text: d(v,

text: w)

text: k

text: and,

text: say,

text: v

text: V(H

text: then

text: w

text: V(H

text: where

text: j

text: is

text: either

text: i

text: j

text: i

text: or

text: i

text: Take

text: u

text: in

text: H

text: or

text: H

text: respectively.

text: Then

text: d(v,

text: u)

text: k.

text: This

text: i

text: proves

text: Note

text: that

text: the

text: structure

text: of

text: the

text: single

text: H

text: ’s

text: is

text: immaterial

text: here,

text: i

text: because

text: two

text: vertices

text: at

text: distance

text: at

text: least

text: k

text: from

text: each

text: other

text: must

text: come

text: from

text: a

text: diﬀerent

text: H

text: i

text: Until

text: now,

text: we

text: did

text: not

text: mention

text: that

text: the

text: concerned

text: TF-isomorphisms

text: are

text: non-trivial,

text: so

text: all

text: the

text: above

text: would

text: work

text: ﬁne

text: for

text: the

text: case

text: of

text: isomorphisms

text: too.

text: But

text: adding

text: the

text: hypothesis

text: that

text: at

text: least

text: one

text: of

text: them

text: is

text: non-trivial,

text: the

text: obtained

text: graph

text: G

text: has

text: a

text: non-trivial

text: TF-isomorphism,

text: namely

text: (α,

text: β)

text: as

text: de-

text: scribed

text: above.

text: The

text: statements

text: show

text: that

text: there

text: are

text: unstable

text: graphs

text: of

text: arbitrarily

text: high

text: diameter,

text: where

text: each

text: edge

text: belongs

text: to

text: a

text: triangle.

text: Surowski

text: proved

text: various

text: results

text: concerning

text: graph

text: stability.

text: In

text: Proposition

text: he

text: claims

text: that

text: if

text: G

text: is

text: a

text: connected

text: graph

text: of

text: diameter

text: d

text: in

text: which

text: every

text: edge

text: lies

text: in

text: a

text: triangle,

text: then

text: G

text: is

text: stable.

text: However,

text: by

text: taking

text: m

text: in

text: Theorem

text: we

text: get

text: inﬁnitely

text: many

text: counterexamples

text: to

text: this

text: claim

text: by

text: taking

text: all

text: the

text: H

text: isomorphic

text: to

text: the

text: same

text: vertex-determining

text: i

text: bipartite

text: graph,

text: because

text: such

text: a

text: vertex-transitive

text: graph

text: is

text: unstable,

text: therefore


text: H 1 7 H 0 1 6 2 12 8 5 3 11 9 4 10 H H 7 proepssriebsleen etsd gaells 2 43 between H1 and H2 13 48 44 18 14 47 45 17 15 46 16 H H 6 3 37 19 42 38 24 20 41 39 23 21 40 22 H 31 25 H 5 4 36 32 30 26 35 33 29 27 34 28 Figure 10: A graph constructed using Theorem 5.1. one can ﬁnd a non-trivial TF-isomorphism from H to H , and since these i i+1 H are isomorphic, the resulting graph G is vertex-determining, and has i ≥ diameter k 4 and unstable. One of these counterexamples is illustrated in Figure 10. We detected one possible ﬂaw in Surowski’s proof. It is claimed in [10] − that whenever an automorphism of B(G) ﬁxes (v, 1) it also ﬁxes (v, 1). We have not seen a proof of this result. Besides, in our last example, G has a non-trivial ﬁxed-point-free TF-automorphism, which implies that B(G) has a ﬁxed-point-free automorphism that that ﬁxes the colour classes. This claim is also used in [10] Proposition 2.2 which states that if G is a strongly (cid:54) ≥ regular graph with k > µ = λ 1, then G is stable. Hence, we believe that at this point, the stability of strongly regular graphs with these parameters requires further investigation. 15 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: H

text: H

text: H

text: H

text: proepssriebsleen

text: etsd

text: gaells

text: between

text: and

text: H

text: H

text: H

text: H

text: Figure

text: A

text: graph

text: constructed

text: using

text: Theorem

text: one

text: can

text: ﬁnd

text: a

text: non-trivial

text: TF-isomorphism

text: from

text: H

text: to

text: H

text: and

text: since

text: these

text: i

text: H

text: are

text: isomorphic,

text: the

text: resulting

text: graph

text: G

text: is

text: vertex-determining,

text: and

text: has

text: i

text: diameter

text: k

text: and

text: unstable.

text: One

text: of

text: these

text: counterexamples

text: is

text: illustrated

text: in

text: Figure

text: We

text: detected

text: one

text: possible

text: ﬂaw

text: in

text: Surowski’s

text: proof.

text: It

text: is

text: claimed

text: in

text: that

text: whenever

text: an

text: automorphism

text: of

text: B(G)

text: ﬁxes

text: (v,

text: it

text: also

text: ﬁxes

text: (v,

text: We

text: have

text: not

text: seen

text: a

text: proof

text: of

text: this

text: result.

text: Besides,

text: in

text: our

text: last

text: example,

text: G

text: has

text: a

text: non-trivial

text: ﬁxed-point-free

text: TF-automorphism,

text: which

text: implies

text: that

text: B(G)

text: has

text: a

text: ﬁxed-point-free

text: automorphism

text: that

text: that

text: ﬁxes

text: the

text: colour

text: classes.

text: This

text: claim

text: is

text: also

text: used

text: in

text: Proposition

text: which

text: states

text: that

text: if

text: G

text: is

text: a

text: strongly

text: regular

text: graph

text: with

text: k

text: µ

text: λ

text: then

text: G

text: is

text: stable.

text: Hence,

text: we

text: believe

text: that

text: at

text: this

text: point,

text: the

text: stability

text: of

text: strongly

text: regular

text: graphs

text: with

text: these

text: parameters

text: requires

text: further

text: investigation.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##