text: 3 order convergence and to indicate the level of truncation error. We also demonstrate how well the simulation bal- ances self-gravity against pressure for an isolated main- 0.04 ) sequence star solution in Figure 2, where we compare the M∗ 0.02 initial solution to one evolved with the lower resolution /H 0.00 for 10R∗/cs, where cs is the sound speed at the center 2MB −0.02 Low res. of the star. (For comparison, for the simulation with ×( Medium Res. the main-sequence star and M /M∗ = 106, the radial C2−0.04 High res. BH free-fall time from the initial separation of d = 200M −0.06 ≈ BH −40 −20 0 20 40 60 used here is 4R∗/c .) For the 3D simulations, the res- 1e−3 s 4 olution is equivalent to the lower value of resolution, and ) since the same coordinates, etc., are used, the truncation M∗ 2 / error is comparable. H 0 2B 3. CASES M ×( −2 We consider two diﬀerent types of stars. To ini- C2−4 (Low−Med.)×1.57 tially represent a main-sequence star, we choose the ∆ (Med.−High)×1.8 Γ = 5/3 polytropic Tolman–Oppenheimer–Volkoﬀ so- −−640 −20 0 20 40 60 lution with ≈compaction M∗/R∗ = 2 × 10−6 (equiva- (t−tmax)/MBH lent to R∗ (M∗/M(cid:12))R(cid:12) ). To represent a white dwarf, we choose the Γ = 5/3 polytropic solution with × − ≈ 0.03 compaction×M∗/R∗ = 2.5 10 4 (equivalent to R∗ ) 0.02 (M∗/M(cid:12))6 103 km ). As described in Cook et al. M∗ 0.01 (1994), for such polytropic solutions one has the freedom /H 0.00 2B to choose any overall length scale (or equivalently, mass M −0.01 Low res. scale), though when considering the potential GW de- ×( −0.02 Medium Res. tectability in Section , we will consider diﬀerent speciﬁc C3 −0.03 High res. choices of physical mass. −0.04 −40 −20 0 20 40 60 Because of the computational expense of the 3D cases, 81e−3 we restrict most of our simulations to head-on collisions. )∗ 6 (Low−Med.)×1.57 We consider the head-on collisions of main-sequence stars M 4 (Med.−High)×1.8 / with nonspinnin(cid:0)g BHs×with ma(cid:1)ss ratios MBH/M∗ = 1, 2MBH 2 0.5, 0.25, and 0.125 106. The main-sequence star 0 ×( hras/Ma no=min5a0l MNe∗w/Mtonia×n 1t0id6a2l/3d.isr(uWpteionnotreadthiuast ionf ∆C3−−42 T BH BH ≈ × −6 geometric units 106 M(cid:12) is equivalent to 1.5 106 −40 −20 0 20 40 60 ≈ (t−t )/M km or 5 s.) In order to probe any eﬀects of BH max BH spin, we also consider the collisions of main-sequence stars with BHs with dimensionless spin a = 0.99, where the spin and collision axes are aligned. For the spin- ) ning cases, which have more high-frequency power, we M∗ 0.01 / consider a slightly higher range×of mass ratios given by 2BH 0.00 MhBeHBadHw-/oiMtnh∗cmo=lalis2ss,ior1na,st0ioo.5fs,aManwdhi/0tMe.2(cid:0)∗5dw=a1r40f,6w2W,it1he, aaalnsndo(cid:1)ons0ts.u5pdi×nyn1ti0nh3ge. ×C(M4−0.01 LMHoiegwdh ir ureemss. .Res. BH −0.02 The white dwarf has a nominal Newtonian tidal disrup- −40 −20 0 20 40 60 tion radius of r /M = 40 M∗/M × 103 2/3. With 1.01e−2 T BH BH ) 0.8 these parameters, for the same value of the tidal disrup- ∗ M 0.6 tion radius in units of BH mass (i.e., the same value of / 0.4 H rT /MBH), the white dwarf’s size compared to the BH 2MB 00..02 (i.e., R∗/MBH) will be larger than for main-sequence ×( −0.2 stars. Hence, we can anticipate that GW suppression C4−0.4 (Low−Med.)×1.57 due to decoherence will happen at somewhat smaller ∆ −0.6 (Med.−High)×1.8 −0.8 values of r /M for the white dwarf compared to the −40 −20 0 20 40 60 T BH (t−t )/M main-sequence star. max BH Finally, we consider two of the more computation- ally expensive nonzero-angular-momentum collisions Fig. 1.— F−rom top to bottom, the (cid:96) = 2, 3, and 4 (with m = 0) of a main-sequence star and a nonspinning BH with spin weight 2 spherical harmonics of rΨ4 from a nonspinning, M /M∗ = 106. These cases have reduced orbital angu- MBH/M∗ = 106, head-on collision simulation performed at three BH resolutions. In each case, the lower panel shows the diﬀerence be- lar momentums of L˜/MBH = 2 and 3.5, which should be tween the diﬀerent resolutions, scaled assuming second-order con- compared to the maximum angular momentum for the vergence. star to fall into the BH of L˜ /M = 4. We do not cap BH ##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## ##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## ##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## ##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## ##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## ##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: 9 L˜=0, a=0 L˜=3.5MBH, a=0 10-19 MBH=4×103 M∗ 10-19 L˜=0, a=0.99 LISA MBH=2×103 M∗ L˜=2MBH, a=0 MBH=103 M∗ 10-20 10-20 MAdBvH. =LI5G00OM∗ KAGRA 10-21 hc hc 10-21 10-22 10-23 10-22 10-24 10-3 10-2 10-1 100 101 102 f (Hz) f (Hz) Fig. 12.— Characteristic GW strain for m×ain-sequence star–BH Fig. 13.— Characteristic GW strain for head-on white dwarf– collisions with M∗ = 2 M(cid:12) and MBH = 2 106 M(cid:12) assumed to BH collisions with M∗ = M(cid:12) and various values of MBH, assumed be observed with optimal orientation at a distance of d = 15 Mpc. to be observed with optimal orientation at a distance of d = 1 Mpc. For comparison, we also show a proposed LISA noise curve. For comparison, we also show the Advanced LIGO and KAGRA noise curve. × − the choice M∗/R∗ = 2 10 6 for a main-sequence star is appropriate for M∗ ∼ M(cid:12) and more massive stars will lower), while scaling to M∗ = 1 M(cid:12) roughly halves the S/N. be slightly more compact, while less massive stars will be The frequency of the GWs produced by the white less compact, but we ignore that weak dependence here.) dwarf–BH systems considered here would fall at higher In Figure 12, we show the characteristic GW strain h c from the four simulations with M /M∗ = 106, assum- frequencies, outside the range of a LI(cid:46)SA-like instrument. ing M∗ = 2 M(cid:12) and an optimallyBHoriented and located The ringdown of BHs with masses 103 M(cid:12) will oc- cur in the Advanced LIGO sensitivity range; however, source at a distance of 15 Mpc. The characteristic strain is deﬁned as h = |h˜|f where h˜ is the Fourier trans- though those with higher masses will fall in the inter- c mediate regime where neither instrument is sensitive. form of the strain and f is frequency. For comparison, Gravitational wave detectors utilizing atom interferome- we also show a proposed LISA noise curve1 where for a √ ters could potentially ﬁll this frequency gap (Dimopoulos noise power spectral density S (f), we deﬁne the char- (cid:2)(cid:82) (cid:3) n et al. 2008), as could other proposed space-based laser acteristic strain as h = S f. With these deﬁnitions, n n interferometers such as DECIGO (Kawamura et al. 2006) the signal-to-noise of a match ﬁltered signal is given by or BBO (Phinney et al. 2003). In Figure 13, we show the 1/2 S/N= 2 h /h d ln(f) , i.e., is given by integrating characteristic GW strain from the white-dwarf simula- c n the ratio of characteristic strain signal- to-noise over a tions, assuming M∗ = M(cid:12) and observation with optimal logarithmic frequency interval. In Figure 12, we see that orientation at a distance of d = 1 Mpc. For comparison, BHs with spin have an additional signal at high frequen- we also show the proposed broadband Advanced LIGO cies compared to nonspinning BHs, and that signals from noise curve (Shoemaker 2011) and the broadband KA- collisions with more angular momentum are signiﬁcantly GRA noise curve (LCGT Special Working Group 2009). ≈ stronger. For example, for L˜/M = 3.5, S/N 18, We can see that for this particular choice of physical BH while t∼he L˜/MBH = 2 and head-on case have, respec- ma×ss, the str≈ongest signal occurs for MBH/M∗ = 1 to tively, 0.5 and 0.4 the S/N of the head-on case for these 2 103 (S/N 3 for the KAGRA noise curve) since for parameters. Non-head-on collisions with spinning BHs or higher BH mass the frequency of the signal is too low, collisions with somewhat higher L˜ than considered here and for lower masses, tidal eﬀects strongly suppress the (the results of Section suggest L˜/M ≤ 3.75 will not be GW signal. The curves shown are all for head-on col- BH lisions, though, of course, generically the collision will strongly suppressed) could potentially produce stronger occur with angular momentum. As mentioned in Sec- signals. The signal-to-noise is not extremely sensitive to × tion , the geodesic model for the M /M∗ = 2 103 pMh(cid:12)ysiincastlemadasosfinMth∗i=s r2anMge(cid:12). SinccarleinagsetshtehreesSu/ltNs tboyM∼∗7=0%4 mass ratio suggests a collision with L˜B/HMBH ∼ 2 should (this doubles the overall amplitude, but shifts the signal be similarly coherent as the head-on case, thus angular to lower frequencies where the sensitivity is somewhat momentum could boost the detectability by a factor of a few. A collision with a spinning BH would also be 1 We use the LISA noise curve from Larson (accessed June 2014) somewhat easier to detect because of the extra power at that includes both pointing and shot noise, but remove the all-sky high frequencies. and polarization averaging factor as in Berti et al. (2006). We 7. CONCLUSION also note that the proposed eLISA design is less sensitive at lower frequencies than the original LISA design (Amaro-Seoane et al. In this paper, we explored the GWs produced by colli- 2013). sions of strongly tidally aﬀected stars with massive BHs. ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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label: paragraph

font: default

text: M∗

text: LISA

text: M∗

text: M∗

text: MAdBvH.

text: KAGRA

text: hc

text: hc

text: f

text: (Hz)

text: f

text: (Hz)

text: Fig.

text: Characteristic

label: caption

text: GW

text: strain

text: for

text: m×ain-sequence

text: star–BH

text: Fig.

text: Characteristic

text: GW

text: strain

text: for

text: head-on

text: white

text: dwarf–

text: collisions

text: with

text: M∗

text: and

text: MBH

text: assumed

text: to

text: BH

text: collisions

text: with

text: M∗

text: and

text: various

text: values

text: of

text: MBH,

text: assumed

text: be

text: observed

text: with

text: optimal

text: orientation

text: at

text: a

text: distance

text: of

text: d

text: Mpc.

text: to

text: be

text: observed

text: with

text: optimal

text: orientation

text: at

text: a

text: distance

text: of

text: d

text: Mpc.

text: For

text: comparison,

text: we

text: also

text: show

text: a

text: proposed

text: LISA

text: noise

text: curve.

text: For

text: comparison,

text: we

text: also

text: show

text: the

text: Advanced

text: LIGO

text: and

text: KAGRA

text: noise

text: curve.

text: the

text: choice

text: M∗/R∗

text: for

text: a

text: main-sequence

text: star

text: is

text: appropriate

text: for

text: M∗

text: and

text: more

text: massive

text: stars

text: will

text: lower),

text: while

text: scaling

text: to

text: M∗

text: roughly

text: halves

text: the

text: S/N.

text: be

text: slightly

text: more

text: compact,

text: while

text: less

text: massive

text: stars

text: will

text: be

text: The

text: frequency

text: of

text: the

text: GWs

text: produced

text: by

text: the

text: white

text: less

text: compact,

text: but

text: we

text: ignore

text: that

text: weak

text: dependence

text: here.)

text: dwarf–BH

text: systems

text: considered

text: here

text: would

text: fall

text: at

text: higher

text: In

text: Figure

text: we

text: show

text: the

text: characteristic

text: GW

text: strain

text: h

text: c

text: from

text: the

text: four

text: simulations

text: with

text: M

text: /M∗

text: assum-

text: frequencies,

text: outside

text: the

text: range

text: of

text: a

text: instrument.

text: ing

text: M∗

text: and

text: an

text: optimallyBHoriented

text: and

text: located

text: The

text: ringdown

text: of

text: BHs

text: with

text: masses

text: will

text: oc-

text: cur

text: in

text: the

text: Advanced

text: LIGO

text: sensitivity

text: range;

text: however,

text: source

text: at

text: a

text: distance

text: of

text: Mpc.

text: The

text: characteristic

text: strain

text: is

text: deﬁned

text: as

text: h

text: |h˜|f

text: where

text: h˜

text: is

text: the

text: Fourier

text: trans-

text: though

text: those

text: with

text: higher

text: masses

text: will

text: fall

text: in

text: the

text: inter-

text: c

text: mediate

text: regime

text: where

text: neither

text: instrument

text: is

text: sensitive.

text: form

text: of

text: the

text: strain

text: and

text: f

text: is

text: frequency.

text: For

text: comparison,

text: Gravitational

text: wave

text: detectors

text: utilizing

text: atom

text: interferome-

text: we

text: also

text: show

text: a

text: proposed

text: LISA

text: noise

text: where

text: for

text: a

text: ters

text: could

text: potentially

text: ﬁll

text: this

text: frequency

text: gap

text: (Dimopoulos

text: noise

text: power

text: spectral

text: density

text: S

text: (f),

text: we

text: deﬁne

text: the

text: char-

text: n

text: et

text: al.

text: as

text: could

text: other

text: proposed

text: space-based

text: laser

text: acteristic

text: strain

text: as

text: h

text: S

text: f.

text: With

text: these

text: deﬁnitions,

text: n

text: n

text: interferometers

text: such

text: as

text: DECIGO

text: (Kawamura

text: et

text: al.

text: the

text: signal-to-noise

text: of

text: a

text: match

text: ﬁltered

text: signal

text: is

text: given

text: by

text: or

text: BBO

text: (Phinney

text: et

text: al.

text: In

text: Figure

text: we

text: show

text: the

text: S/N=

text: h

text: /h

text: d

text: ln(f)

text: i.e.,

text: is

text: given

text: by

text: integrating

text: characteristic

text: GW

text: strain

text: from

text: the

text: white-dwarf

text: simula-

text: c

text: n

text: the

text: ratio

text: of

text: characteristic

text: strain

text: signal-

text: to-noise

text: over

text: a

text: tions,

text: assuming

text: M∗

text: and

text: observation

text: with

text: optimal

text: logarithmic

text: frequency

text: interval.

text: In

text: Figure

text: we

text: see

text: that

text: orientation

text: at

text: a

text: distance

text: of

text: d

text: Mpc.

text: For

text: comparison,

text: BHs

text: with

text: spin

text: have

text: an

text: additional

text: signal

text: at

text: high

text: frequen-

text: we

text: also

text: show

text: the

text: proposed

text: broadband

text: Advanced

text: LIGO

text: cies

text: compared

text: to

text: nonspinning

text: BHs,

text: and

text: that

text: signals

text: from

text: noise

text: curve

text: (Shoemaker

text: and

text: the

text: broadband

text: KA-

text: collisions

text: with

text: more

text: angular

text: momentum

text: are

text: signiﬁcantly

text: GRA

text: noise

text: curve

text: (LCGT

text: Special

text: Working

text: Group

text: stronger.

text: For

text: example,

text: for

text: L˜/M

text: S/N

text: We

text: can

text: see

text: that

text: for

text: this

text: particular

text: choice

text: of

text: physical

text: BH

text: while

text: t∼he

text: L˜/MBH

text: and

text: head-on

text: case

text: have,

text: respec-

text: ma×ss,

text: the

text: str≈ongest

text: signal

text: occurs

text: for

text: MBH/M∗

text: to

text: tively,

text: and

text: the

text: S/N

text: of

text: the

text: head-on

text: case

text: for

text: these

text: (S/N

text: for

text: the

text: KAGRA

text: noise

text: curve)

text: since

text: for

text: parameters.

text: Non-head-on

text: collisions

text: with

text: spinning

text: BHs

text: or

text: higher

text: BH

text: mass

text: the

text: frequency

text: of

text: the

text: signal

text: is

text: too

text: low,

text: collisions

text: with

text: somewhat

text: higher

text: L˜

text: than

text: considered

text: here

text: and

text: for

text: lower

text: masses,

text: tidal

text: eﬀects

text: strongly

text: suppress

text: the

text: (the

text: results

text: of

text: Section

text: suggest

text: L˜/M

text: will

text: not

text: be

text: GW

text: signal.

text: The

text: curves

text: shown

text: are

text: all

text: for

text: head-on

text: col-

text: BH

text: lisions,

text: though,

text: of

text: course,

text: generically

text: the

text: collision

text: will

text: strongly

text: suppressed)

text: could

text: potentially

text: produce

text: stronger

text: occur

text: with

text: angular

text: momentum.

text: As

text: mentioned

text: in

text: Sec-

text: signals.

text: The

text: signal-to-noise

text: is

text: not

text: extremely

text: sensitive

text: to

text: tion

text: the

text: geodesic

text: model

text: for

text: the

text: M

text: /M∗

text: SinccarleinagsetshtehreesSu/ltNs

text: mass

text: ratio

text: suggests

text: a

text: collision

text: with

text: L˜B/HMBH

text: should

text: (this

text: doubles

text: the

text: overall

text: amplitude,

text: but

text: shifts

text: the

text: signal

text: be

text: similarly

text: coherent

text: as

text: the

text: head-on

text: case,

text: thus

text: angular

text: to

text: lower

text: frequencies

text: where

text: the

text: sensitivity

text: is

text: somewhat

text: momentum

text: could

text: boost

text: the

text: detectability

text: by

text: a

text: factor

text: of

text: a

text: few.

text: A

text: collision

text: with

text: a

text: spinning

text: BH

text: would

text: also

text: be

text: We

text: use

text: the

text: LISA

text: noise

text: curve

text: from

text: Larson

text: (accessed

text: June

text: somewhat

text: easier

text: to

text: detect

text: because

text: of

text: the

text: extra

text: power

text: at

text: that

text: includes

text: both

text: pointing

text: and

text: shot

text: noise,

text: but

text: remove

text: the

text: all-sky

text: high

text: frequencies.

text: and

text: polarization

text: averaging

text: factor

text: as

text: in

text: Berti

text: et

text: al.

text: We

text: CONCLUSION

text: also

text: note

text: that

text: the

text: proposed

text: eLISA

text: design

text: is

text: less

text: sensitive

text: at

text: lower

text: frequencies

text: than

text: the

text: original

text: LISA

text: design

text: (Amaro-Seoane

text: et

text: al.

text: In

text: this

text: paper,

text: we

text: explored

text: the

text: GWs

text: produced

text: by

text: colli-

text: sions

text: of

text: strongly

text: tidally

text: aﬀected

text: stars

text: with

text: massive

text: BHs.

text: ##LTFigure##

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text: 10 Taking into account the dynamics of the tidally per- presses the gravitational radiation. While these sources turbed star collision process and the distribution of the will most likely not have very high rates for near-future star’s mass, we found that nonnegligible GWs are pro- GW detectors, these results do illustrate that such events duced well into the regime where the star is colliding can be direct probes of the BH’s characteristics if they with the BH over a timescale comparable to the period occur suﬃciently close, or if they are targeted by a sub- of the GWs. We also found that collisions with angular sequent generation of detectors. momentum can substantially boost the GW strength, as Though neutron stars have signiﬁcantly higher com- can BH spin. The main features of our results were well- pactions than the cases considered here, these results captured by a simple model based on geodesics. might also be used to inform an understanding of the ≥ For main-sequence stars and BHs with M 106M∗, GW signal produced when a neutron star merges with BH we found that for most of the range of merging values a stellar-mass BH and how tidal eﬀects may suppress of angular momentum, the waveform is still similar to the merger-ringdown signal. For future study, it would that of a point particle. Tidal disruption event rates also be interesting to explore high-angular-momentum per galaxy are estimated to be roughly in the range of collisions with spinning BHs since the point-particle pre- ∼ − − − 10 5–10 3 yr 1 (Wang & Merritt 2004; Gezari et al. diction suggests that spin could enhance the strength 2008; van Velzen et al. 2011; Kesden 2012) and to vary of the GW signal signiﬁcantly (Kojima & Nakamura inversely with BH mass. The results found here suggest 1983a). In upcoming work, we will use these same that collisions occurring in the Virgo cluster, which con- numerical methods to explore noncollisional tidal dis- ∼ tains 103 galaxies, could be visible by a LISA-like GW ruption events (W. E. East & F. Pretorius 2014, in detector for a range of parameters. While event rates in- preparation). volving intermediate-mass BHs are not well-known, our results also suggest that there is a small window of pa- rameter space for a white dwarf collision where the BH is I thank Frans Pretorius and Nick Stone for valuable not so massive that the ringdown signal falls outside the comments on this work. Simulations were run on the frequency range of a ground-based detector, but still not Bullet cluster at SLAC and the Orbital cluster at small enough in mass that decoherence completely sup- Princeton University. REFERENCES Abramovici, A., Althouse, W. E., Drever, R. W., Gursel, Y., Haugan, M. P., Shapiro, S. 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text: East

text: F.

text: Pretorius

text: in

text: detector

text: for

text: a

text: range

text: of

text: parameters.

text: While

text: event

text: rates

text: in-

text: preparation).

text: volving

text: intermediate-mass

text: BHs

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text: rameter

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text: BH

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text: • ± One charged Higgs boson H . where the neutral sectors are admixtures of doublet and singlet Higgs ﬁelds. × M The 2 2 mass matrix for the CP-odd Higgs bosons 2 has in the basis (A , S ) the P MSSM I elements M 2 µ (A + κs) 2 = eﬀ λ , P,11 sin 2β M − 2 = λ(A /s + 4κ)v v 3κA s , P,22 λ u d κ M − (cid:112) 2 = λ(A 2κs) v (4) P,12 λ where v , v denote the vevs of H , H , respectively, v = v2 + v2 and, as usual, tan β = u d u d u d M v /v . The entry 2 would resemble the mass of the MSSM-like CP-odd scalar. Note, u d P,11 that the singlet like-state can be relatively light. We order the mass eigenstates according ≤ to m m and apply this also to the scalar sector. A1 A2 In the CP-even sector, three states H (i = 1, 2, 3) are admixtures of the real components i H , H and S. The state H , which could be either H or H , with the nearly SM-like u d SM 1 2 coupling to the electroweak gauge bosons has a mass [26, 27] m2 = M2 cos2 2β + λ2v2 sin2 2β + radcorrs + ∆ . (5) HSM Z mix The term ∆ represents the singlet-doublet mixing mix − (cid:39) 4λ2s2v2(λ κ sin 2β)2 ∆mix 2 − (6) M M2 HSM S 2 where M is m2 without the mixing term and M2 is the mass of singlet like Higgs HSM HSM S boson. In scenarios as considered here where all Higgs states are light enough, one can still have signiﬁcant mixing among all these states. If the singlet like state M2 is lighter/heavier S 2 than M , then ∆ can even produce signiﬁcant positive/negative contributions to the HSM mix lightest Higgs state. (cid:18) (cid:19) The mass of the charged Higgs scalar is given by g2 − M2 ± = M2 + v2 2 λ2 . (7) H A 2 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: list

text: One

text: charged

text: Higgs

text: boson

text: H

text: where

text: the

text: neutral

text: sectors

text: are

text: admixtures

text: of

text: doublet

text: and

text: singlet

text: Higgs

text: ﬁelds.

text: M

text: The

text: mass

text: matrix

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text: the

text: CP-odd

text: Higgs

text: bosons

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text: in

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text: basis

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text: S

text: the

text: P

text: MSSM

text: I

text: elements

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text: κs)

text: eﬀ

text: λ

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text: 2β

text: M

text: λ(A

text: /s

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text: λ

text: u

text: d

text: κ

text: M

text: λ(A

text: v

text: λ

text: where

text: v

text: v

text: denote

text: the

text: vevs

text: of

text: H

text: H

text: respectively,

text: v

text: and,

text: as

text: usual,

text: tan

text: β

text: u

text: d

text: u

text: d

text: u

text: d

text: M

text: v

text: /v

text: The

text: entry

text: would

text: resemble

text: the

text: mass

text: of

text: the

text: MSSM-like

text: CP-odd

text: scalar.

text: Note,

text: u

text: d

text: that

text: the

text: singlet

text: like-state

text: can

text: be

text: relatively

text: light.

text: We

text: order

text: the

text: mass

text: eigenstates

text: according

text: to

text: m

text: m

text: and

text: apply

text: this

text: also

text: to

text: the

text: scalar

text: sector.

text: In

text: the

text: CP-even

text: sector,

text: three

text: states

text: H

text: (i

text: are

text: admixtures

text: of

text: the

text: real

text: components

text: i

text: H

text: H

text: and

text: S.

text: The

text: state

text: H

text: which

text: could

text: be

text: either

text: H

text: or

text: H

text: with

text: the

text: nearly

text: SM-like

text: u

text: d

text: SM

text: coupling

text: to

text: the

text: electroweak

text: gauge

text: bosons

text: has

text: a

text: mass

text: 2β

text: 2β

text: radcorrs

text: HSM

text: Z

text: mix

text: The

text: term

text: represents

text: the

text: singlet-doublet

text: mixing

text: mix

text: κ

text: sin

text: ∆mix

text: M

text: HSM

text: S

text: where

text: M

text: is

text: without

text: the

text: mixing

text: term

text: and

text: is

text: the

text: mass

text: of

text: singlet

text: like

text: Higgs

text: HSM

text: HSM

text: S

text: boson.

text: In

text: scenarios

text: as

text: considered

text: here

text: where

text: all

text: Higgs

text: states

text: are

text: light

text: enough,

text: one

text: can

text: still

text: have

text: signiﬁcant

text: mixing

text: among

text: all

text: these

text: states.

text: If

text: the

text: singlet

text: like

text: state

text: is

text: lighter/heavier

text: S

text: than

text: M

text: then

text: can

text: even

text: produce

text: signiﬁcant

text: positive/negative

text: contributions

text: to

text: the

text: HSM

text: mix

text: lightest

text: Higgs

text: state.

text: The

text: mass

text: of

text: the

text: charged

text: Higgs

text: scalar

text: is

text: given

text: by

text: λ2

text: H

text: A

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: parameter BMP-A BMP-B Branching ratios BMP-A BMP-B → × − × − tan β 1.68 1.45 Br(t bH+) 3.3 10 3 1.8 10 2 → κ 0.596 0.94 Br(H+ W +H ) 0.68 0 1 → λ 0.596 0.62 Br(H+ W +A ) 0 0.86 1 → × − × − µ 131.5 143.7 Br(H γγ) 6.0 10 3 1.76 10 3 eff 1 → × − × − m 73.8 126.8 Br(A γγ) 2.6 10 5 1.0 10 4 H0 1 1 → m 126.7 172.0 Br(H b¯b) 0.24 0.72 H0 1 2 → m 192.6 364.1 Br(A b¯b) 0.11 0.86 H0 1 3 → m 155.4 66.6 Br(H cc¯) 0.32 0 A0 1 1 → m 428.4 402.0 Br(A cc¯) 0 0 A0 1 2 → m ± 161.9 149.4 Br(H g g) 0.4 0.02 H 1 → m 645.4 656.2 Br(A g g) 0 0.02 t˜1 1 m 883.5 879.0 ˜b1 TABLE I. Relevant masses and branching fractions of the benchmark points used for our simulation. All masses are in GeV. | | | | | | p (j) p (γ) p ((cid:96)) η (j) η (γ) η ((cid:96)) ∆R Tmin Tmin Tmin max max max 20 GeV 10 GeV 10 GeV 5 2.5 2.5 0.4 TABLE II. The default cut setup in MadGraph 5.1.5.13 where p is the transverse momentum, η T the pseudorapidity and ∆R for the angular distance between two objects comprised of leptons, jets and photons. × → × → × → − σΦ = 2 σ(pp tt¯) Br(t bH+) Br(t¯ ¯bW ) 2b+2W+2γ × → × → Br(H+ W +Φ) Br(Φ γγ) (8) with Φ = H , A . For a qualititative understanding we calculate the eﬀective signal events, 1 1 S ≡ × L L as deﬁned by H1(A1) σH1(A1) , where represents the integrated luminosity for 2b+2W+2γ 8 ##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## ##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##

text: parameter

label: table

text: BMP-A

text: BMP-B

text: Branching

text: ratios

text: BMP-A

text: BMP-B

text: tan

text: β

text: Br(t

text: bH+)

text: κ

text: Br(H+

text: W

text: +H

text: λ

text: Br(H+

text: W

text: +A

text: µ

text: Br(H

text: γγ)

text: eff

text: m

text: Br(A

text: γγ)

text: m

text: Br(H

text: b¯b)

text: m

text: Br(A

text: b¯b)

text: m

text: Br(H

text: cc¯)

text: m

text: Br(A

text: cc¯)

text: m

text: Br(H

text: g

text: g)

text: H

text: m

text: Br(A

text: g

text: g)

text: m

text: TABLE

text: I.

text: Relevant

text: masses

text: and

text: branching

text: fractions

text: of

text: the

text: benchmark

text: points

text: used

text: for

text: our

text: simulation.

text: All

text: masses

text: are

text: in

text: GeV.

text: p

text: (j)

text: p

text: (γ)

text: p

text: η

text: (j)

text: η

text: (γ)

text: η

text: ∆R

text: Tmin

text: Tmin

text: Tmin

text: max

text: max

text: max

text: GeV

text: GeV

text: GeV

text: TABLE

text: II.

text: The

text: default

text: cut

text: setup

text: in

text: MadGraph

text: where

text: p

text: is

text: the

text: transverse

text: momentum,

text: η

text: T

text: the

text: pseudorapidity

text: and

text: ∆R

text: for

text: the

text: angular

text: distance

text: between

text: two

text: objects

text: comprised

text: of

text: leptons,

text: jets

text: and

text: photons.

text: σΦ

text: σ(pp

text: tt¯)

text: Br(t

text: bH+)

text: Br(t¯

text: ¯bW

text: Br(H+

text: W

text: +Φ)

text: Br(Φ

text: γγ)

text: with

text: Φ

text: H

text: A

text: For

text: a

text: qualititative

text: understanding

text: we

text: calculate

text: the

text: eﬀective

text: signal

text: events,

text: S

text: L

text: L

text: as

text: deﬁned

text: by

text: where

text: represents

text: the

text: integrated

text: luminosity

text: for

text: ##LTLine##

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text: 14 PERIODIC WAVES FOR THE LOG-KDV EQUATION Since the solitary-wave solution (1.3) satisﬁes (cid:48) −(φ )2 ω + 1 − 1 + φ2 φ2 log φ2 = 0, 2 2 2 E (cid:48) E we deduce that (φ, φ ) = (0, 0) = 0. This means, we can take the closed curve Γ, in assumption (H2), to be the orbit of the soliton (1.3) together with the equilibrium point (r , 0) = (0, 0) (see Figure 4.1). Since the origin belongs to Γ, it is clear that g is smooth 1 (cid:48) − ∈ R ω in the region inside Γ and g (r ) = 2 < 0, for all ω . As a conclusion, g satisﬁes ω P2 R ω assumptions (H1)-(H3) with = . Figure 4.1. Left: Graphs of the functions g = g and G = G for ω = 1. − (cid:48)(cid:48) − ω ω Right: Phase space of the equation φ + ωφ φ log(φ2) = 0. The orbits in blue are those for which φ is periodic and does not change sign. (cid:112) Theorem 3.1 and Corollary 3.4 can now be applied to prove the existence of positive ∞ √ L-periodic solutions, where the period L ranges over the interval (α, + ), with α = − (cid:48) 2π/ g (r ) = π 2. More precisely, we have. ω 2 √ ∈ ∞ ∈ R Proposition 4.1. Let L ( 2π, + ) be ﬁxed. Then, for any ω , equation − (cid:48)(cid:48) − φ + ωφ φ log φ2 = 0 (4.2) possesses an L-periodic solution, which is even and strictly positive. (cid:48) The initial conditions (φ(0), φ (0)) = (φ(0), 0) that give rise to positive periodic solutions are in the range r = eω/2 < φ(0) < e(ω+1)/2, where the constant e(ω+1)/2 is the intersection 2 of Γ with the axis φ (in the phase space). ∈ R (cid:55)→ Note that since the L-periodic solutions are strictly positive the function x (cid:48) − − g (φ(x)) = ω 2 log φ2(x) is of class C1. Hence, assumption (H4) also holds and we ω can apply all the results in Section . In particular, to construct a smooth family of L- periodic solution, one can use Theorem 3.5. However, with the equation we have in hand, we observe if φ is an L-periodic solution of 0 − (cid:48)(cid:48) φ + φ log φ2 = 0, (4.3) 0 0 0 ∈ R then ψ = eω/2φ is an L-periodic solution of (4.2) for any ω . As a consequence, ω 0 we have proved the following. ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: PERIODIC

text: WAVES

text: FOR

text: THE

text: LOG-KDV

text: EQUATION

text: Since

text: the

text: solitary-wave

text: solution

text: satisﬁes

text: −(φ

text: ω

text: φ2

text: φ2

text: log

text: φ2

text: E

text: E

text: we

text: deduce

text: that

text: (φ,

text: φ

text: This

text: means,

text: we

text: can

text: take

text: the

text: closed

text: curve

text: Γ,

text: in

text: assumption

text: to

text: be

text: the

text: orbit

text: of

text: the

text: soliton

text: together

text: with

text: the

text: equilibrium

text: point

text: (r

text: (see

text: Figure

text: Since

text: the

text: origin

text: belongs

text: to

text: Γ,

text: it

text: is

text: clear

text: that

text: g

text: is

text: smooth

text: R

text: ω

text: in

text: the

text: region

text: inside

text: Γ

text: and

text: g

text: (r

text: for

text: all

text: ω

text: As

text: a

text: conclusion,

text: g

text: satisﬁes

text: ω

text: R

text: ω

text: assumptions

text: with

text: Figure

text: Left:

text: Graphs

text: of

text: the

text: functions

text: g

text: g

text: and

text: G

text: G

text: for

text: ω

text: ω

text: ω

text: Right:

text: Phase

text: space

text: of

text: the

text: equation

text: φ

text: ωφ

text: φ

text: The

text: orbits

text: in

text: blue

text: are

text: those

text: for

text: which

text: φ

text: is

text: periodic

text: and

text: does

text: not

text: change

text: sign.

text: Theorem

text: and

text: Corollary

text: can

text: now

text: be

text: applied

text: to

text: prove

text: the

text: existence

text: of

text: positive

text: L-periodic

text: solutions,

text: where

text: the

text: period

text: L

text: ranges

text: over

text: the

text: interval

text: (α,

text: with

text: α

text: g

text: (r

text: π

text: More

text: precisely,

text: we

text: have.

text: ω

text: R

text: Proposition

text: Let

text: L

text: be

text: ﬁxed.

text: Then,

text: for

text: any

text: ω

text: equation

text: φ

text: ωφ

text: φ

text: log

text: φ2

text: possesses

text: an

text: L-periodic

text: solution,

text: which

text: is

text: even

text: and

text: strictly

text: positive.

text: The

text: initial

text: conditions

text: φ

text: that

text: give

text: rise

text: to

text: positive

text: periodic

text: solutions

text: are

text: in

text: the

text: range

text: r

text: where

text: the

text: constant

text: is

text: the

text: intersection

text: of

text: Γ

text: with

text: the

text: axis

text: φ

text: (in

text: the

text: phase

text: space).

text: R

text: Note

text: that

text: since

text: the

text: L-periodic

text: solutions

text: are

text: strictly

text: positive

text: the

text: function

text: x

text: g

text: (φ(x))

text: ω

text: log

text: is

text: of

text: class

text: Hence,

text: assumption

text: also

text: holds

text: and

text: we

text: ω

text: can

text: apply

text: all

text: the

text: results

text: in

text: Section

text: In

text: particular,

text: to

text: construct

text: a

text: smooth

text: family

text: of

text: L-

text: periodic

text: solution,

text: one

text: can

text: use

text: Theorem

text: However,

text: with

text: the

text: equation

text: we

text: have

text: in

text: hand,

text: we

text: observe

text: if

text: φ

text: is

text: an

text: L-periodic

text: solution

text: of

text: φ

text: φ

text: log

text: φ2

text: R

text: then

text: ψ

text: is

text: an

text: L-periodic

text: solution

text: of

text: for

text: any

text: ω

text: As

text: a

text: consequence,

text: ω

text: we

text: have

text: proved

text: the

text: following.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: PERIODIC WAVES FOR THE LOG-KDV EQUATION 15 √ ∈ ∞ Proposition 4.2. Let L ( 2π, + ) be ﬁxed. Let φ be the positive L-periodic solution 0 obtained in Proposition 4.1 with ω = 0. Then, ∈ R (cid:55)→ ∈ ω ψ = eω/2φ H2 ([0, L]) ω 0 per is a smooth family of positive L-periodic solutions for (4.2). − It is easily seen that if φ is a solution of (4.2), so is φ. Thus, in view of Proposition 4.2 we may also obtain a smooth curve of negative L-periodic solutions. More precisely. √ ∈ ∞ Proposition 4.3. Let L ( 2π, + ) be ﬁxed. Let φ be the positive L-periodic solution 0 obtained in Proposition 4.1 with ω = 0. Then, ∈ R (cid:55)→ − ∈ ω ψ = eω/2φ H2 ([0, L]) ω 0 per is a smooth family of negative L-periodic solutions for (4.2). √ Attention is now turned to the orbital stability of the periodic traveling waves in Propo- ∈ ∞ sitions 4.2 and 4.3. So, in what follows in this section, we ﬁx L ( 2π, + ) and let ψ , ∈ R ω ω , be either a positive solution as in Proposition 4.2 or a negative solution as in Proposition 4.3. Recall that the quantities E and F , deﬁned in (1.6) and (1.7), are con- served by the ﬂow of (1.1) and are invariant under the action of the group of translations · · ∈ R T (s)f( ) = f( + s), s . Note also that functional E is not smooth at the origin on H1 ([0, L]). Nevertheless, the arguments above imply that ψ2 is strictly positive, for any p∈erR ω ω , guaranteeing thus the smoothness of the functional E around any periodic trav- eling wave ψ . This is enough to apply the abstract theory in [20], because the orbital ω stability is determined for initial data suﬃciently close to ψ . ω The space we shall be working with is the Hilbert space X := H1 ([0, L]). Before per going into details, let us recall that periodic traveling-wave solutions are of the form − ∈ R u(x, t) = φ(x ωt), ω , where φ is a solution of (1.2). Now, we present the deﬁnition of orbital stability Deﬁnition 4.4. We say that an L-periodic solution φ is orbitally stable in X, by the ∈ periodic ﬂow of (1.1), if for any ε > 0 there exists δ > 0 such that for any u X (cid:107) − (cid:107) 0 satisfying u φ < δ, the solution of (1.1) with initial data u exists globally and 0 X 0 satisﬁes (cid:107) · − · (cid:107) inf u( , t) φ( + y) < ε, ∈R X y ∈ R for all t . Roughly speaking, we say that φ is orbitally stable if for any initial data close enough to φ, the corresponding solution remains close enough to the orbit of φ generated by translations, { · ∈ R} O := φ( + y); y . (4.4) φ Deﬁne the functional H = E + ωF . Thus, in a neighborhood of ψ , H is smooth. This ω allows us to calculate the Fr´echet derivative of H at ψ to deduce, from (1.2), that ψ is ω ω a critical point of H, that is, (cid:48) (cid:48) − (cid:48)(cid:48) − H (ψ ) = (E + ωF ) (ψ ) = ψ + ωψ ψ log ψ2 = 0. ω ω ω ω ω ω ##LTLine## ##LTLine## ##LTLine##

text: PERIODIC

text: WAVES

text: FOR

text: THE

text: LOG-KDV

text: EQUATION

text: Proposition

text: Let

text: L

text: be

text: ﬁxed.

text: Let

text: φ

text: be

text: the

text: positive

text: L-periodic

text: solution

text: obtained

text: in

text: Proposition

text: with

text: ω

text: Then,

text: R

text: ω

text: ψ

text: L])

text: ω

text: per

text: is

text: a

text: smooth

text: family

text: of

text: positive

text: L-periodic

text: solutions

text: for

text: It

text: is

text: easily

text: seen

text: that

text: if

text: φ

text: is

text: a

text: solution

text: of

text: so

text: is

text: φ.

text: Thus,

text: in

text: view

text: of

text: Proposition

text: we

text: may

text: also

text: obtain

text: a

text: smooth

text: curve

text: of

text: negative

text: L-periodic

text: solutions.

text: More

text: precisely.

text: Proposition

text: Let

text: L

text: be

text: ﬁxed.

text: Let

text: φ

text: be

text: the

text: positive

text: L-periodic

text: solution

text: obtained

text: in

text: Proposition

text: with

text: ω

text: Then,

text: R

text: ω

text: ψ

text: L])

text: ω

text: per

text: is

text: a

text: smooth

text: family

text: of

text: negative

text: L-periodic

text: solutions

text: for

text: Attention

text: is

text: now

text: turned

text: to

text: the

text: orbital

text: stability

text: of

text: the

text: periodic

text: traveling

text: waves

text: in

text: Propo-

text: sitions

text: and

text: So,

text: in

text: what

text: follows

text: in

text: this

text: section,

text: we

text: ﬁx

text: L

text: and

text: let

text: ψ

text: R

text: ω

text: ω

text: be

text: either

text: a

text: positive

text: solution

text: as

text: in

text: Proposition

text: or

text: a

text: negative

text: solution

text: as

text: in

text: Proposition

text: Recall

text: that

text: the

text: quantities

text: E

text: and

text: F

text: deﬁned

text: in

text: and

text: are

text: con-

text: served

text: by

text: the

text: ﬂow

text: of

text: and

text: are

text: invariant

text: under

text: the

text: action

text: of

text: the

text: group

text: of

text: translations

text: R

text: T

text: (s)f(

text: f(

text: s),

text: s

text: Note

text: also

text: that

text: functional

text: E

text: is

text: not

text: smooth

text: at

text: the

text: origin

text: on

text: L]).

text: Nevertheless,

text: the

text: arguments

text: above

text: imply

text: that

text: ψ2

text: is

text: strictly

text: positive,

text: for

text: any

text: p∈erR

text: ω

text: ω

text: guaranteeing

text: thus

text: the

text: smoothness

text: of

text: the

text: functional

text: E

text: around

text: any

text: periodic

text: trav-

text: eling

text: wave

text: ψ

text: This

text: is

text: enough

text: to

text: apply

text: the

text: abstract

text: theory

text: in

text: because

text: the

text: orbital

text: ω

text: stability

text: is

text: determined

text: for

text: initial

text: data

text: suﬃciently

text: close

text: to

text: ψ

text: ω

text: The

text: space

text: we

text: shall

text: be

text: working

text: with

text: is

text: the

text: Hilbert

text: space

text: X

text: L]).

text: Before

text: per

text: going

text: into

text: details,

text: let

text: us

text: recall

text: that

text: periodic

text: traveling-wave

text: solutions

text: are

text: of

text: the

text: form

text: R

text: u(x,

text: t)

text: φ(x

text: ωt),

text: ω

text: where

text: φ

text: is

text: a

text: solution

text: of

text: Now,

text: we

text: present

text: the

text: deﬁnition

text: of

text: orbital

text: stability

text: Deﬁnition

text: We

text: say

text: that

text: an

text: L-periodic

text: solution

text: φ

text: is

text: orbitally

text: stable

text: in

text: X,

text: by

text: the

text: periodic

text: ﬂow

text: of

text: if

text: for

text: any

text: ε

text: there

text: exists

text: δ

text: such

text: that

text: for

text: any

text: u

text: X

text: satisfying

text: u

text: φ

text: δ,

text: the

text: solution

text: of

text: with

text: initial

text: data

text: u

text: exists

text: globally

text: and

text: X

text: satisﬁes

text: inf

text: u(

text: t)

text: φ(

text: y)

text: ε,

text: ∈R

text: X

text: y

text: R

text: for

text: all

text: t

text: Roughly

text: speaking,

text: we

text: say

text: that

text: φ

text: is

text: orbitally

text: stable

text: if

text: for

text: any

text: initial

text: data

text: close

text: enough

text: to

text: φ,

text: the

text: corresponding

text: solution

text: remains

text: close

text: enough

text: to

text: the

text: orbit

text: of

text: φ

text: generated

text: by

text: translations,

text: R}

text: O

text: φ(

text: y);

text: y

text: φ

text: Deﬁne

text: the

text: functional

text: H

text: E

text: ωF

text: Thus,

text: in

text: a

text: neighborhood

text: of

text: ψ

text: H

text: is

text: smooth.

text: This

text: ω

text: allows

text: us

text: to

text: calculate

text: the

text: Fr´echet

text: derivative

text: of

text: H

text: at

text: ψ

text: to

text: deduce,

text: from

text: that

text: ψ

text: is

text: ω

text: ω

text: a

text: critical

text: point

text: of

text: H,

text: that

text: is,

text: H

text: (ψ

text: (E

text: ωF

text: (ψ

text: ψ

text: ωψ

text: ψ

text: log

text: ψ2

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##