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Adams, K. L.; King, John R.; Tew, R. H.

doi:10.1023/a:1022600411856

Cited by 33 publications

(13 citation statements)

References 21 publications

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“…We note that beyond-all-order techniques have been applied in a few instances to multiple-scale problems: gravity-capillary solitary waves [12], oscillating shock solutions of the Kuramoto-Sivashinsky equation [13,14], and strut buckling on an elastic foundation [15]. The last analysis is close in spirit to the present one but breaks down near the Maxwell point.…”

supporting

confidence: 52%

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Asymptotics of Large Bound States of Localized Structures

Kozyreff

Chapman²

2006

Phys. Rev. Lett.

110

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We analyze stationary fronts connecting uniform and periodic states emerging from a pattern-forming instability. The size of the resulting periodic domains cannot be predicted with weakly nonlinear methods.We show that what determine this size are exponentially small (but exponentially growing in space) terms. These can only be computed by going beyond all orders of the usual multiple-scale expansion. We apply the method to the Swift-Hohenberg equation and derive analytically a snaking bifurcation curve. At each fold of this bifurcation curve, a new pair of peaks is added to the periodic domain, which can thus be seen as a bound state of localized structures. Such scenarios have been reported with optical localized structures in nonlinear cavities and localized buckling.

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supporting

confidence: 52%

“…With this parametrization, X is on the Stokes line when 0. The large-n terms of ff n;k g are then found to produce (11) in rhs N , and the key observation [14] is that, at optimal truncation, where N=2 1 r=" 2 , the fraction above is asymptotic to…”

Section: 044502 (2006) P H Y S I C a L R E V I E W L E T T E R S mentioning

confidence: 99%

Asymptotics of Large Bound States of Localized Structures

Kozyreff

Chapman²

2006

Phys. Rev. Lett.

110

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“…In Appendix A, we show how the above equation naturally arises in electromagnetic, elastic, and plasma wave propagation. The fifth-order KdV equation, allowing waves only in one direction, is in fact not included in (14). The agreement with the calculations of Calvo, Yang, and Akylas on that equation support the claim that the results presented here are general and not tied to a particular wave model.…”

Section: A General Wave Modelsupporting

confidence: 78%

“…with ω 2 (k) = (T k 2 + Bk 4 + f 1 )/ρ. This is of the same form as (14), except for the exchange of coordinate x and t and the replacement β(ω) → ω(k). A minimum of v p is obtained at the wave number k 0 = ( f 1 /B) 1/4 .…”

Section: Appendix A: Derivation Of the Wavementioning

confidence: 99%

“…To compensate for this correction, a second exponentially growing term is born that is proportional to ẍ0 so as to cancel the effect of the former. This summarises in a few words the technique of beyond-all-orders asymptotics, which has been applied to multiple-scale problems in only a few instances [8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Among these works, the one by Yang and Akylas [8] stands as particularly relevant.…”

Section: Introductionmentioning

confidence: 99%

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Speed of wave packets and the nonlinear Schrödinger equation

Kozyreff¹

2023

Phys. Rev. E

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The universal theory of weakly nonlinear wave packets given by the nonlinear Schrödinger equation is revisited. In the limit where the group and phase velocities are very close together, a multiple-scale analysis carried out beyond all orders reveals that a single soliton, bright or dark, can travel at a different speed than the group velocity. In an exponentially small but finite range of parameters, the envelope of the soliton is locked to the rapid oscillations of the carrier wave. Eventually, the dynamics is governed by an equation analogous to that of a pendulum, in which the center of mass of the soliton is subjected to a periodic potential. Consequently, the soliton speed is not constant and generally contains a periodic component. Furthermore, the interaction between two distant solitons can in principle be profoundly altered by the aforementioned effective periodic potential and we conjecture the existence of new bound states. These results are derived on a wide class of wave models and in such a general way that they are believed to be of universal validity.

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Study for the Sea-Air Oscillator Model of Decadal Variations in Subtropical Cells and the Sea Surface Temperature (SST) of Equatorial Pacific

Jia-Qi

Lin

Wang

2006

Chinese J. Geophys.

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A time delay equation for the sea‐air oscillator model is studied. The aim is to create an asymptotic solving method of nonlinear equation for the sea‐air oscillator model. Employing the method of homotopic mapping, the approximation solution of corresponding problem is studied. This homotopic method is an approximate analytic method, which can be used for further analyzing other behavior of the sea surface temperature anomaly of the atmosphere‐ocean oscillator model.

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Cited by 33 publications

References 21 publications

Asymptotics of Large Bound States of Localized Structures

Asymptotics of Large Bound States of Localized Structures

Speed of wave packets and the nonlinear Schrödinger equation

Study for the Sea-Air Oscillator Model of Decadal Variations in Subtropical Cells and the Sea Surface Temperature (SST) of Equatorial Pacific

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