Nonlinear Waves, Solitons and Chaos

Infeld, E.; Rowlands, G.

doi:10.1017/cbo9781139171281

Cited by 626 publications

(673 citation statements)

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“…It has been reported also in several systems, but the four virtual soliton resonance does not seem to have been done for KP-II [6] prior to our work. Recently we realized that resonance solitons for KP-II have been constructed independently also by Biondini and Kodama [2,8] using Sato's theory.…”

Section: Resonance Interaction Of Planar Solitonsmentioning

confidence: 82%

Dissipative hierarchies and resonance solitons for KP-II and MKP-II

Francisco

Lee

Pashaev

2007

Mathematics and Computers in Simulation

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We show that dissipative solitons (dissipatons) of the second and the third members of SL(2,R) AKNS hierarchy give rise to the real solitons of KP-II, while for SL(2,R) Kaup-Newell hierarchy they give solitons of MKP-II. By the Hirota bilinear form for both flows, we find new bilinear system for these equations, and one-and two-soliton solutions. For special values of parameters our solutions show resonance behaviour with creation of four virtual solitons. We first time created four virtual soliton resonance solution for KP-II and established relations of it with degenerate four-soliton solution in the Hirota-Satsuma bilinear form for KP-II. Our approach allows one to interpret the resonance soliton as a composite object of two dissipative solitons in 1 + 1 dimensions.

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Section: Resonance Interaction Of Planar Solitonsmentioning

confidence: 82%

Dissipative hierarchies and resonance solitons for KP-II and MKP-II

Francisco

Lee

Pashaev

2007

Mathematics and Computers in Simulation

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“…In that last case, there will be no wavefront dislocation and the Chu-Mei quotient will be finite. A combination 1 Some authors call this quotient the 'Fornberg-Whitham term' [27], referring to [28]. However, throughout this Letter we call it the 'Chu-Mei quotient', since they introduced it for the first time [29,30] when they derived the modulation equations of Whitham's theory [31] for slowly varying Stokes waves.…”

Section: Basic Notionsmentioning

confidence: 99%

Note on wavefront dislocation in surface water waves

Karjanto¹,

Groesen²

2007

Physics Letters A

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At singular points of a wave field, where the amplitude vanishes, the phase may become singular and wavefront dislocation may occur. In this Letter we investigate for wave fields in one spatial dimension the appearance of these essentially linear phenomena. We introduce the Chu-Mei quotient as it is known to appear in the 'nonlinear dispersion relation' for wave groups as a consequence of the nonlinear transformation of the complex amplitude to real phase-amplitude variables. We show that unboundedness of this quotient at a singular point, related to unboundedness of the local wavenumber and frequency, is a generic property and that it is necessary for the occurrence of phase singularity and wavefront dislocation, while these phenomena are generic too. We also show that the 'soliton on finite background', an explicit solution of the NLS equation and a model for modulational instability leading to extreme waves, possesses wavefront dislocations and unboundedness of the Chu-Mei quotient.

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“…Soliton solutions are relevant to nonlinear models in various areas of physics which deal with the propagation of intensive wave fields in dispersive media: optical pulses and beams in fibers and spatial waveguides, electromagnetic waves in plasma, surface waves on deep water, etc. [1][2][3][4][5][6][7]. Recently, solitons have also drawn a great deal of interest in plasmonics [8][9][10].…”

Section: Introductionmentioning

confidence: 99%

“…The dynamics of narrow HF wave packets is described by the third-order nonlinear dispersive wave theory [1], which takes into account the nonlinear dispersion (self-steeping) [14], stimulated Raman scattering (SRS) [15][16][17] and third-order dispersion (TOD). The basic equation of the theory is the extended NLSE [17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Damped solitons in an extended nonlinear Schrödinger equation with a spatial stimulated Raman scattering and decreasing dispersion

Gromov

Malomed

2014

Optics Communications

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Dynamics of solitons is considered in the framework of an extended nonlinear Schrödinger equation (NLSE), which is derived from a system of the Zakharov's type for the interaction between high-and low-frequency (HF and LF) waves. The resulting NLSE includes a pseudo-stimulatedRaman-scattering (pseudo-SRS) term, i.e., a spatial-domain counterpart of the SRS term, which is a known ingredient of the temporal-domain NLSE in optics. Also included is inhomogeneity of the spatial second-order dispersion (SOD) and linear losses of HF waves. It is shown that wavenumber downshift by the pseudo-SRS may be compensated by upshift provided by SOD whose local strength is an exponentially decaying function of the coordinate. An analytical soliton solution with a permanent shape is found in an approximate form, and is verified by comparison with numerical results. Keywords: Extended Nonlinear Schrödinger Equation, Damped Soliton Solution, StimulatedScattering, Damping Low-Frequency Waves, Linear Loss High-Frequency Waves, Inhomogeneity Highlights >Dynamics of damped solitons is studied in an extended inhomogeneous nonlinear Schrödinger equation. >We consider media with stimulated-scattering damping acting on low-frequency waves and inhomogeneous spatial dispersion. >An approximate analytical solution for damped solitons is found in an explicit form.

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Nonlinear Waves, Solitons and Chaos

Cited by 626 publications

References 0 publications

Dissipative hierarchies and resonance solitons for KP-II and MKP-II

Dissipative hierarchies and resonance solitons for KP-II and MKP-II

Note on wavefront dislocation in surface water waves

Damped solitons in an extended nonlinear Schrödinger equation with a spatial stimulated Raman scattering and decreasing dispersion

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