Interactions and motions of double-solitons with opposite polarity in a parametrically driven system

Wang, Xinlong; Wei, Rongjue

doi:10.1016/s0375-9601(97)00022-4

Cited by 13 publications

(1 citation statement)

References 12 publications

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“…It describes the dynamics of small-amplitude breathers of the easy-plane ferromagnet and the long Josephson junction (driven, damped sine-Gordon equation) [11]. Moreover it has been used to model the nonlinear Faraday resonance in a vertically oscillating uid layer [12][13][14][15][16][17][18][19][20][21] and the effect of phasesensitive parametric ampli ers on solitons in optical bres [22][23][24][25]. It also serves as a continuum limit for small-amplitude excitations in the parametrically driven Frenkel-Kontorova system [26,27].…”

Section: Introductionmentioning

confidence: 99%

Empirical stability criteria for parametrically driven solitons of the nonlinear Schrödinger equation

Mertens¹,

Quintero²

2020

J. Phys. A: Math. Theor.

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For the parametrically driven nonlinear Schrödinger equation, two criteria for the dynamical stability of solitons are presented, using the driving force f(x, t) = a exp[2 i K(t) x]. Both criteria are based on the results of a collective coordinate theory. In this theory, the parameters of the 1-soliton solution are allowed to depend on time and a variational approach yields a set of coupled nonlinear ODEs for the collective coordinates. The solutions of these ODEs are used to compute the soliton's normalized momentum p(t), velocity v(t), and a parametric plot p(v). The rst criterion states that a suf cient condition for instability is ful lled if the curve p(v), or a piece of it, has a negative slope. If the slope is positive then a necessary condition for stability is ful lled. For constant K, a second criterion is presented. Here a complex wavefunction is calculated and a parametric plot of its imaginary part vs its real part is made. This yields a closed orbit in the complex plane. If this orbit has a negative sense of rotation with respect to a xed point, then a suf cient condition for instability is ful lled, whereas a positive sense is a necessary condition for stability. The ensemble of orbits and xed points is a so-called phase portrait, which is very helpful for the classi cation of the different types of solutions of the ODEs. Finally, the validity of the two criteria is tested by comparing with simulations for the driven nonlinear Schrödinger equation. Both criteria make correct predictions about the stability and instability of the solitons, with the exception of a few cases at the border between stability and instability regions.

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Section: Introductionmentioning

confidence: 99%

Empirical stability criteria for parametrically driven solitons of the nonlinear Schrödinger equation

Mertens¹,

Quintero²

2020

J. Phys. A: Math. Theor.

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Soliton–antisoliton interaction in a parametrically driven easy-plane magnetic wire

2014

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Soliton–radiation coupling in the parametrically driven, damped nonlinear Schrödinger equation

Shchesnovich

Barashenkov

2002

Physica D: Nonlinear Phenomena

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We use the Riemann-Hilbert problem to study the interaction of the soliton with radiation in the parametrically driven, damped nonlinear Schrödinger equation. The analysis is reduced to the study of a finite-dimensional dynamical system for the amplitude and phase of the soliton and the complex amplitude of the long-wavelength radiation. In contrast to previously utilised Inverse Scattering-based perturbation techniques, our approach is valid for arbitrarily large driving strengths and damping coefficients. We show that, contrary to suggestions made in literature, the complexity observed in the soliton's dynamics cannot be accounted for just by its coupling to the long-wavelength radiation.

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Interactions and motions of double-solitons with opposite polarity in a parametrically driven system

Cited by 13 publications

References 12 publications

Empirical stability criteria for parametrically driven solitons of the nonlinear Schrödinger equation

Empirical stability criteria for parametrically driven solitons of the nonlinear Schrödinger equation

Soliton–antisoliton interaction in a parametrically driven easy-plane magnetic wire

Soliton–radiation coupling in the parametrically driven, damped nonlinear Schrödinger equation

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