Evolution of near-soliton initial conditions in non-linear wave equations through their Bäcklund transforms

Tsigaridas, G.; Fragos, A.; Polyzos, Ioannis; Fakis, Mihalis; Ioannou, Andreas; Giannetas, V.; Persephonis, P.

doi:10.1016/j.chaos.2004.07.010

Cited by 13 publications

(4 citation statements)

References 18 publications

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“…These exact solutions have been derived by Shang 42 using the extended hyperbolic function method, which describes the resonance interaction between the long wave and the short wave. Let us now apply the MVIM to the initial value problem 1.4 and 5.21 : x ξ 1 2 e i kx ξ 0 , u 2 x, t 2t 2 k β i C − k 2 2 x ξ 1 4ik…”

Section: Approximate Solution Of the Nonlinear Long And Short Wave Rementioning

confidence: 99%

Approximate Solutions for Nonlinear Initial Value Problems Using the Modified Variational Iteration Method

Nofal

2012

Journal of Applied Mathematics

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We have used the modified variational iteration method (MVIM) to find the approximate solutions for some nonlinear initial value problems in the mathematical physics, via the Burgers-Fisher equation, the Kuramoto-Sivashinsky equation, the coupled Schrodinger-KdV equations, and the long-short wave resonance equations together with initial conditions. The results of these problems reveal that the modified variational iteration method is very powerful, effective, convenient, and quite accurate to systems of nonlinear equations. It is predicted that this method can be found widely applicable in engineering and physics.

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Section: Approximate Solution Of the Nonlinear Long And Short Wave Rementioning

confidence: 99%

Approximate Solutions for Nonlinear Initial Value Problems Using the Modified Variational Iteration Method

Nofal

2012

Journal of Applied Mathematics

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“…Thus, in [10], Mann used a linearized Bäcklund transformation to compute the Green's function for the KdV equation linearized about the soliton solution and then in turn used this to study the evolution of initial conditions close to the soliton. Likewise, Tsigaridas, et al [20] make a more general study of this same question and apply these ideas to compute approximate solutions of both the nonlinear Schrödinger equation and KdV equations with the aid of linearized Bäcklund transformations.…”

Section: Introductionmentioning

confidence: 99%

Orbital stability of localized structures via Bäcklund transfomations

Hoffman¹,

Wayne²

2013

Differential Integral Equations

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The Bäcklund Transform, first developed in the context of differential geometry, has been classically used to obtain multi-soliton states in completely integrable infinite dimensional dynamical systems. It has recently been used to study the stability of these special solutions. We offer here a dynamical perspective on the Bäcklund Transform, prove an abstract orbital stability theorem, and demonstrate its utility by applying it to the sine-Gordon equation and the Toda lattice.

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“…Thus, in [15], Mann used a linearized Bäcklund transformation to compute the Green's function for the KdV equation linearized about the soliton solution and then in turn used this to study the evolution of initial conditions close to the soliton. Likewise, Tsigaridas, et al [16] make a more general study of this same question and apply these ideas to compute approximate solutions of both the nonlinear Schrödinger equation and KdV equations with the aid of linearized Bäcklund transformations.…”

Section: Introductionmentioning

confidence: 99%