Resonance and web structure in discrete soliton systems: the two-dimensional Toda lattice and its fully discrete and ultra-discrete analogues

Maruno, Ken Ichi; Biondini, Gino

doi:10.1088/0305-4470/37/49/005

Cited by 28 publications

(18 citation statements)

References 19 publications

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“…[10,11] for a coupled KP system, and similar solutions were also found in Ref. [14] in discrete soliton systems such as the two-dimensional Toda lattice, together with its fully-discrete and ultra-discrete analogues. In other words, the existence of these solutions appears to be a rather common feature of (2+1)-dimensional integrable systems.…”

Section: Introductionsupporting

confidence: 82%

Soliton solutions of the Kadomtsev-Petviashvili II equation

Biondini

Chakravarty

2006

Journal of Mathematical Physics

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112

654

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We study a general class of line-soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation by investigating the Wronskian form of its tau-function. We show that, in addition to previously known line-soliton solutions, this class also contains a large variety of new multi-soliton solutions, many of which exhibit nontrivial spatial interaction patterns. We also show that, in general, such solutions consist of unequal numbers of incoming and outgoing line solitons. From the asymptotic analysis of the taufunction, we explicitly characterize the incoming and outgoing line-solitons of this class of solutions. We illustrate these results by discussing several examples.

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Section: Introductionsupporting

confidence: 82%

Soliton solutions of the Kadomtsev-Petviashvili II equation

Biondini

Chakravarty

2006

Journal of Mathematical Physics

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112

654

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“…[2,3,5,6,20,21,31,35,36,42] and references therein). The purpose of this work is to show that, mutatis mutandis, an approach similar to that for PDEs can also be used to solve IBVPs for linear and integrable nonlinear differential-difference equations (DDEs).…”

Section: Introduction and Outlinementioning

confidence: 99%

Initial-boundary-value problems for discrete evolution equations: discrete linear Schrödinger and integrable discrete nonlinear Schrödinger equations

Biondini

Hwang

2008

Inverse Problems

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Abstract. We present a method to solve initial-boundary value problems for linear and integrable nonlinear differential-difference evolution equations. The method is the discrete version of the one developed by A. S. Fokas to solve initial-boundary value problems for linear and integrable nonlinear partial differential equations via an extension of the inverse scattering transform. The method takes advantage of the Lax pair formulation for both linear and nonlinear equations, and is based on the simultaneous spectral analysis of both parts of the Lax pair. A key role is also played by the global algebraic relation that couples all known and unknown boundary values. Even though additional technical complications arise in discrete problems compared to continuum ones, we show that a similar approach can also solve initial-boundary value problems for linear and integrable nonlinear differential-difference equations. We demonstrate the method by solving initial-boundary value problems for the discrete analogue of both the linear and the nonlinear Schrödinger equations, comparing the solution to those of the corresponding continuum problems. In the linear case we also explicitly discuss Robin-type boundary conditions not solvable by Fourier series. In the nonlinear case we also identify the linearizable boundary conditions, we discuss the elimination of the unknown boundary datum, we obtain explicitly the linear and continuum limit of the solution, and we write down the soliton solutions.28 December 2017

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“…When b(1, 2, 1 * , 2 * ) = 0, that is, b(1, 2) = 0 in (55), resonant interactions can happen. The resonant interactions in this case are called the "minus resonance" [36,37], namely, after the solitons interact with each other, the amplitudes decrease, sometimes the amplitudes can even reach zero. The resonant situation here is similar as the continuous case.…”

Section: Dynamic Propertiesmentioning

confidence: 99%

Dynamics of a differential-difference integrable(2+1)-dimensional system

2015

Phys. Rev. E

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A Kadomtsev-Petviashvili- (KP-) type equation appears in fluid mechanics, plasma physics, and gas dynamics. In this paper, we propose an integrable semidiscrete analog of a coupled (2+1)-dimensional system which is related to the KP equation and the Zakharov equation. N-soliton solutions of the discrete equation are presented. Some interesting examples of soliton resonance related to the two-soliton and three-soliton solutions are investigated. Numerical computations using the integrable semidiscrete equation are performed. It is shown that the integrable semidiscrete equation gives very accurate numerical results in the cases of one-soliton evolution and soliton interactions.

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Resonance and web structure in discrete soliton systems: the two-dimensional Toda lattice and its fully discrete and ultra-discrete analogues

Cited by 28 publications

References 19 publications

Soliton solutions of the Kadomtsev-Petviashvili II equation

Soliton solutions of the Kadomtsev-Petviashvili II equation

Initial-boundary-value problems for discrete evolution equations: discrete linear Schrödinger and integrable discrete nonlinear Schrödinger equations

Dynamics of a differential-difference integrable(2+1)-dimensional system

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