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

Abstract

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.

An integrable semi-discretization of the modified Camassa-Holm equation with linear dispersion term

“…Starting from the discrete Kadomtsev-Petviashvili (KP) equation, or the so-called Hirota-Miwa (HW) equation [31,32] , Shi et al have derived discrete KdV equation and discrete potential mKdV equation, as well as their Lax pairs and multi-soliton solutions [33,34] . The authors have done a series of work in finding integrable discretizations of soliton equations such as the short pulse equation [35,36] , (2+1)-dimensional Zakharov equation [37] , the Camassa-Holm equation [38,39] and the Degasperis-Proceli equaiton [40] . Therefore, it is a natural but definitely not a trivial problem for us to construct an integrable discrete analogue of the mCH equation (1).…”

On a new integrable generalization of the Toda lattice and a discrete Yajima-Oikawa system

“…(2.12) and (2.13)). Note that the generalized Toda hierarchy is different from, and apparently has no relation to, the discrete Yajima-Oikawa hierarchy recently studied in [11,13] (also see [12] for a (2 + 1)-dimensional version [23,32,33] of the discrete Yajima-Oikawa system).…”

“…This generalization provides a discrete analog of the generalization of the KdV hierarchy to a long wave-short wave interaction hierarchy, called the Yajima-Oikawa hierarchy [9,10], in the continuous case. A space discretization of the Yajima-Oikawa system was already proposed in the recent paper [11] (also see [12]) and its Lax pair as well as the next higher symmetry was presented in [13]. The first flow of the generalized Toda hierarchy proposed in this paper provides, in a special case, a new integrable discretization of the Yajima-Oikawa system, which is essentially different from the discrete Yajima-Oikawa system studied in [11,13] and has its own advantages; in particular, the discrete Yajima-Oikawa system in this paper possesses not only a Lax pair and an infinite number of conservation laws but also a simple Hamiltonian structure, so the higher flows of the hierarchy can easily be constructed.…”

Hybrid waves consisting of the solitons, breathers and lumps for a (2+1)-dimensional extended shallow water wave equation

“…Based on Conditions (9) and taking k 4ζ1−2 → −k 4ζ1−3 , we can Figs. 4 The same as Figs. 3 except that ϕ(y) = 0.5 sin(0.66y).…”

“…Nonlinear evolution equations (NLEEs) have been used to describe the wave propagations in fluid mechanics, plasma physics and nonlinear optics [1][2][3][4][5][6][7]. Solitions have been found that their shapes and amplitudes unchanged during the propagations [8].…”