Reverse space and time nonlocal coupled dispersionless equation and its solutions

Abstract

Coupled dispersionless (CD) equation is an important integrable model since it describes the current-fed string in a certain external magnetic field. Recently, Ablowitz and Musslimani introduced a class of reverse space, reverse time and reverse space and time nonlocal integrable equations, including nonlocal nonlinear Schrödinger equation, nonlocal sine-Gordon equation and nonlocal Davey-Stewartson equation etc. In this paper we study an integrable reverse space and time nonlocal CD equation. By applying the Darboux transformation, we present the one-soliton and twosoliton solutions for the nonlocal CD equation. We also show the asymptotic analysis of the one-soliton solution from nonzero seed and two-soliton solutions.

Bi-Dbar-Approach for a Coupled Shifted Nonlocal Dispersionless System

“…The solution can be obtained by assuming the c j and d j in (63) to c j = ic j , d j = i dj , where cj , dj are real functions of b j and j as well as x 0 , t 0 . For example, to equation (83), the amplitude-changing one-soltion [35] and a dark single-soliton where X 1 and X1 are defined in (78). In additon, the two-solition (74) of the CCSND system reduces to that of the real reverse space-time shifted nonlocal sine-Gordon (83), which takes the following form where Xj , (j = 1, 2) are defined in (78) and (83) q tx (x, t) + 2w(x, t)q(x, t) = 0, w x (x, t) − q(x, t)q(x 0 − x, t 0 − t) t = 0, The other explicit solutions can be obtained similarly from those of the CCSND system.…”

“…The O(k −1 ) terms of the expansion of ( 30), or n = 1 in (32), takes the form which can be rewritten as the off-diagonal part and the diagonal part Substituting ( 29) into (35), we obtain which gives the second conservation law. Here and after, we take the integral constants of a [d] n to be zero.…”

Multi-place physics and multi-place nonlocal systems

“…In addition to the nonlocal NLS system (1), there are many other types of two-place nonlocal models, such as the NLS equations with different non-localities [27], the nonlocal KdV systems [25,26,28,29], nonlocal modified KdV (MKdV) systems [25,[30][31][32][33][34][35], nonlocal discrete NLS systems [36][37][38], nonlocal coupled NLS systems [39][40][41][42][43][44][45], nonlocal derivative NLS equation [46], nonlocal Davey-Stewartson systems [47][48][49][50], generalized nonlocal NLS equation [51], nonlocal nonautonomous KdV equation [52], nonlocal peakon systems [53], nonlocal KP systems, nonlocal sine Gordon systems, nonlocal Toda systems [25,26], nonlocal Sawada-Kortera equations [54], nonlocal Kaup-Kupershmidt equations [54] and many others [55][56][57][58][59][60][61][62][63][64][65].…”