Keywords: (3 + 1)-dimensional Jimbo–Miwa equation Hirota bilinear form Wronskian determinant solution a b s t r a c t In this paper, we obtain two kinds of sufficient conditions consisting of systems of linear partial differential equations, which guarantee that the corresponding Wroskian determinant solves the (3 + 1)-dimensional Jimbo–Miwa equation in the Hirota bilinear form. Our results sug...

Conservation laws of the nonlinear Schrödinger equation are studied in the presence of higher-order nonlinear optical effects including the third-order dispersion and the selfsteepening. In a context of group theory, we derive a general expression for infinitely many conserved currents and charges of the coupled higher-order nonlinear Schrödinger equation. The first few currents and charges are...

Small-amplitude waves in the Fermi-Pasta-Ulam (FPU) lattice with weakly anharmonic interaction potentials are described by the generalized Korteweg-de Vries (KdV) equation. Justification of the small-amplitude approximation is usually performed on the time scale, for which dynamics of the KdV equation is defined. We show how to extend justification analysis on longer time intervals provided dyn...

which can be considered as a coupling between the KdV (with respect to u) and the mKdV (with respect to v) equations. The coupled KdV-mKdV equations were proposed by Kersten and Krasil’shchik [1] and originate from a supersymmetric extension of the classical KdV [2]. It also can be considered as a coupling between the KdV and mKdV equations: By setting v = 0 we obtain the KdV equation ut + uxxx...

General breather solution to the Sasa–Satsuma equation (SSE) is systematically investigated in this paper. We firstly transform SSE into a set of three Hirota bilinear equations under proper plane wave boundary condition. Starting from specially arranged tau-function Kadomtsev–Petviashvili hierarchy and 11 satisfied, we implement series steps reduction procedure, i.e. C-type reduction, dimensio...

We construct a one-parameter family of N=3 supersymmetric extensions of the KdV equation as a Hamiltonian flow on N=3 superconformal algebra and argue that it is non-integrable for any choice of the parameter. Then we propose a modified N=3 super KdV equation which possesses the higher order conserved quantities and so is a candidate for an integrable system. Upon reduction to N=2, it yields th...

We study the traveling waves of the Nonlinear Schrödinger Equation in dimension one. Through various model cases, we show that for nonlinearities having the same qualitative behaviour as the standard Gross-Pitaevkii one, the traveling waves may have rather different properties. In particular, our examples exhibit multiplicity or nonexistence results, cusps (as for the Jones-Roberts curve in the...

It is shown that the generalized N=4 supersymmetric Toda lattice hierarchy contains the N=4 super-KdV hierarchy with the first flow time in the role of space coordinate. Two different N=2 superfield form of the generalized N=4 TL equation which are useful when solving the N=4 KdV and (1,1)-GNLS hierarchies are discussed.