Rational solutions of the fourth order analogue to the Painlevé equations are classified. Special polynomials associated with the rational solutions are introduced. The structure of the polynomials is found. Formulas for their coefficients and degrees are derived. It is shown that special solutions of the Fordy Gibbons, the Caudrey Dodd Gibbon and the Kaup Kupershmidt equations can be expressed...

We propose integrable discretizations of derivative nonlinear Schrödinger (DNLS) equations such as the Kaup–Newell equation, the Chen–Lee–Liu equation and the Gerdjikov–Ivanov equation by constructing Lax pairs. The discrete DNLS systems admit the reduction of complex conjugation between two dependent variables and possess bi-Hamiltonian structure. Through transformations of variables and reduc...

This paper extends the dual-Petrov-Galerkin method proposed by Shen [21], further developed by Yuan, Shen and Wu [27] to general fifth-order KdV type equations with various nonlinear terms. These fifth-order equations arise in modeling different wave phenomena. The method is implemented to compute the multi-soliton solutions of two representative fifth-order KdV equations: the Kaup-Kupershmidt ...

Based on the WTC truncation method and the general variable separation approach (GVSA), we have first found a general solution including three arbitrary functions for the (2+1)-dimensional simplified generalized Broer-Kaup (GBK) system (B=0). A class of double periodic wave solutions is obtained by selecting these arbitrary functions appropriately. The interaction properties of the periodic wav...

In this paper, we investigate local and nonlocal reductions of a discrete negative order Ablowitz–Kaup–Newell–Segur equation. By the bilinearization reduction method, construct exact solutions in double Casoratian form to reduced sine-Gordon equations. Then, semidiscrete equations their are obtained through continuum limits. The dynamics soliton analyzed illustrated by asymptotic analysis. rese...

The interaction and propagation of optical pulses in a nonlinear waveguide array is described by the discrete nonlinear Schrödinger equation i∂zψn = −D(ψn+1 + ψn−1 − 2ψn) − γ|ψn|ψn, (1) whereD is a dispersion (or diffraction) coefficient, and γ is a measure of the nonlinearity. By means of the variational approximation, we study the discrete soliton solutions of this equation. We use a trial fu...