Jacobian elliptic function solutions of the discrete cubic–quintic nonlinear Schrödinger equation

Abstract

The study of solitary wave solutions is of prime significance for the nonlinear Schrödinger equation with higher order dispersion and/or higher degree nonlinearities in nonlinear physical systems. We derive the discrete cubic–quintic nonlinear Schrödinger equation from a Hamiltonian using different Poisson brackets. By using the extended Jacobian elliptic function approach, we investigate the abundant exact stationary solitons and periodic waves solution of this equation. These solutions include, Jacobian periodic solutions, alternating phase Jacobi periodic solution, kink and bubble soliton solutions, alternating phase kink soliton solution and alternating phase bubble soliton solution, provided that coefficients are bound by special relation. And then with the aid of symbolic computation, we present in explicit form these solutions. The stability of bubble and kink soliton as well as alternating kink and alternating bubble soliton are also investigated.