The direct algebra method to the Hamiltonian amplitude equation and the higher-order nonlinear Schr¨odinger equation

Abstract

By using the direct algebra method, the traveling wave solutions for the Hamiltonian amplitude equation and the higher-order nonlinear Schr\{o}dinger equation are constructed. The obtained results include complex exponential function solutions, complex traveling solitary wave solutions, complex periodic wave solutions. The power of this manageable method is confirmed. The Hamiltonian amplitude equation is an equation which governs certain instabilities of modulated wave trains, with the additional term $-\epsilon u_{xt}$ overcoming the ill-posedness of the unstable nonlinear Schr\{o}dinger equation. It is a Hamiltonian analogue of the Kuramoto-Sivashinski equation which arises in dissipative systems and is apparently not integrable.