EXACT-SOLUTIONS FOR A HIGHER-ORDER NONLINEAR SCHRODINGER-EQUATION

We performed a systematic analysis of exact solutions for the higher-order nonlinear Schrödinger equation iX+TT=a12+a24+ia3 (2)T+(a4+ia5)(2)T that describes wave propagation in nonlinear dispersive media. The method consists of the determination of all transformations that reduce the equation to ordinary differential equations that are solved whenever possible. All obtained solutions fall into one of the following categories: bright or dark solitary waves, solitonic waves, regular and singular periodic waves, shock waves, accelerating waves, and self-similar solutions. They are expressed in terms of simple functions except for few cases given in terms of the less-known Painlevé transcendents. © 1990 The American Physical Society.