Exact solutions for a higher-order nonlinear Schrödinger equation

Abstract

We performed a systematic analysis of exact solutions for the higher-order nonlinear Schr\"odinger equation i${\mathrm{\ensuremath{\psi}}}_{\mathit{X}}$+${\mathrm{\ensuremath{\psi}}}_{\mathit{T}\mathit{T}}$=${\mathit{a}}_{1}$\ensuremath{\psi}\ensuremath{\Vert}\ensuremath{\psi}${\mathrm{\ensuremath{\Vert}}}^{2}$+${\mathit{a}}_{2}$\ensuremath{\psi}\ensuremath{\Vert}\ensuremath{\psi}${\mathrm{\ensuremath{\Vert}}}^{4}$+${\mathit{ia}}_{3}$ (\ensuremath{\psi}\ensuremath{\Vert}\ensuremath{\psi}${\mathrm{\ensuremath{\Vert}}}^{2}$${)}_{\mathit{T}}$+(${\mathit{a}}_{4}$+${\mathit{ia}}_{5}$)\ensuremath{\psi}(\ensuremath{\Vert}\ensuremath{\psi}${\mathrm{\ensuremath{\Vert}}}^{2}$${)}_{\mathit{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 Painlev\'e transcendents.