Exact solutions of the cubic and quintic nonlinear Schrödinger equation for a cylindrical geometry

Abstract

Exact solutions of the nonlinear Schr\"odinger equation ${\mathit{i}\mathit{\ensuremath{\psi}}}_{\mathrm{t}}$+\ensuremath{\Delta}\ensuremath{\psi}=${\mathrm{a}}_{0}$\ensuremath{\psi}/B +${a}_{1}$\ensuremath{\psi}\ensuremath{\Vert}\ensuremath{\psi}${\ensuremath{\Vert}}^{2}$+${a}_{2}$\ensuremath{\psi}\ensuremath{\Vert}\ensuremath{\psi}${\ensuremath{\Vert}}^{4}$, for which initial conditions can be imposed on a cylinder, are presented. A symmetry group of the equation is used to reduce it to an ordinary differential equation which is then solved with the help of a singularity analysis. Solutions are obtained in terms of elementary functions, Jacobi elliptic functions, and Painlev\'e transcendents.