Exact<i>N</i>-soliton solutions of the extended nonlinear Schrödinger equation

Abstract

By use of Hirota's direct method and a simple transformation, we obtain the exact N-soliton solutions of the extended nonlinear Schr\"odinger equation, i \ensuremath{\partial}q / \ensuremath{\partial}z - k'' / 2 ${\mathrm{\ensuremath{\partial}}}^{2}$q / \ensuremath{\partial}${\mathit{t}}^{2}$ +\ensuremath{\beta}\ensuremath{\Vert}q${\mathrm{\ensuremath{\Vert}}}^{2}$q+i\ensuremath{\gamma} \ensuremath{\partial}(\ensuremath{\Vert}q${\mathrm{\ensuremath{\Vert}}}^{2}$q) / \ensuremath{\partial}t -ik''' / 6 ${\mathrm{\ensuremath{\partial}}}^{3}$q / \ensuremath{\partial}${\mathit{t}}^{3}$ =0 , under the conditions 3k''\ensuremath{\gamma}=\ensuremath{\beta}k''' and k''\ensuremath{\gamma}=\ensuremath{\beta}k''', respectively. The features of the solutions are discussed.