ExactN-soliton solutions of the extended nonlinear Schrödinger equation

By use of Hirota’s direct method and a simple transformation, we obtain the exact N-soliton solutions of the extended nonlinear Schrödinger equation, i ∂q / ∂z - k’ ’ / 2 ${\mathrm{\partial }}^2$q / ∂${\mathit{t}}^2$ +β‖q${\mathrm{\Vert }}^2$q+iγ ∂(‖q${\mathrm{\Vert }}^2$q) / ∂t -ik’ ’ ’ / 6 ${\mathrm{\partial }}^3$q / ∂${\mathit{t}}^3$ =0 , under the conditions 3k’ ’γ=βk’ ’ ’ and k’ ’γ=βk’ ’ ’, respectively. The features of the solutions are discussed.