Traveling-wave method for solving the modified nonlinear Schrödinger equation describing soliton propagation along optical fibers

We give a traveling-wave method for obtaining exact solutions of the modified nonlinear Schrödinger equation ${\mathit{iu}}_{\mathit{t}}$+ɛ${\mathit{u}}_{\mathit{x}\mathit{x}}$+2p‖u${\mathrm{\Vert }}^2$u +2iq(‖u${\mathrm{\Vert }}^2$u${)}_{\mathit{x}}$=0, describing the propagation of light pulses in optical fibers, where u represents a normalized complex amplitude of a pulse envelope, t is the normalized distance along a fiber, and x is the normalized time within the frame of reference moving along the fiber at the group velocity. With the help of the ‘‘potential function’’ we obtained by this method, we find a family of solutions that are finite everywhere, particularly including periodic solutions expressed in terms of Jacobi elliptic functions, stationary periodic solutions, and ‘‘algebraic’’ soliton solutions. Compared with previous work [D. Mihalache and N. C. Panoiu, J. Math. Phys. 33, 2323 (1992)] in which two kinds of the simplest solution were given, the physical meaning of the integration constants in the potential function we give is clearer and more easily fixed with the initial parameters of the light pulse.