BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS FOR THE GENERALIZED NONLINEAR SCHRÖDINGER EQUATION WITH WAVE OPERATOR

Abstract

<abstract> In this paper, we investigate the dynamical bifurcations and exact traveling wave solutions for the generalized nonlinear Schrödinger equation with wave operator under different parametric conditions by means of the theory of singular system. We analyse the high order equilibrium point and give the phase portraits. We obtain many results under different values of the parameter $ p $ reflecting the strength of the nonlinearity in the model. For $ p = 1 $, we find explicit exact solutions of Jacobian elliptic functions type which is corresponding to the curves given by $ H(\phi, y) = h $. According to the qualitative analysis of the phase portraits, we give the conclusions on the existence of solitary wave solutions and periodic wave solutions when $ p\geq\frac{1}{2} $. In addition, we obtain the only explicit exact solitary wave solution corresponding to the curves given by $ H(\phi, y) = 0 $ for any $ p $. Especially, we obtain some explicit exact double periodic solutions of elliptic functions type for $ p = \frac{1}{2} $. </abstract>