Soliton solutions and traveling wave solutions for the two-dimensional generalized nonlinear Schr\"odinger equations

In this paper, we present the two-dimensional generalized nonlinear Schr\"odinger equations with the Lax pair. These equations are related to many physical phenomena in the Bose-Einstein condensates, surface waves in deep water and nonlinear optics. The existence of the Lax pair defines integrability for the partial differential equation, so the two-dimensional generalized nonlinear Schr\"odinger equations are integrable. We obtain bilinear forms of the two-dimensional GNLS equations. One- and two-soliton solutions are derived via the Hirota bilinear method, a procedure quite useful in the solution of nonlinear partial differential equations. We apply the extended tanh method in order to construct new exact traveling wave solutions. Through 3D plots, we show the dynamical behavior of the obtained solutions.