  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.
