Benjamin–Feir instability in nonlinear dispersive waves

In this paper, the authors extended the derivation to the nonlinear Schrödinger equation in two-dimensions, modified by the effect of non-uniformity. The authors derived several classes of soliton solutions in 2+1 dimensions. When the solution is assumed to depend on space and time only through a single argument of the function, they showed that the two-dimensional nonlinear Schrödinger equation is reduced either to the sine-Gordon for the hyperbolic case or sinh-Gordon equations for the elliptic case. Moreover, the authors extended this method to obtain analytical solutions to the nonlinear Schrödinger equation in two space dimensions plus time. This contains some interesting solutions such as the plane solitons, the N multiple solitons, the propagating breathers and quadratic solitons. The authors displayed graphically the obtained solutions by using the software Mathematica 5.