Numerical simulations of singular solutions of the nonlinear schrödinger equations

The nonlinear Schrödinger equation (NLS) with cubic nonlinearity (equation found) arises in various physical contest as an amplitude equation for weakly nonlinear waves [23]. For a certain class of initial conditions, namely those for which the invariant (equation found) is negative, NLS has solutions that become singular in a finite time when the dimension of the space d is larger than or equal to two [32], [15]. In two space dimensions d = 2, NLS is also the model equation for the propagation of cw (continuous wave) laser beams in Kerr media, where ψ is the electric field envelope, t is axial distance in the direction of beam propagation, and Δψ is the diffraction term. It is also well known that when the power, or L² norm, of the input beam is sufficiently high, solutions of eq. (1.1) can self-focus and become singular in finite t [8], [29].