ABSTRACT
Xiao-Ping Wang, The Hong Kong University of Science and Technology, mawang@ust.hk
The nonlinear Schro¨dinger equation (NLS) with cubic nonlinearity
iψt + ∆ψ + |ψ|2ψ = 0 t > 0, (1.1)
ψ(0,x) = ψ0(x), x ∈ Rd
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 H =
∫ t 0 (|∇ψ|2 − 12 |ψ|2)dx 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 ∆ψ 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].
