Mathematics – Analysis of PDEs
Scientific paper
2011-07-17
Mathematics
Analysis of PDEs
Scientific paper
We study the nonlinear-damping continuation of singular solutions of the critical and supercritical NLS. Our simulations suggest that for generic initial conditions that lead to collapse in the undamped NLS, the solution of the weakly-damped NLS $$ i\psi_t(t,\X)+\Delta\psi+|\psi|^{p-1}\psi+i\delta|\psi|^{q-1}\psi=0,\qquad0<\delta \ll 1, $$ is highly asymmetric with respect to the singularity time, and the post-collapse defocusing velocity of the singular core goes to infinity as the damping coefficient $\delta$ goes to zero. In the special case of the minimal-power blowup solutions of the critical NLS, the continuation is a minimal-power solution with a higher (but finite) defocusing velocity, whose magnitude increases monotonically with the nonlinear damping exponent $q$.
Fibich Gadi
Klein Marian
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