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We study the nonlinear-damping continuation of singular solutions of thecritical and supercritical NLS. Our simulations suggest that for genericinitial conditions that lead to collapse in the undamped NLS, the solution ofthe 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 thepost-collapse defocusing velocity of the singular core goes to infinity as thedamping coefficient $\delta$ goes to zero. In the special case of theminimal-power blowup solutions of the critical NLS, the continuation is aminimal-power solution with a higher (but finite) defocusing velocity, whosemagnitude increases monotonically with the nonlinear damping exponent $q$.

Nonlinear-damping continuation of the nonlinear Schr\"odinger equation - a numerical study