Singular solutions of the biharmonic Nonlinear Schrodinger equation

Abstract

We consider singular solutions of the biharmonic NLS. In the L^2-critical case, the blowup rate is bounded by a quartic-root power law, the solution approaches a self-similar profile, and a finite amount of L^2-norm, which is no less than the critical power, concentrates into the singularity ("strong collapse"). In the L^2-critical and supercritical cases, we use asymptotic analysis and numerical simulations to characterize singular solutions with a peak-type self-similar collapsing core. In the critical case, the blowup rate is slightly faster than a quartic-root, and the self-similar profile is given by the standing-wave ground-state. In the supercritical case, the blowup rate is exactly a quartic-root, and the self-similar profile is a zero-Hamiltonian solution of a nonlinear eigenvalue problem. These findings are verified numerically (up to focusing levels of 10^8) using an adaptive grid method. We also calculate the ground states of the standing-wave equations and the critical power for collapse in two and three dimensions.