Singular solutions of the biharmonic Nonlinear Schrodinger equation

TL;DR

This paper investigates singular solutions of the biharmonic nonlinear Schrödinger equation, analyzing blowup rates, profiles, and collapse behavior in critical and supercritical cases through asymptotic analysis and numerical simulations.

Contribution

It provides a detailed characterization of blowup dynamics and profiles for singular solutions in the biharmonic NLS, including new insights into the critical and supercritical regimes.

Findings

Blowup rate in critical case is slightly faster than quartic-root.

Self-similar profile in critical case is the standing-wave ground state.

Supercritical case exhibits a quartic-root blowup rate with a zero-Hamiltonian profile.

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.