Ring-type singular solutions of the biharmonic nonlinear Schrodinger equation

TL;DR

This paper introduces new ring-type singular solutions for the biharmonic nonlinear Schrödinger equation, describing their collapse behavior, profile, and blowup rates in various dimensions and nonlinearities.

Contribution

It presents the first known singular solutions with ring profiles for the biharmonic nonlinear Schrödinger equation, detailing their collapse dynamics and blowup rates.

Findings

Solutions collapse with a quasi self-similar ring profile.

Ring width vanishes at singularity, with radius proportional to L^.

Blowup rate varies with  and = (4-)/((d-1)).

Abstract

We present new singular solutions of the biharmonic nonlinear Schrodinger equation in dimension d and nonlinearity exponent 2\sigma+1. These solutions collapse with the quasi self-similar ring profile, with ring width L(t) that vanishes at singularity, and radius proportional to L^\alpha, where \alpha=(4-\sigma)/(\sigma(d-1)). The blowup rate of these solutions is 1/(3+\alpha) for 4/d\le\sigma<4, and slightly faster than 1/4 for \sigma=4. These solutions are analogous to the ring-type solutions of the nonlinear Schrodinger equation.