Zero Energy Self-Similar Solutions Describing Singularity Formation In The Nonlinear Schrodinger Equation In Dimension N=3

TL;DR

This paper proves that for the cubic nonlinear Schrödinger equation in three dimensions, self-similar solutions describing finite-time singularities exist for all positive parameters, advancing understanding of wave collapse phenomena.

Contribution

It establishes the existence of self-similar solutions satisfying the zero-energy constraint for all positive parameters, resolving a long-standing open problem since 1972.

Findings

Existence of self-similar solutions for all positive parameters a and Q(0).

Validation of the zero-energy integral constraint in the model.

Advancement in mathematical understanding of singularity formation in NLS.

Abstract

In dimension N=3 the cubic nonlinear Schrodinger equation has solutions which become singular, i.e. at a spatial point they blow up to infinity in finite time. In 1972 Zakharov famously investigated finite time singularity formation in the cubic nonlinear Schrodinger equation as a model for spatial collapse of Langmuir waves in plasma, the most abundant form of observed matter in the universe. Zakharov assumed that (NLS) blow up of solutions is self-similar and radially symmetric, and that singularity formation can be modeled by a solution of an associated self-similar, complex ordinary differential equation~(ODE). A parameter a>0 appears in the ODE, and the dependent variable, Q, satisfies (Q(0),Q'(0))=(Q_{0},0), where Q(0)>0. A fundamentally important step towards putting the Zakharov model on a firm mathematical footing is to prove, when N=3, whether values a>0 and Q_{0}>0 exist such that Q also satisfies the physically important `zero-energy' integral constraint. Since 1972 this has remained an open problem. Here, we resolve this issue by proving that for every a>0 and Q(0)>0, Q satisfies the the `zero-energy' integral constraint.