A Normal Form for the Onset of Collapse: the Prototypical Example of the Nonlinear Schrodinger Equation

TL;DR

This paper derives a comprehensive normal form for the onset of finite-time collapse in the nonlinear Schrödinger equation, unifying various bifurcation scenarios and providing accurate predictions across different regimes and domain types.

Contribution

It introduces a novel normal form for blowup solutions in the nonlinear Schrödinger equation, integrating previous bifurcation analyses and applicable universally.

Findings

Normal form accurately predicts collapse behavior.

Unifies dimension-dependent and power-law bifurcations.

Valid across all regimes and domain types.

Abstract

The study of nonlinear waves that collapse in finite time is a theme of universal interest, e.g. within optical, atomic, plasma physics, and nonlinear dynamics. Here we revisit the quintessential example of the nonlinear Schrodinger equation and systematically derive a normal form for the emergence of blowup solutions from stationary ones. While this is an extensively studied problem, such a normal form, based on the methodology of asymptotics beyond all algebraic orders, unifies both the dimension-dependent and power-law-dependent bifurcations previously studied; it yields excellent agreement with numerics in both leading and higher-order effects; it is applicable to both infinite and finite domains; and it is valid in all (subcritical, critical and supercritical) regimes.