Normal form for the onset of collapse: The prototypical example of the nonlinear Schrödinger equation.

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 Schrödinger equation and systematically derive a normal form for the emergence of radially symmetric 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, applies to 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 both critical and supercritical regimes.