Mechanisms of unstable blowup in a quadratic nonlinear Schr\"odinger equation

TL;DR

This paper investigates the blowup behavior in a quadratic nonlinear Schrödinger equation, identifying initial data that lead to finite-time blowup and exploring the underlying stable manifold structure.

Contribution

It provides the first numerical evidence of blowup solutions contradicting previous conjectures and links blowup to the stable manifold of a related heat equation.

Findings

Numerical solutions blow up in finite time, contradicting prior conjectures.

Blowup solutions exhibit self-similar behavior with complex dynamics.

The blowup set likely forms a codimension-1 manifold related to the heat equation's stable manifold.

Abstract

In the work Cho et al. [Jpn. J. Ind. Appl. Math. 33 (2016): 145-166] the authors conjecture that the quadratic nonlinear Schr\"odinger equation (NLS) $i u_t = u_{xx} + u^2 $ for $ x \in \mathbb{T}$ is globally well-posed for real initial data. We identify initial data whose numerical solution blows up in contradiction of this conjecture. The solution exhibits self-similar blowup and potentially nontrivial self-similar dynamics, however the proper scaling ansatz remains elusive. Furthermore, the set of real initial data which blows up under the NLS dynamics appears to occur on a codimension-1 manifold, and we conjecture that it is precisely the stable manifold of the zero equilibrium for the nonlinear heat equation $u_t = u_{xx} + u^2 $. We apply the parameterization method to study the internal dynamics of this manifold, offering a heuristic argument in support of our conjecture.