A Spectral Analysis of the Nonlinear Schroedinger Equation in the Co-Exploding Frame

TL;DR

This paper conducts a spectral analysis of the nonlinear Schrödinger equation in a co-exploding frame, revealing the stability properties of self-similar solutions through spectral characteristics of the linearized operator.

Contribution

It provides a detailed spectral analysis of the linearized problem around self-similar solutions, identifying eigenvalues related to symmetries and stability in a complex dispersive wave context.

Findings

Two eigenvalues are positive but due to symmetries, not instabilities.

Three eigenvalues are negative and real, indicating stability aspects.

The continuous spectrum lies nearly on the left-half plane, suggesting spectral stability.

Abstract

The nonlinear Schroedinger model is a prototypical dispersive wave equation that features finite time blowup, either for supercritical exponents (for fixed dimension) or for supercritical dimensions (for fixed nonlinearity exponent). Upon identifying the self-similar solutions in the so-called "co-exploding frame", a dynamical systems analysis of their stability is natural, yet is complicated by the mixed Hamiltonian-dissipative character of the relevant frame. In the present work, we study the spectral picture of the relevant linearized problem. We examine the point spectrum of 3 eigenvalue pairs associated with translation, $U(1)$ and conformal invariances, as well as the continuous spectrum. We find that two eigenvalues become positive, yet are attributed to symmetries and are thus not associated with instabilities. In addition to a vanishing eigenvalue, 3 more are found to be negative and real, while the continuous spectrum is nearly vertical and on the left-half (spectral) plane. Finally, the subtle effects of the boundaries are also assessed and their role in the observed weak eigenvalue oscillations is clarified.