Self-Similar Singular Solutions to the Nonlinear Schr\"odinger and the Complex Ginzburg-Landau Equations

TL;DR

This paper establishes the existence of self-similar singular solutions for the nonlinear Schrödinger and complex Ginzburg-Landau equations, using a combination of analytical bounds and computer-assisted methods.

Contribution

It proves the existence of radial self-similar singular solutions far from the critical regime and controls their monotonicity, extending previous results to new parameter regimes.

Findings

Existence of self-similar singular solutions for NLS and Ginzburg-Landau equations.

Construction of solutions with controlled monotonicity properties.

Application of combined analytical and computational techniques for solution matching.

Abstract

We prove the existence of radial self-similar singular solutions for the mass supercritical Nonlinear Schr\"odinger Equation far from the critical regime and, more generally, branches of such solutions for the Complex Ginzburg-Landau Equation. We are also able to control their monotone index (number of monotone intervals). In particular, we prove the existence of monotone radial self-similar singular solutions for the three dimensional cubic Nonlinear Schr\"odinger Equation. The paper combines sharp analytic bounds of the self-similar profile at infinity with computer assisted bounds around zero and their matching at an intermediate value.