Bounded solutions for an ordinary differential system from the Ginzburg-Landau theory

TL;DR

This paper investigates the existence of bounded solutions in a linear ODE system derived from the Ginzburg-Landau equation, proving non-existence in certain cases through eigenvalue analysis.

Contribution

It establishes the absence of bounded solutions in specific cases of the Ginzburg-Landau derived system, extending previous knowledge about solution boundedness.

Findings

Bounded solutions exist due to invariance properties.

In certain cases, no bounded solutions are possible.

Bounded solution problem linked to eigenvalue problem.

Abstract

In this paper, we look at a linear system of ordinary differential equations as derived from the two-dimensional Ginzburg-Landau equation. In two cases, it is known that this system admits bounded solutions coming from the invariance of the Ginzburg-Landau equation by translations and rotations. The specific contribution of our work is to prove that in the other cases, the system does not admit any bounded solutions. We show that this bounded solution problem is related to an eigenvalue problem.