KdV equation in the quarter--plane: evolution of the Weyl functions and unbounded solutions

TL;DR

This paper studies the matrix KdV equation in the quarter-plane, deriving the evolution of Weyl functions and demonstrating unbounded solutions for certain initial-boundary conditions.

Contribution

It introduces an evolution law for Weyl functions in the quarter-plane and links low energy asymptotics to unbounded solutions, advancing understanding of matrix KdV behavior.

Findings

Derived the evolution law for Weyl functions in the quarter-plane

Showed unbounded solutions exist for specific initial-boundary conditions

Connected low energy asymptotics to solution unboundedness

Abstract

The matrix KdV equation with a negative dispersion term is considered in the right upper quarter--plane. The evolution law is derived for the Weyl function of a corresponding auxiliary linear system. Using the low energy asymptotics of the Weyl functions, the unboundedness of solutions is obtained for some classes of the initial--boundary conditions.