Explicit solutions of Schr\"odinger and KdV equations in terms of square roots of the generalised matrix eigenvalues

TL;DR

This paper develops explicit solutions for matrix Schr"odinger and KdV equations using a novel GBDT-based B"acklund--Darboux transformation involving square roots of generalized matrix eigenvalues, including singular potentials.

Contribution

It introduces a new method to explicitly solve matrix Schr"odinger and KdV equations using GBDT and eigenvalue square roots, extending previous approaches.

Findings

Explicit solutions constructed for matrix Schr"odinger and KdV equations

Method applicable to strongly singular potentials

Includes several illustrative examples

Abstract

In this paper, we consider matrix Schr\"odinger equation, dynamical Schr\"odinger equation and matrix KdV. We construct their explicit solutions using our GBDT version of B\"acklund--Darboux transformation and square roots of the generalised matrix eigenvalues. A separate section is dedicated to several examples including the case of strongly singular potentials.