On the dynamics of the singularities of the solutions of some non-linear integrable differential equations

TL;DR

This paper investigates the behavior and dynamics of singularities in solutions of certain non-linear integrable differential equations, using operator identities to analyze and solve related direct and inverse problems.

Contribution

It introduces a novel application of the Method of Operator Identities to study singular solutions and their dynamics in integrable equations like sinh-Gordon, NLS, and mKdV.

Findings

Singularities can be modeled as interacting particles.

Differential equations for singularity dynamics are derived and solved.

Numerous examples illustrate the methodology's effectiveness.

Abstract

This paper concerns with some of the results related to the singular solutions of certain types of non-linear integrable differential equations (NIDE) and behavior of the singularities of those equations. The approach heavily relies on the Method of Operator Identities which proved to be a powerful tool in different areas such as interpolation problems, spectral analysis, inverse spectral problems, dynamic systems, non-linear equations. We formulate and solve a number of problems (direct and inverse) related to the singular solutions of sinh-Gordon, non-linear Schr\"{o}dinger and modified Korteweg - de Vries equations. Dynamics of the singularities of these solutions suggests that they can be interpreted in terms of particles interacting through the fields surrounding them. We derive differential equations describing the dynamics of the singularities and solve some of the related problems. The developed methodologies are illustrated by numerous examples.