Existence of solutions to non-homogeneous higher order differential equation in the Schwartz space

TL;DR

This paper establishes necessary and sufficient conditions for the existence of solutions to higher even order non-homogeneous differential equations within the Schwartz space, relevant to soliton studies in integrable PDEs.

Contribution

It provides a rigorous proof using Fourier transform and pseudodifferential operators for the existence criteria of solutions in the Schwartz space.

Findings

Derived necessary and sufficient conditions for solutions

Applied Fourier transform and pseudodifferential operator theory

Focused on higher even order differential equations

Abstract

There is studied problem on existence of solutions to non-homogeneous differential equation of higher even order. Similar problem arises while studying soliton and soliton-like solutions to partial differential equations of integrable type. By means of Fourier transform and theory of pseudodifferential operators there is proved the theorem on necessary and sufficient conditions on existence of solutions to linear non-homogeneous differential equation of higher even order in the Schwartz space.