Multitime Rayleigh Solitons

TL;DR

This paper introduces multitime Rayleigh PDEs and constructs soliton solutions demonstrating complex deformation behaviors, extending classical single-time PDE theory to multiple evolution variables with stability analysis.

Contribution

It formulates multitime Rayleigh PDEs, constructs their soliton solutions, and analyzes their stability, advancing the understanding of multitime evolution equations.

Findings

Derived multitime Rayleigh PDEs with geometric methods

Constructed stable multitime Rayleigh solitons

Extended PDE theory from single-time to multitime cases

Abstract

Multitime evolution PDEs for Rayleigh waves are considered, using geometrical ingredients capable to build an ultra-parabolic-hyperbolic differential operator. Their soliton solutions are found based on appropriate hypotheses and specific ODEs. These multitime solitons develop complex behavior of deformation phenomena. The original results include: the form of multitime Rayleigh PDEs, the construction of multitime Rayleigh solitons via some significant amounts of analysis and the stability of multitime Rayleigh solitons, which are stable enough to persist indefinitely. In this context we survey some of the highlights of multitime PDEs theory, from the more classical single-time case, to the more recent multitime case, as well as current developments in using this theory to rigorously prove the sense for several evolution variables and PDEs.