Dispersionless Limits of Some Integrable Equations

TL;DR

This paper provides a pedagogical overview of dispersionless integrable equations across multiple dimensions, summarizes known results, introduces new dispersionless magnetic equations, and discusses their Lax representations for beginners.

Contribution

It offers a comprehensive, accessible introduction to dispersionless integrable systems, including new results on magnetic equations and detailed Lax formulations.

Findings

Detailed exposition of classical dispersionless systems like dKdV, dNLSE, dKPE, dDSE

Introduction of new dispersionless magnetic equations

Development of Lax representation for dispersionless equations

Abstract

This is a write-up of the lectures on dispersionless equations in 1+1, 2+1 and 3+1 dimensions presented by one of us (RM) at "Eurasian International Center for Theoretical Physics" (EICTP). We provide pedagogical introduction to the subject and summarize well-known results and some recent developments in theory of integrable dispersionless equations. Almost all results presented are available in the literatures. We just add some new results related to dispersionless limits of integrable magnetic equations. On the contrary, some of the basic tools of the integrable dispersionless systems and related theoretical techniques are not available in a pedagogical format. For this reason we think that it seems worthwhile to present basics of theory of dispersionless equations here for the benefit of beginner students. We begin with a detailed exposition of the well-known dispersionless systems like dKdVE, dNLSE, dKPE, dDSE and other classical soliton equations. We present in detail some new dispersionless systems. Next we develop the dispersionless limits of known magnetic equations. Lastly, we discuss in full detail the Lax representation formulation of some presented dispersionless equations. On the basis of the material presented here one can proceed smoothly to read the recent developments in this field of integrable dispersionless equations and related topics.