Dispersionless Limits of Some Integrable 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.

Dispersionless Davey-Stewartson system: Lie symmetry algebra, symmetry group and exact solutions

“…The Benney system of equations, originally introduced in [15], has been obtained in [16,17] as the quasiclassical limit of NLS equations, which are integrable by the inverse scattering method. Dispersionless limits of several equations in (1+1), (2+1) and (3+1) dimensions including Heisenberg ferromagnet equation, Landau-Lifshitz equation, KdV, NLS and DS equations are listed in [18]. A discussion of dispersionless limits of integrable systems can be found in [19], which involves Kadomtsev-Petviashvili (KP), generalized Benney and Davey-Stewartson (DS) equations.…”