Nonlinear evolution equations, rescalings, model PDES and their integrability: I

Abstract

The authors study the problem of wave modulation for a large and quite general class of nonlinear evolution equations. They demonstrate that only a very limited number of 'universal' model equations, on relevant time and space scales, describe the phenomena of interest under all circumstances. Classical among the model equations is of course the nonlinear Schrodinger equation (NLS); however, under certain conditions, modulations occur on shorter time and space scales than those relevant for the NLS. On the other hand, if the NLS becomes linear by cancellation of terms, then appropriate model equations exist on longer time and space scales. The limited number of model equations suggests that they should have wide applicability. The method of analysis consists basically of Fourier decomposition followed by rescalings and appropriate limits. If one adheres to the principle that integrability properties are inherited through such limit procedures, then the model equations should be integrable (in some sense) under a wide range of conditions. Through their investigation of integrability properties, they find this expectation largely confirmed.

Classification of polynomial integrable systems of mixed scalar and vector evolution equations: I

“…This system is identical to (12) in [25]. Foursov claimed in [25] that this system was either reduced to the representative case α = 0 (a = 1) or decoupled by a linear change of dependent variables.…”

“…12 The computation was too memory demanding to be performed on the computers available to one of the authors in 2000.…”

“…Systems of this sort are believed to be linearizable by an appropriate Change of variables and, if so, said to be "C-integrable" in the terminology of Calogero [12,13].…”

Multiscale Expansion and Integrability Properties of the Lattice Potential KdV Equation

“…Reductive perturbation techniques [19,20] have proved to be important tools for finding approximate solutions of many physical problems, by reducing a given nonlinear partial differential equation to a simpler equation, often integrable [3], and for proving integrability [3][4][5]10,21]. Recently, after various attempts to carry over this approach to partial difference equations [1,11,13] we have presented a procedure for carrying out a multiscale expansion on the lattice [7,12,14] which seems to preserve the integrability properties [8].…”

The Matrix Kadomtsev­Petviashvili Equation as a Source of Integrable Nonlinear Equations

“…Moreover, the same model equations obtained in this way appear in many applicative situations (for instance in plasma physics, nonlinear optics, hydrodynamics, etc. ), where weakly nonlinear effects are important [2][3][4][5].…”