In this paper the results of a search for complex bilinear equations with two‐soliton solutions are presented. The following basic types are discussed: (a) the nonlinear Schrodinger equation B(Dx, ...)G⋅F=0, A(Dx,Dt) F⋅F=GG*, and (b) the Benjamin–Ono equation P(Dx, ...)F⋅F*=0. It is found that the existence of two‐soliton solutions is not automatic, but introduces conditions that are like the usual three‐ and four‐soliton conditions. The search was limited by the degree of A=2, and by degree of P≤4. The main results are the following: (1) (iaD3x+DxDt +iDy+b)G⋅F=0, D2xF⋅F=GG*; (2) (D2x+aD2y +iDt+b)G⋅F =0, DxDy F⋅F=GG*; (3) (iaD3x+D2x +iDt)F⋅F*=0; and (4) (DxDt+i(aDx +bDt))F⋅F*=0.

A search for bilinear equations passing Hirota's three-soliton condition. II. mKdV-type bilinear equations

http://mathsci.ucd.ie/~dias/ZakharovDiasPushkarev.pdf

One-dimensional wave turbulence

Interaction Equations for Short and Long Dispersive Waves

Theoretical descriptions of solitons and weakly nonlinear waves propagating in plasma media are reviewed, with particular attention to the Korteweg-de Vries (KDV) equation and the Nonlinear Schrodinger equation (NLS). The modifications of these basic equations due to the effects of resonant particles and external magnetic fields are discussed.

Solitons and Weakly Nonlinear Waves in Plasmas

We study well-posedness of the Cauchy problem associated to the Schrodinger-Korteweg-de Vries system. We obtain local well-posedness for weak initial data, where the best result obtained is for data in the Sobolev space L 2 (R) x H -3 4+ . This result implies in particular the global well-posedness in the energy space H 1 (R) x H 1 (R). Both results considerably improve the previous ones by Bekiranov, Ogawa and Ponce (1997), Guo and Miao (1999), and Tsutsumi (1993).

https://www.ams.org/tran/2007-359-09/S0002-9947-07-04239-0/S0002-9947-07-04239-0.pdf

Well-posedness for the Schrödinger-Korteweg-de Vries system

To the integrability of the equations describing the Langmuir-wave-ion-acoustic-wave interaction

The perturbed Kadomtsev-Petviashvili equation is analysed for media with a split dispersion law For arbitrary perturbations the method of asymptotic description of the equation solutions is constructed Applications of this method are given for problems in hydrodynamics and plasma physics

S.P. Burtsev

Papers

To the integrability of the equations describing the Langmuir-wave-ion-acoustic-wave interaction

Perturbation theory for two-dimensional solitons