A search for bilinear equations passing Hirota’s three-soliton condition. IV. Complex bilinear equations

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Abstract
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 Schrödinger 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.