A search for bilinear equations passing Hirota’s three-soliton condition. III. Sine–Gordon-type bilinear equations

Abstract

In this paper the results of a search for pairs of bilinear equations of the type Ai(Dx,Dt)F⋅F +Bi(Dx,Dt)G⋅F +Ci(Dx,Dt)G⋅G=0, i=1,2, which have standard type three-soliton solutions, are presented. The freedom to rotate in (F,G) space is fixed by the one-soliton ansatz F=1, G=en, then the Bi determine the dispersion manifold while Ai and Ci are auxiliary functions. In this paper it is assumed that B1 and B2 are even and proportional, and that Ai and Ci are quadratic. As new results, B1=aD3x Dt +DtDy+b, A2=−C2=DxDt, and generalizations of the sine–Gordon model B1=DxDt+a with a family of auxiliary functions Ai and Ci are obtained.