A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations

Abstract

In this paper the results of a search for bilinear equations of the type P(Dx, Dt)F ⋅ F=0, which have three-soliton solutions, are presented. Polynomials up to order 8 have been studied. In addition to the previously known cases of KP, BKP, and DKP equations and their reductions, a new polynomial P=DxDt(Dx2 +√3DxDt+Dt2) +aDx2+bDxDt +cDt2 has been found. Its complete integrability is not known, but it has three-soliton solutions. Infinite sequences of models with linear dispersion manifolds have also been found, e.g., P=DxMDtNDyP, if some powers are odd, and P=DxMDtN(Dx2 −1)P, if M and N are odd.