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A search for bilinear equations passing Hirota’s three-soliton condition. I: KdV-type bilinear equations. (English) Zbl 0641.35073
The results of a search for bilinear equations of the type \(P(D_ x,D_ t)F\cdot F=0\), 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 \[ P=D_ xD_ t(D^ 2_ x+\sqrt{3D_ xD_ t}+D^ 2_ t)+aD^ 2_ x+bD_ xD_ t+cD^ 2_ t \] 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., \(P=D^ M_ x D^ N_ t D^ P_ y\), if some powers are odd, and \(P=D^ M_ x D^ N_ t(D^ 2_ x-1)^ P\), if M and N are odd.

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35G20 Nonlinear higher-order PDEs
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