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A search for bilinear equations passing Hirota’s three-soliton condition. II. MKdV-type bilinear equations. (English) Zbl 0658.35081
[For part I see ibid. 28, 1732-1742 (1987; Zbl 0641.35073).]
In this paper (second in a series) the search for bilinear equations having three-soliton solutions continues. This time pairs of bilinear equations of type \(P_ 1(D_ x,D_ t)F\cdot G=0\), \(P_ 2(D_ x,D_ t)F\cdot G=0\), where \(P_ 1\) is an odd polynomial and \(P_ 2\) is quadratic, are considered. The main results are the following new bilinear systems: \[ P_ 1=aD^ 7_ x+bD^ 5_ x+D^ 2_ x D_ t+D_ y,\quad P_ 2=D^ 2_ x;\quad P_ 1=aD^ 3_ x+bD^ 3_ 1+D_ y,\quad P_ 2=D_ xD_ t; \] and \[ P_ 1=D_ xD_ tD_ y+aD_ x+bD_ t,\quad P_ 2=D_ xD_ t. \] In addition to these, several models with linear dispersion manifolds were obtained, as before.

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
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[1] DOI: 10.1063/1.527815 · Zbl 0641.35073 · doi:10.1063/1.527815
[2] DOI: 10.1143/JPSJ.33.1456 · doi:10.1143/JPSJ.33.1456
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