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A search for bilinear equations passing Hirota’s three-soliton condition. IV: Complex bilinear equations. (English) Zbl 0684.35082
Summary: [For part III see ibid. 28, 2586-2592 (1987; Zbl 0658.35082).]
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 $$B(D_ x,...)G\cdot F=0,$$ $$A(D_ x,D_ t)F\cdot F=GG^*,$$ and
(b) the Benjamin-Ono equation $$P(D_ x,...)F\cdot F^*=0.$$
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 $$P\leq 4$$. The main results are the following:
(1) $$(iaD^ 3_ x+D_ xD_ t+iD_ y+b)G\cdot F=0$$, $$D^ 2_ xF\cdot F=GG^*;$$
(2) $$(D^ 2_ x+aD^ 2_ y+iD_ t+b)G\cdot F=0$$, $$D_ xD_ yF\cdot F=GG^*;$$
(3) $$(iaD^ 3_ x+D^ 2_ x+iD_ t)F\cdot F^*=0$$; and
(4) $$(D_ xD_ t+i(aD_ x+bD_ t))F\cdot F^*=0$$.

##### MSC:
 35Q99 Partial differential equations of mathematical physics and other areas of application
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##### References:
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