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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:
[1] DOI: 10.1063/1.527815 · Zbl 0641.35073 · doi:10.1063/1.527815
[2] DOI: 10.1063/1.527421 · Zbl 0658.35081 · doi:10.1063/1.527421
[3] DOI: 10.1063/1.527750 · Zbl 0658.35082 · doi:10.1063/1.527750
[4] DOI: 10.1063/1.1666399 · Zbl 0257.35052 · doi:10.1063/1.1666399
[5] DOI: 10.1063/1.524208 · Zbl 0422.35014 · doi:10.1063/1.524208
[6] DOI: 10.1143/JPSJ.52.3713 · doi:10.1143/JPSJ.52.3713
[7] DOI: 10.1063/1.862493 · Zbl 0408.76009 · doi:10.1063/1.862493
[8] DOI: 10.1143/JPSJ.52.2277 · doi:10.1143/JPSJ.52.2277
[9] DOI: 10.1143/JPSJ.53.1221 · doi:10.1143/JPSJ.53.1221
[10] DOI: 10.1143/JPSJ.53.1634 · doi:10.1143/JPSJ.53.1634
[11] DOI: 10.1002/sapm197960173 · Zbl 0412.35075 · doi:10.1002/sapm197960173
[12] DOI: 10.1088/0305-4470/16/16/002 · Zbl 0542.35066 · doi:10.1088/0305-4470/16/16/002
[13] DOI: 10.1016/0375-9601(83)90865-4 · doi:10.1016/0375-9601(83)90865-4
[14] DOI: 10.1016/0375-9601(83)90865-4 · doi:10.1016/0375-9601(83)90865-4
[15] DOI: 10.1016/0375-9601(81)90423-0 · doi:10.1016/0375-9601(81)90423-0
[16] DOI: 10.1088/0305-4470/12/4/019 · Zbl 0399.45014 · doi:10.1088/0305-4470/12/4/019
[17] DOI: 10.1143/JPSJ.46.681 · doi:10.1143/JPSJ.46.681
[18] DOI: 10.1103/PhysRevLett.43.264 · doi:10.1103/PhysRevLett.43.264
[19] DOI: 10.1016/0375-9601(79)90779-5 · doi:10.1016/0375-9601(79)90779-5
[20] DOI: 10.1016/0375-9601(83)90944-1 · doi:10.1016/0375-9601(83)90944-1
[21] DOI: 10.1063/1.528134 · Zbl 0648.76006 · doi:10.1063/1.528134
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