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A search for bilinear equations passing Hirota’s three-soliton condition. III: Sine-Gordon-type bilinear equations. (English) Zbl 0658.35082
[For part II see the preceding review.]
The results of a search for pairs of bilinear equations of the type \[ A^ i(D_ x,D_ t)F\cdot F+B^ i(D_ x,D_ t)G\cdot F+C^ i(D_ x,D_ t)G\cdot G=0,\quad i=1,2, \] which have standard type three- soliton solutions, are presented. The freedom to rotate in (F,G) space is fixed by the one-soliton ansatz \(F=1\), \(G=e^ n\), then the \(B^ i\) determine the dispersion manifold while \(A^ i\) and \(C^ i\) are auxiliary functions. It is assumed that \(B^ 1\) and \(B^ 2\) are even and proportional, and that \(A^ i\) and \(C^ i\) are quadratic. As new results, \(B^ 1=aD^ 3_ xD_ t+D_ tD_ y+b,\quad A^ 2=-C^ 2=D_ xD_ t,\) and generalizations of the sine-Gordon model \(B^ 1=D_ xD_ t+a\) with a family of auxiliary functions \(A^ i\) and \(C^ i\) are obtained.

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
[2] DOI: 10.1063/1.527421 · Zbl 0658.35081
[3] DOI: 10.1143/JPSJ.33.1459
[4] DOI: 10.1143/PTP.52.1498 · Zbl 1168.37322
[5] DOI: 10.1143/PTP.57.797 · Zbl 1098.81547
[6] DOI: 10.1143/JPSJ.41.1091 · Zbl 1334.35284
[7] DOI: 10.1143/JPSJ.41.1091 · Zbl 1334.35284
[8] DOI: 10.1143/JPSJ.41.1091 · Zbl 1334.35284
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[10] DOI: 10.1143/JPSJ.40.611 · Zbl 1334.76016
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