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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 [9] DOI: 10.1143/JPSJ.40.611 · Zbl 1334.76016 [10] DOI: 10.1143/JPSJ.40.611 · Zbl 1334.76016
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