Hietarinta, Jarmo A search for bilinear equations passing Hirota’s three-soliton condition. III: Sine-Gordon-type bilinear equations. (English) Zbl 0658.35082 J. Math. Phys. 28, 2586-2592 (1987). [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. Cited in 2 ReviewsCited in 22 Documents MSC: 35Q99 Partial differential equations of mathematical physics and other areas of application Keywords:bilinear equations; Hirota’s three-soliton condition; sine-Gordon-type bilinear equations; three-soliton solutions; one-soliton ansatz; dispersion manifold; sine-Gordon model Citations:Zbl 0641.35073; Zbl 0658.35081 PDFBibTeX XMLCite \textit{J. Hietarinta}, J. Math. Phys. 28, 2586--2592 (1987; Zbl 0658.35082) Full Text: DOI 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: 10.1143/JPSJ.33.1459 · doi:10.1143/JPSJ.33.1459 [4] DOI: 10.1143/PTP.52.1498 · Zbl 1168.37322 · doi:10.1143/PTP.52.1498 [5] DOI: 10.1143/PTP.57.797 · Zbl 1098.81547 · doi:10.1143/PTP.57.797 [6] DOI: 10.1143/JPSJ.41.1091 · Zbl 1334.35284 · doi:10.1143/JPSJ.41.1091 [7] DOI: 10.1143/JPSJ.41.1091 · Zbl 1334.35284 · doi:10.1143/JPSJ.41.1091 [8] DOI: 10.1143/JPSJ.41.1091 · Zbl 1334.35284 · doi:10.1143/JPSJ.41.1091 [9] DOI: 10.1143/JPSJ.40.611 · Zbl 1334.76016 · doi:10.1143/JPSJ.40.611 [10] DOI: 10.1143/JPSJ.40.611 · Zbl 1334.76016 · doi:10.1143/JPSJ.40.611 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.