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Symmetry analysis and exact solutions of the $2+1$ dimensional sine-Gordon system. (English) Zbl 0976.35067
Summary: According to the space variable exchange symmetry, we change the $2+ 1$-dimensional sine-Gordon system obtained by Konopelchenko and Rogers to a variant form. Some types of similarity reductions are obtained by using some Lie symmetry analysis. From these similarity reductions, we find that the soliton structure of the system possesses quite a rich structure. The line solitons parallel to the lines $x+ y=0$ and $x-y= 0$ may have an arbitrary shape. In addition to the well-known dromion solution, which is constructed by two line solitons, one may find many other kinds of soliton solutions localized in all directions. For instance, some kinds of ring type (basin-like, plateau-like and bowl-like) and instanton type soliton solutions, can be found directly by selecting the arbitrary functions included in the “single” soliton solution. The dromion solutions may also be constructed by straight line and curved line solitons.

MSC:
35Q53KdV-like (Korteweg-de Vries) equations
37K40Soliton theory, asymptotic behavior of solutions
35L70Nonlinear second-order hyperbolic equations
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