Abstract
Soliton-like solution in the scalar -theory that is called kink (antikink) belongs to the most widely studied solutions on nonlinear equations of mathematical physics. This is a non-singular stable configuration that is conserved in collisions and has a finite energy. Kink (antikink) may be characterized by means of conserved topological index (topological charge) that is determined by behavior of solution at space infinities. The equation of motion in (1þ 1)-dimensional case in -theory has a form tt xx m þ 3 1⁄4 0; ð1Þ where m and are positive constants, tt 1⁄4 @ =@t and so on. Kink (antikink) is the alone solution for this equation. The multi-kink solutions are absent. Hence, it is a nonintegrable equation. The equation of motion in the theory belongs to the non-integrable equations of Klein– Gordon type. Only two equations of such a type (sineGordon equation and Tzitzeica equation) have the Backlund transformations that generate the multi-kink solutions. If in the -theory take a damping into account by introducing the term proportional to t, the equation of motion takes a form
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