Solitone solutions complexifications of the Korteweg - de Vriz equation

Abstract

The Hirota method for construction of soliton solutions is applied to the complexification of the Korteweg-de Vries equation. To use the method, the complex equation is replaced by a system of two third-order equations into two real functions, which, using the Hirota differential operator, is reduced to a bilinear form that is quadratic in the functions considered. The existence of a one-soliton solution is proved, the real part of which has the form of a soliton, and the imaginary part is a kink. It is proved that the use of the classical perturbation theory approach does not make it possible to construct a two-soliton solution. A special connection between unknown functions is found, which made it possible to reduce the system to a single bilinear equation for which a two-soliton solution is constructed. It is shown that the obtained Hirota polynomial does not satisfy the required properties, which led to the impossibility of constructing a three-soliton solution.