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The two loop soliton solution of the Vakhnenko equation. (English) Zbl 0914.35115

Summary: An exact two loop soliton solution to the Vakhnenko equation

x t+u xu+u=0

is found. The key step in finding this solution is to transform the independent variables in the equation. This leads to a transformed equation for which it is straightforward to find an exact explicit 2-soliton solution by use of Hirota’s method. The exact two loop soliton solution to the Vakhnenko equation is then found in implicit form by means of a transformation back to the original independent variables. The nature of the interaction between the two loop solitons depends on the ratio of their amplitudes.

MSC:
35Q51Soliton-like equations
35C05Solutions of PDE in closed form