The two loop soliton solution of the Vakhnenko equation

Abstract

An exact two loop soliton solution to the Vakhnenko equation 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.