The two loop soliton solution of the Vakhnenko equation.

*(English)*Zbl 0914.35115Summary: An exact two loop soliton solution to the Vakhnenko equation

$$\frac{\partial}{\partial x}\left(\frac{\partial}{\partial t}+u\frac{\partial}{\partial x}\right)u+u=0$$

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.