Solution of the KdV equation using evolutionary vessels

Abstract

In this work we present a new method for solving of the Korteweg-de Vries (KdV) equation q'_t = - \dfrac{3}{2} q q'_x + \dfrac{1}{4} q"'_{xxx}. The proposed method is a particular case of the theory of evolutionary vessels, developed in this work. Inverse scattering of the Sturm-Liouville operator and evolution of its potential are the basic ingredients, similar to the existing methods developed by Gardner-Greene-Kruskal-Miura (1967), Zacharov-Shabbath (1974) and Peter Lax (1977). Evolutionary KdV vessel may be considered as a generalization of these previous works. The advantage of the new method is that it produces a unified approach to existing solutions of the KdV equation. For example, odd or even analytic, periodic, almost periodic solutions are shown to be particular cases of this theory. Generalizing this method we can also produce many PDEs, associated with integrable systems, in an arbitrary number of variables (in the spirit of Zakarov-Shabat).