On completely integrable polynomial PDEs arising from Sturm-Liouville differential equation using evolutionary vessels. KdV Hierarchy

TL;DR

This paper develops a unified scheme for constructing solutions to polynomial integrable PDEs, generalizing inverse scattering methods for Sturm-Liouville problems, and introduces a hierarchy related to the KdV equation with explicit soliton solutions.

Contribution

It presents a novel scheme for solving polynomial integrable PDEs using evolutionary vessels, extending the inverse scattering approach to a broader class of equations including a KdV hierarchy.

Findings

Complete solution for type 0 equations.

Introduction of a KdV hierarchy related to polynomial PDEs.

Explicit soliton solutions for each evolutionary equation.

Abstract

In this work we present a scheme for construction of solutions for evolutionary PDEs of some polynomial types q'_t = P(q,q'_x,...), where P is a polynomial in a finite number of variables. This scheme is a generalization of the existing technique for solution of completely integrable PDEs using Inverse Scattering of the Sturm-Liouville differential equation. The KdV equation q'_t = - 3/2 q q'_x + 1/4 q"'_{xxx} is a special case, corresponding to type 1 evolutionary equations. We present a complete solution of type 0, and present a KdV hierarchy corresponding to infinite number of polynomial evolutionary equations rather for \beta = 1/2 \int_0^x q(y,t)dy then for q(x,t) itself, of the form \beta'_t = i^n b_n(\beta_x'), where b_0 = -1/4 \beta"'_{xxx} + 3/2 (\beta'_x)^2 corresponds to the KdV equation and 4 (b_{n+1})'_x = -i (b_n)_{xxx}"' + 4i (\beta'_xb_n)'_x. Soliton solutions (i.e. involving pure exponents only) are presented for each such evolutionary equation, demonstrating a "simplicity" of the solutions construction.