Solution of the Boussinesq equation using evolutionary vessels

TL;DR

This paper introduces a novel approach using evolutionary vessels to solve the Boussinesq equation, producing a wide variety of solutions including solitons, Schwartz class solutions, and those with singularities, extending existing theories.

Contribution

It develops a more flexible theory of non-symmetric evolutionary vessels for complex solutions and broadens the class of solutions for the Boussinesq equation beyond traditional methods.

Findings

Derived formulas include solitons and Schwartz class solutions

Constructed solutions with singularities on a closed set Z

Extended the theory of evolutionary vessels for complex-valued solutions

Abstract

In this work we present a solution of the Boussinesq equation. The derived formulas include solitons, Schwartz class solutions and solutions, possessing singularities on a closed set Z of the (x,t) domain, obtained from the zeros of the tau function. The idea for solving the Boussinesq equation is identical to the (unified) idea of solving the KdV and the evolutionary NLS equations: we use a theory of evolutionary vessels. But a more powerful theory of non-symmetric evolutionary vessels is presented, inserting flexibility into the construction and allowing to deal with complex-valued solutions. A powerful scattering theory of Deift-Tomei-Trubowitz for a three dimensional operator, which is used to solve the Boussinesq equation, fits into our setting only in a particular case. On the other hand, we create a much wider class of solutions of the Boussinesq equation with singularities on a closed set $Z$.