Kadomtsev-Petviashvili system and reduction: generalized Cauchy matrix approach

TL;DR

This paper introduces a generalized Cauchy matrix approach using the Sylvester equation to find exact solutions for various Kadomtsev-Petviashvili (KP) equations and their reductions to other integrable systems.

Contribution

It develops a novel matrix-based method for solving KP equations and their reductions, expanding the toolkit for integrable systems analysis.

Findings

Derived explicit solutions for KP equations using the approach.

Established reduction techniques to KdV, Boussinesq, and extended Boussinesq systems.

Connected the matrix M to tau-functions via determinant expressions.

Abstract

By the Sylvester equation $\bL\bM-\bM\bK=\br\bs^{\st}$ together with an evolution equation set of $\br$ and $\bs$, generalized Cauchy matrix approach is established to investigate exact solutions for Kadomtsev-Petviashvili system, including Kadomtsev-Petviashvili equation, modified Kadomtsev-Petviashvili equation and Schwarzian Kadomtsev-Petviashvili equation. The matrix $\bM$ provides $\tau$-function by $\tau=|\bI+\bM\bC|$. With the help of some recurrence relations, the reduction to Korteweg-de Vries system, Boussinesq system and extended Boussinesq system are also discussed.