A new bidirectional generalization of (2+1)-dimensional matrix k-constrained KP hierarchy

TL;DR

This paper introduces a comprehensive bidirectional generalization of the (2+1)-dimensional matrix k-constrained KP hierarchy, unifying various integrable systems and providing methods for exact solution construction, including multi-soliton solutions.

Contribution

It presents a new hierarchy (2+1)-BDk-cKPH that generalizes existing hierarchies and includes new matrix (2+1)-dimensional integrable systems.

Findings

Contains matrix generalizations of DS, mKdV, and Nizhnik equations.

Includes new matrix (2+1)-dimensional generalizations of Yajima-Oikawa and Melnikov systems.

Provides a binary Darboux transformation method for exact solutions, including multi-soliton solutions.

Abstract

We introduce a new bidirectional generalization of (2+1)-dimensional k-constrained KP hierarchy ((2+1)-BDk-cKPH). This new hierarchy generalizes (2+1)-dimensional k-cKP hierarchy, $(t_A,\tau_B)$ and $(\gamma_A,\sigma_B)$ matrix hierarchies. (2+1)-BDk-cKPH contains a new matrix (1+1)-k-constrained KP hierarchy. Some members of (2+1)-BDk-cKPH are also listed. In particular, it contains matrix generalizations of DS systems, (2+1)-dimensional modified Korteweg-de Vries equation and the Nizhnik equation. (2+1)-BDk-cKPH also includes new matrix (2+1)-dimensional generalizations of the Yajima-Oikawa and Melnikov systems. Binary Darboux Transformation Dressing Method is also proposed for construction of exact solutions for equations from (2+1)-BDk-cKPH. As an example the exact form of multi-soliton solutions for vector generalization of the Davey-Stewartson system is given.