Enlarging Maurer-Cartan form via Kronecker product and construction of Coupled Integrable systems by Nilpotent, Hadamard, Idempotent and K-idempotent matrix

TL;DR

This paper introduces a novel method for constructing coupled integrable systems using Kronecker products of special matrices and Lie algebraic elements, enabling flexible generation of multi-equation systems with Hamiltonian structure.

Contribution

The paper presents a new approach to generate coupled integrable systems via Kronecker products of nilpotent, Hadamard, idempotent, and k-idempotent matrices, expanding the toolkit for integrability construction.

Findings

Constructed coupled systems with closure properties of matrices.

Demonstrated Hamiltonian formulation using trace identity.

Established hereditary operators indicating integrability.

Abstract

Coupled nonlinear integrable systems are generated from usual zero curvature equation. The relevant Maurer-Cartan forms are constructed by combining suitably chosen matrices (nilpotent, Hadamard, idempotent and k-idempotent) and Lie algebraic elements via Kronecker product. In each case a closure type property among the matrices chosen is found to be playing a key role to produce both the coupling and nonlinearity present in the system of equations obtained. The method is highly flexible and can be used to construct general systems containing 'p' number of equations. It is also shown that these new equations can be written in the Hamiltonian form (with a preassigned symplectic operator) with the trace identity introduced by Tu. Since the Lax operator is known one can obtain the hereditary operators signifying the complete integrability. Various properties of Kronecker product are found to be useful in our construction.