A combined Liouville integrable hierarchy associated with a sixth-order matrix spectral problem and its integrable reductions

Yadong Zhong, Jingjing Ge, Yi Zhang

School of Mathematical Sciences, Zhejiang Normal University, Jinhua, Zhejiang, China

Abstract: We solve a unique matrix spectral problem that encompasses eight distinct potentials and subsequently derive a corresponding soliton hierarchy using the zero curvature representation. Moreover, we establish a bi-Hamiltonian framework by applying the trace identity, thereby emphasizing the Liouville integrability of the derived soliton hierarchy. Two illustrative examples are provided, including generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations, to showcase the applicability and significance of the proposed methodology. Finally, we obtain some integrable reductions within the derived soliton hierarchy.

Keywords: matrix eigenvalue problem, zero curvature equation, integrable hierarchy, integrable reduction.

Received: 09.01.2025
Revised: 09.01.2025

DOI: 10.4213/tmf10888

 English version: Theoretical and Mathematical Physics, 2025, 225:3, 2134–2166

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