Multi-component Integrable Couplings and Bi-Hamiltonian Structures of the Modified Jaulent-Miodek Hierarchy

Abstract

In this paper, we derive a nonisospectral modified Jaulent-Miodek (mJM) hierarchy and construct the bi-Hamiltonian structure of its corresponding isospectral hierarchy via the trace identity. To generate integrable couplings of the mJM hierarchy, we introduce a new matrix Lie algebra g and its loop algebra

\tilde{g}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tilde{g}$$\end{document}

, and further obtain the bi-Hamiltonian structure of the resulting integrable couplings using the quadratic-form identity. Moreover, by extending the matrix Lie algebra g to a non-semisimple Lie algebra

\widehat{g}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\widehat{g}$$\end{document}

, we derive a nonisospectral multi-component integrable coupling hierarchy, and establish the Hamiltonian structure of its corresponding isospectral hierarchy using the

Z_N^{\epsilon }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Z_{N}^{\varepsilon }$$\end{document}

-trace identity.

Keywords

Nonisospectral integrable hierarchy / Hamiltonian structure / Lie algebra / Multi-component integrable hierarchy / 37K05 / 37K10 / 37K40 / 35Q53

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Funding

National natural Science Foundation of China(11971475)

RIGHTS & PERMISSIONS

Shanghai University

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