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<p>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 <i>g</i> and its loop algebra <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\tilde{g}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>g</mi> <mo stretchy="false">~</mo> </mover> </math></EquationSource> </InlineEquation>, and further obtain the bi-Hamiltonian structure of the resulting integrable couplings using the quadratic-form identity. Moreover, by extending the matrix Lie algebra <i>g</i> to a non-semisimple Lie algebra <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\widehat{g}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>g</mi> <mo stretchy="true">^</mo> </mover> </math></EquationSource> </InlineEquation>, we derive a nonisospectral multi-component integrable coupling hierarchy, and establish the Hamiltonian structure of its corresponding isospectral hierarchy using the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(Z_{N}^{\varepsilon }\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>Z</mi> <mrow> <mi>N</mi> </mrow> <mi>ε</mi> </msubsup> </math></EquationSource> </InlineEquation>-trace identity.</p>

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

  • 期刊论文

  • 发表日期：

    2025年12月13日

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  • Zhenbo Wang,
  • Yufeng Zhang

摘要

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 \(\tilde{g}\) g ~ , 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 \(\widehat{g}\) g ^ , we derive a nonisospectral multi-component integrable coupling hierarchy, and establish the Hamiltonian structure of its corresponding isospectral hierarchy using the \(Z_{N}^{\varepsilon }\) Z N ε -trace identity.