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<p>We find two Lax representations for the reduced magnetohydrodynamics equations (<span>rmhd</span>) and construct a local variational Poisson structure (a Hamiltonian operator) for them. Its inverse defines a nonlocal symplectic structure for the same equations. We describe the action of both operators on the second-order cosymmetries and on the infinitesimal contact symmetries of <span>rmhd</span>, respectively. The reduction of <span>rmhd</span> by the symmetry of shifts along the <i>z</i>-axis coincides with the equations of two-dimensional ideal magnetohydrodynamics (<span>imhd</span>). Applied to the Lax representations and the variational Poisson structure of <span>rmhd</span>, the reduction provides analogous constructions for <span>imhd</span>.</p>

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Lax representations and variational Poisson structures for magnetohydrodynamics equations

  • 期刊论文

  • 发表日期：

    2025年8月28日

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  • Oleg I. Morozov

摘要

We find two Lax representations for the reduced magnetohydrodynamics equations (rmhd) and construct a local variational Poisson structure (a Hamiltonian operator) for them. Its inverse defines a nonlocal symplectic structure for the same equations. We describe the action of both operators on the second-order cosymmetries and on the infinitesimal contact symmetries of rmhd, respectively. The reduction of rmhd by the symmetry of shifts along the z-axis coincides with the equations of two-dimensional ideal magnetohydrodynamics (imhd). Applied to the Lax representations and the variational Poisson structure of rmhd, the reduction provides analogous constructions for imhd.