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arXiv · 2409.05752

A Lax representation, symmetries, and conservation laws for the three-dimensional Euler--Helmholtz equations

Oleg I. Morozov
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Abstract

We consider the three-dimensional Euler--Helmholtz equations for an inviscid incompressible fluid. Under the Poincar{\'{e}} lemma assumption, the incompressibility condition is resolved by introducing a vector potential, leading to a vorticity-type reformulation of the system. The main result is the construction of a Lax representation for the obtained system. We compute the Lie algebra of point symmetries and find the zeroth-order cosymmetries together with the associated local conservation laws. Using the Lax representation, we derive a shadow of cosymmetry and find nonlocal conservation laws using the construction of the canonical conservation law on the Whitney sum of the tangent and the cotangent coverings. Finally, we establish a B\"acklund transformation between the tangent and the cotangent coverings and describe its action on symmetries and cosymmetries.

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Oleg I. Morozov. 2024-09-09. A Lax representation, symmetries, and conservation laws for the three-dimensional Euler--Helmholtz equations. https://arxiv.org/abs/2409.05752

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