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Oleg I. Morozov

Publications and source records attributed to Oleg I. Morozov.

At least 19 recordsLinked to original sources
arXiv2026

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

We consider the three-dimensional Euler--Helmholtz equations for an inviscid incompressible fluid. Under the Poincar{é} 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 Bäcklund transformation between the tangent and the cotangent coverings and describe its action on symmetries and cosymmetries.

nlin.SI↗
arXiv2025

Nonlocal conservation laws for the two-dimensional Euler equation in vorticity form

We combine the construction of the canonical conservation law and the nonlocal cosymmetry to derive a collection of nonlocal conservation laws for the two-dimensional Euler equation in vorticity form. For computational convenience and simplicity of presentation of the results we perform a complex rotation of the independent variables.

math.AP↗
arXiv2025

Lax representations for the magnetohydrodynamics equations

We find two Lax representations for the reduced magnetohydrodynamics equations ({\sc rmhd}) and derive a B{ä}cklund transformation between the tangent and the cotangent coverings of these equations. Then, we study the action of the B{ä}cklund transformation on the second-order cosymmetries and the action of its inverse on the Lie symmetries of {\sc rmhd}. The action of the inverse transformation produces another Lax representation for {\sc rmhd}. The reduction of {\sc rmhd} by the symmetry of shifts along the $z$-axis coincides with the equations of two-dimensional ideal magnetohydrodynamics ({\sc imhd}). Applied to the Lax representations and the B{ä}cklund transformation of {\sc rmhd}, the reduction provides analogous constructions for {\sc imhd}. The action of the inverse B{ä}cklund transformation on the Lie symmetries of {\sc imhd} is expressed in terms of a new four-parameter Lax representation of these equations.

nlin.SI↗
arXiv2024

Extensions of the symmetry algebra and Lax representations for the two-dimensional Euler equation

We find the twisted extensions of the symmetry algebra of the 2D Euler equation in the vorticity form and use them to construct new Lax representation for this equation. Then we generalize this result by considering the transformation Lie--Rinehart algebras generated by finite-dimensional subalgebras of the symmetry algebra and derive a family of Lax representations for the Euler equation. The family depends on functional parameters and contains a non-removable spectral parameter.

nlin.SI↗
arXiv2022

Lax representations via twisted extensions of infinite-dimensional Lie algebras: some new results

We apply the technique of twisted extensions of infinite-dimensional Lie algebras to find new 3D integrable {\sc pde}s related to the deformations of Lie algebra $\mathbb{R}_N[s]\otimes \mathfrak{w}$ with $N=1, 2$ as well as to the Lie algebra $\mathfrak{h} \oplus \mathfrak{w}$, where $\mathbb{R}_N[s]$ is the algebra of truncated polynomials of degree $N$, $\mathfrak{w}$ is the Lie algebra of polynomial vector fields on $\mathbb{R}$ and $\mathfrak{h}$ is the Lie algebra of polynomial Hamiltonian vector fields on $\mathbb{R}^2$.

nlin.SI↗
arXiv2020

Isospectral deformation of the reduced quasi-classical self-dual Yang--Mills equation

We derive new four-dimensional partial differential equation with the isospectral Lax representation by shrinking the symmetry algebra of the reduced quasi-classical self-dual Yang--Mills equation. Then we find a recursion operator for the obtained equation and construct B{ä}cklund transformations between this equation and the reduced quasi-classical self-dual Yang--Mills equation as well as the four-dimensional Mart{\'ı}nez Alonso--Shabat equation

nlin.SI↗
arXiv2020

Integrability structures of the generalized Hunter--Saxton equation

We consider integrability structures of the generalized Hunter--Saxton equation. In particular, we obtain the Lax representation with nonremovable spectral parameter, find local recursion operators for symmetries and cosymmetries, generate an infinite-dimensional Lie algebra of higher symmetries, and prove existence of infinite number of cosymmetries of higher order. Further, we give an example of employing the higher order symmetry to constructing exact globally defined solutions for the generalized Hunter--Saxton equation.

nlin.SI↗
arXiv2019

Lax representations with non-removable parameters and integrable hierarchies of PDEs via exotic cohomology of symmetry algebras

This paper develops the technique of constructing Lax representations for PDEs via non-central extensions generated by non-triivial exotic 2-cocycles of their contact symmetry algebras. We show that the method is applicable to the Lax representations with non-removable spectral parameters. Also we demonstrate that natural extensions of the symmetry algebras produce the integrable hierarchies associated to their PDEs.

nlin.SI↗
arXiv2018

Deformations of infinite-dimensional Lie algebras, exotic cohomology and integrable nonlinear partial differential equations. II

We consider the four-dimensional reduced quasi-classical self-dual Yang--Mills equation and show that non-triviality of the second exotic cohomology group of its symmetry algebra implies existence of a two-component integrable generalization of this equation. The sequence of natural extensions of this symmetry algebra generate an integrable hierarchy of multi-dimensional nonlinear PDEs. We write out the first three elements of this hierarchy.

nlin.SI↗
arXiv2017

Deformations of infinite-dimensional Lie algebras, exotic cohomology, and integrable nonlinear partial differential equations

The important unsolved problem in theory of integrable systems is to find conditions guaranteeing existence of a Lax representation for a given PDE. The use of the exotic cohomology of the symmetry algebras opens a way to formulate such conditions in internal terms of the PDEs under the study. In this paper we consider certain examples of infinite-dimensional Lie algebras with nontrivial second exotic cohomology groups and show that the Maurer-Cartan forms of the associated extensions of these Lie algebras generate Lax representations for integrable systems, both known and new ones.

nlin.SI↗
arXiv2016

The Gibbons--Tsarev equation: symmetries, invariant solutions, and applications

In this paper we present the full classification of the symmetry-invariant solutions for the Gibbons--Tsarev equation. Then we use these solutions to construct explicit expressions for reductions of Benney's moments equations, to get solutions of Pavlov's equation, and to find integrable reductions of the Ferapontov--Huard--Zhang system, which describes implicit two-phase solutions of the dKP equation.

nlin.SI↗
arXiv2014

The four-dimensional Martinez Alonso - Shabat equation: differential coverings and recursion operators

We apply Cartan's method of equivalence to find a contact integrable extension for the structure equations of the symmetry pseudo-group of the four-dimensional Martinez Alonso - Shabat equation. From the extension we derive two differential coverings including coverings with one and two non-removable parameters. Then we apply the same approach to construct a recursion operator for symmetries of the equation under study.

nlin.SI↗