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Alexander Zhalij

Publications and source records attributed to Alexander Zhalij.

10 recordsLinked to original sources
arXiv2026

Extended group analysis and conservation laws of a class of variable coefficient generalized Kawahara equations

We review and extend the results on the group analysis of a class of generalized Kawahara equations with time-dependent coefficients. First, we provide an overview of the existing literature on Lie symmetries and Lie-invariant solutions of such equations. We then present a complete description of their transformation properties, including admissible, equivalence, and Lie symmetry transformations. For practical applications, we further extend these results by presenting a complete Lie symmetry classification without simplifying the coefficients via equivalence transformations. Lie reductions are then systematically performed, and several exact solutions are constructed. Low-order local conservation laws are exhaustively classified: every equation in this class admits conservation of mass and the squared $L^2$ norm, whereas energy-type conservation laws exist only for specific coefficient branches that align with the cases singled out by the symmetry classification. Finally, the classification results are enhanced by a study of contractions, which link cases of Lie symmetry extensions together with the associated reductions and conservation laws.

math-ph↗
arXiv2020

Equivalence groupoid and enhanced group classification of a class of generalized Kawahara equations

Transformation properties of a class of generalized Kawahara equations with time-dependent coefficients are studied. We construct the equivalence groupoid of the class and prove that this class is not normalized but can be presented as a union of two disjoint normalized subclasses. Using the obtained results and properly gauging the arbitrary elements of the class, we carry out its complete group classification, which covers gaps in the previous works on the subject.

math-ph↗
arXiv2018

Classification of reduction operators and exact solutions of variable coefficient Newell-Whitehead-Segel equations

A class of the Newell-Whitehead-Segel equations (also known as generalized Fisher equations and Newell-Whitehead equations) is studied with Lie and "nonclassical" symmetry points of view. The classifications of Lie reduction operators and of regular nonclassical reduction operators are performed. The set of admissible transformations (the equivalence groupoid) of the class is described exhaustively. The criterion of reducibility of variable coefficient Newell-Whitehead-Segel equations to their constant coefficient counterparts is derived. Wide families of exact solutions for such variable coefficient equations are constructed.

math-ph↗
arXiv2009

Quantum integrable systems in three-dimensional magnetic fields: the Cartesian case

In this paper we construct integrable three-dimensional quantum-mechanical systems with magnetic fields, admitting pairs of commuting second-order integrals of motion. The case of Cartesian coordinates is considered. Most of the systems obtained are new and not related to the separation of variables in the corresponding Schrödinger equation.

math-ph↗
arXiv2005

Separable unsteady nonparallel flow stability problems

The so-called 'direct' approach to separation of variables in linear PDEs is applied to the hydrodynamic stability problem. Calculations are made for the complete linear stability equations in cylindrical coordinates. Several classes of the exact solutions of the Navier-Stokes equations describing spatially developing and unsteady flows, for which the linear stability problems can be rigorously reduced to eigenvalue problems of ordinary differential equations, are defined. Those exactly solvable nonparallel and unsteady flow stability problems can be used for testing approximate approaches and the methods based on direct numerical simulations of the (linearized) Navier-Stokes equations. The exact solutions of the viscous incompressible Navier-Stokes equations determined as the basic states, for which the linear stability problem is exactly separable, may be themselves of interest from theoretical and engineering points of view.

physics.flu-dyn↗
arXiv2002

On separable Pauli equations

We classify (1+3)-dimensional Pauli equations for a spin-1/2 particle interacting with the electro-magnetic field, that are solvable by the method of separation of variables. As a result, we obtain the eleven classes of vector-potentials of the electro-magnetic field A(t,x) providing separability of the corresponding Pauli equations. It is established, in particular, that the necessary condition for the Pauli equation to be separable into second-order matrix ordinary differential equations is its equivalence to the system of two uncoupled Schroedinger equations. In addition, the magnetic field has to be independent of spatial variables. We prove that coordinate systems and the vector-potentials of the electro-magnetic field providing the separability of the corresponding Pauli equations coincide with those for the Schroedinger equations. Furthermore, an efficient algorithm for constructing all coordinate systems providing the separability of Pauli equation with a fixed vector-potential of the electro-magnetic field is developed. Finally, we describe all vector-potentials A(t,x) that (a) provide the separability of Pauli equation, (b) satisfy vacuum Maxwell equations without currents, and (c) describe non-zero magnetic field.

math-ph↗
arXiv1999

On separable Schrödinger equations

We classify (1+3)-dimensional Schrödinger equations for a particle interacting with the electromagnetic field that are solvable by the method of separation of variables. As a result, we get eleven classes of the electromagnetic vector potentials of the electromagnetic field $A(t, \vec x)=(A_0(t, \vec x)$, $\vec A(t, \vec x))$ providing separability of the corresponding Schrödinger equations. It is established, in particular, that the necessary condition for the Schrödinger equation to be separable is that the magnetic field must be independent of the spatial variables. Next, we prove that any Schrödinger equation admitting variable separation into second-order ordinary differential equations can be reduced to one of the eleven separable Schrödinger equations mentioned above and carry out variable separation in the latter. Furthermore, we apply the results obtained for separating variables in the Hamilton-Jacobi equation.

math-ph↗
arXiv1999

On separable Fokker-Planck equations with a constant diagonal diffusion matrix

We classify (1+3)-dimensional Fokker-Planck equations with a constant diagonal diffusion matrix that are solvable by the method of separation of variables. As a result, we get possible forms of the drift coefficients $B_1(\vec x),B_2(\vec x),B_3(\vec x)$ providing separability of the corresponding Fokker-Planck equations and carry out variable separation in the latter. It is established, in particular, that the necessary condition for the Fokker-Planck equation to be separable is that the drift coefficients $\vec B(\vec x)$ must be linear. We also find the necessary condition for R-separability of the Fokker-Planck equation. Furthermore, exact solutions of the Fokker-Planck equation with separated variables are constructed

math-ph↗
arXiv1999

Separation of variables in the Kramers equation

We consider the problem of separation of variables in the Kramers equation admitting a non-trivial symmetry group. Provided the external potential $V(x)$ is at most quadratic, a complete solution of the problem of separation of variables is obtained. Furthermore, we construct solutions of the Kramers equation with separated variables in explicit form.

math-ph↗