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Kostya Druzhkov

Publications and source records attributed to Kostya Druzhkov.

9 recordsLinked to original sources
arXiv2026

Computational Algorithms for Invariant Reduction of Variational Forms

Symmetry reductions of partial differential equations (PDEs) inherit more geometric structures than other types of reductions: invariant conservation laws, variational structures, and, under suitable conditions, Hamiltonian-type structures of the original model descend to the reduced model through the mechanism of invariant reduction. This paper develops computational algorithms that carry out such reductions explicitly. We use the interpretation of variational $p$-forms as conservation laws of an enlarged system --- the (degree-shifted) tangent system --- consisting of the original equations together with their linearizations, in which the perturbation variables are treated as anticommuting. This allows us to formulate the algorithms using well-known concepts from the theory of conservation laws, naturally adapted to the graded-commutative setting. The reduction of conservation laws, variational $1$-forms, and presymplectic structures thereby becomes a single algorithmic procedure. We present (i) a homotopy-based reduction algorithm for systems of evolution equations, implemented in Maple; (ii) a descent reduction algorithm applicable to general $\ell$-normal systems for $p>0$; and (iii) a simple reduction algorithm available for point symmetries under suitable conditions, including the description of reductions in terms of systems involving fewer independent variables. Examples include nonlinear evolution equations in one and two spatial dimensions, the Laplace equation, the incompressible Euler equations, and the cotangent system of Pavlov's equation.

math-ph↗
arXiv2026

Invariant Reduction for Partial Differential Equations. III: Poisson Brackets

We show that, under suitable conditions, finite-dimensional systems describing invariant solutions of partial differential equations (PDEs) inherit local Hamiltonian operators through the mechanism of invariant reduction, which applies uniformly to point, contact, and higher symmetries. The inherited operators endow the reduced systems with Poisson bivectors that relate constants of motion to symmetries. Applying the same mechanism to invariant conservation laws, we further show that the induced Poisson brackets agree with those of the original systems, up to sign. The results are illustrated by two examples in which the inherited Poisson brackets and inherited constants of motion yield integrability of the reduced systems. The construction is independent of the choice of an $\ell$-normal inclusion of a PDE system into jet spaces.

nlin.SI↗
arXiv2026

Invariant Reduction for Partial Differential Equations. II: The General Framework

For a system of partial differential equations (PDEs) $F = 0$ admitting a local (point, contact, or higher) symmetry $X$ with the characteristic $φ$, invariant solutions satisfy the reduced system $F = φ= 0$. We propose a framework that allows, for every $X$-invariant conservation law, presymplectic structure, variational principle, or another geometric structure of the given PDE system $F = 0$, to systematically calculate its corresponding reduced form that describes the corresponding structure for the reduced system $F = φ= 0$. In particular, we show in what way Noether's theorem holding for the given PDE system is inherited by the reduced PDE system. We consider several detailed examples, including cases of point and higher symmetry invariance. The proposed framework is directly applicable to a wide range of PDE models, including complex PDE systems of contemporary interest arising across disciplines, where symmetry reduction is essential for analysis and simulation, as well as to integrable, Lagrangian, and gauge systems.

nlin.SI↗
arXiv2026

Invariant Reduction for Partial Differential Equations. IV: Symmetries that Rescale Geometric Structures

For a system of partial differential equations admitting point, contact, or higher symmetries, the framework of invariant reduction systematically computes how invariant geometric structures, such as conservation laws, presymplectic structures, variational principles, and Poisson brackets, are inherited by the systems governing symmetry-invariant solutions. We extend this mechanism to geometric structures that are not invariant but are $\textit{rescaled}$ by a symmetry. Specifically, if $X$ is the symmetry used for reduction, $X_s$ is a symmetry satisfying $[X_s,X]=aX$, and the Lie derivative $\mathcal{L}_{X_s}$ acts on an $X$-invariant element of the Vinogradov $\mathcal{C}$-spectral sequence as multiplication by $b$, then the restricted symmetry $X_s|_{\mathcal{E}_X}$ acts on the corresponding reduction as multiplication by $a+b$. This shift rule gives rise to two phenomena: the $\textit{emergence of invariance}$, where reductions acquire an invariance that was not present at the level of the original structure, and the $\textit{loss of invariance}$, where reductions of invariant structures are no longer invariant. As an application, we describe a class of exact solutions to systems possessing sufficiently many symmetries and conservation laws subject to certain compatibility conditions. These solutions are invariant under pairs of symmetries and are completely determined by explicitly constructed functions that are constant on them; the description is geometric and does not require any integrability-related structures such as Lax pairs. The framework is illustrated by two examples: the Lin--Reissner--Tsien equation of potential nonstationary transonic gas flows, for which closed-form exact solutions are obtained and validated numerically, and the potential Boussinesq system, for which the inherited Poisson bracket is employed to describe solutions determined by algebraic equations.

nlin.SI↗
arXiv2024

Invariant Reduction for Partial Differential Equations. I: Conservation Laws and Systems with Two Independent Variables

For a system of partial differential equations that has an extended Kovalevskaya form, a reduction procedure is presented that allows one to use a local (point, contact, or higher) symmetry of a system and a symmetry-invariant conservation law to algorithmically calculate constants of motion holding for symmetry-invariant solutions. Several examples including cases of point and higher symmetry invariance are presented and discussed. An implementation of the algorithm in Maple is provided.

nlin.SI↗
arXiv2024

Internal Lagrangians and spatial-gauge symmetries

A direct reformulation of the Hamiltonian formalism in terms of the intrinsic geometry of infinitely prolonged differential equations is obtained. Concepts of spatial equation and spatial-gauge symmetry of a Lagrangian system of equations are introduced. A non-covariant canonical variational principle is proposed and demonstrated using the Maxwell equations as an example. A covariant canonical variational principle is formulated. The results obtained are applicable to any variational equations, including those that do not originate in physics.

math-ph↗
arXiv2023

Internal Lagrangians of PDEs as variational principles

A description of how the principle of stationary action reproduces itself in terms of the intrinsic geometry of variational equations is proposed. A notion of stationary points of an internal Lagrangian is introduced. A connection between symmetries, conservation laws and internal Lagrangians is established. Noether's theorem is formulated in terms of internal Lagrangians. A relation between non-degenerate Lagrangians and the corresponding internal Lagrangians is investigated. Several examples are discussed.

math-ph↗
arXiv2022

Lagrangian formalism and the intrinsic geometry of PDEs

A notion of internal Lagrangian for a system of differential equations is introduced. A spectral sequence related to internal Lagrangians is obtained. A connection between internal Lagrangians and presymplectic structures is investigated. An interpretation of the term $E^{3,\, n-2}_2$ of Vinogradov's $\mathcal{C}$-spectral sequence is given for irreducible gauge theories.

math-ph↗