Alexey Shevyakov | Searcharxiv

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

Higher-order Approximate Symmetries

Fushchych and Shtelen defined approximate symmetry by expanding the solution of a perturbed differential equation in the small parameter, replacing the original equation by a triangular system for the expansion coefficients. We extend this construction to arbitrary order. A coefficient-map formulation yields the order-$p$ Fushchych--Shtelen (FS) system and establishes, under explicit hypotheses, a correspondence with the Baikov--Gazizov--Ibragimov (BGI) method. The two frameworks are compared for a perturbed cubic wave equation. Up to equivalences, we classify the nonlinearities admitting an FS dilation through second order. The classification consists of a generic power family and logarithmic branches at exceptional exponents. Among perturbations nontrivial at first order, vertical BGI continuations select a proper subfamily of one FS branch, and a second common branch arises when the perturbation first enters at second order; the two methods are related but not interchangeable. Joint reduction by the Lorentz algebra and the inherited FS dilation integrates all resulting FS normal forms; the explicit solutions of the 1989 Fushchych--Shtelen letter are recovered as the first-order members of this reduction. In the part common to both frameworks, the logarithmic corrections arise from the expansion of a power law with a perturbation-dependent exponent. Finally, periodic travelling waves are used to examine the long-scale validity of the FS expansion. The first correction is obtained by quadrature, the second-order secular structure is isolated, and phase renormalization produces bounded order-consistent approximations. For integer-power members of the generic family, the amplitude dependence of the corrected wavenumber agrees with the scaling weights obtained from the symmetry classification.

math.AP↗