Perturbations of Differential Equations Retaining Conserved Quantities
Tóm tắt
The paper deals with perturbations of the equation that have a number of conservation laws. When a small term is added to the equation its conserved quantities usually decay at individual rates, a phenomenon known as a selective decay. These rates are described by the simple law using the conservation laws’ generating functions and the added term. Yet some perturbation may retain a specific quantity(s), such as energy, momentum and other physically important characteristics of solutions. We introduce a procedure for finding such perturbations and demonstrate it by examples including the KdV–Burgers equation and a system from magnetodynamics. Some interesting properties of solutions of such perturbed equations are revealed and discussed.
Tài liệu tham khảo
A. V. Samokhin, ‘‘Decay velocity of conservation laws for nonevolution equations,’’ Acta Appl. Math. 41, 1–11 (1995).
A. C. Ting, M. H. Matthaeus, and D. Montgomery, ‘‘Turbulent relaxation processes in magnetohydrodynamics,’’ Phys. Fluids 29, 3261–3274 (1986).
E. van Groesen and F. Mainardi, ‘‘Balance laws and centro velocity in dissipative systems,’’ J. Math. Phys. 31, 2136–2140 (1990).
J. B. Taylor, ‘‘Relaxation of toroidal plasma and generation of reverse magnetic fields,’’ Phys. Rev. Lett. 33, 1139–1141 (1974).
R. Brecht, W. Bauer, A. Bihlo, F. Gay-Balmaz, and S. MacLachlan, ‘‘Selective decay for the rotating shallow-water equations with a structure-preserving discretization,’’ Phys. Fluids 33, 116604 (2021). https://doi.org/10.1063/5.0062573
A. V. Samokhin, ‘‘The KdV soliton crosses a dissipative and dispersive border,’’ J. Differ. Geom. Appl. 75, 101723 (2021). https://doi.org/10.1016/j.difgeo.2021.101723
A. V. Samokhin, ‘‘Taylor trick and travelling wave solutions,’’ Lobachevskii J. Math. 43, 2808–2815 (2022). https://doi.org/10.1134/S1995080222130406