First-order differential constraints for the nonlinear wave equation:wave fronts, admissible nonlinearities and exact solutions
Year: 2026
Type: preprint
Abstract: We construct exact solutions of the nonlinear wave equation in three space dimensions by the method of linear determining equations: first-order constraint functions are found as solutions of the linear determining equation, the differential constraints obtained by setting them equal to zero are integrated, and the resulting ansatz is substituted into the equation. Lie point symmetries are a special case, and the solutions obtained generically have no symmetry. The determining equations tie the admissible nonlinearity to the geometry of the constraint: arbitrary on characteristic hyperplanes, cubic on light cones, and, for a pair of constraints, exponential on light cones and quintic on the cylindrical fronts of a line source. Among the solutions are smooth bounded sine-Gordon walls and vortex configurations whose shape varies arbitrarily along a front, the fields of point sources moving on arbitrary trajectories for the cubic and Liouville equations, with an exact nonlinear analogue of the Larmor formula for the power radiated in the cubic case, and smooth finite-energy solutions of the cubic equation with any prescribed number of toroidal nodal surfaces.
Source: arXiv (Cornell University)
Author: Kaptsov, Oleg
Language: English
Topic: Nonlinear Waves and Solitons
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223368285