First-order differential constraints for the nonlinear wave equation:wave fronts, admissible nonlinearities and exact solutions
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.
Related
- Normalized Solutions for a Class of the Nonlinear Elliptic Systems with Mixed-Power Nonlinearities on Bounded Domains
- Existence and Nonexistence of Solutions of Certain type of Nonlinear Differential and Differential-Difference Equations
- Explicit meromorphic solutions of certain types of nonlinear differential-difference equations
- Meromorphic solutions of a nonlinear differential-difference equation with two exponential terms
- Infinitely many normalized solutions of L2-supercritical NLS equations on noncompact metric graphs with localized nonlinearities
BibTeX
@misc{kaptsov2026firstorder,
title = {{First-order differential constraints for the nonlinear wave equation:wave fronts, admissible nonlinearities and exact solutions}},
author = {Oleg Kaptsov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30629},
eprint = {2610.30629},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30629},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - First-order differential constraints for the nonlinear wave equation:wave fronts, admissible nonlinearities and exact solutions
AU - Oleg Kaptsov
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30629
UR - https://arxiv.org/abs/2610.30629
AB - 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.
ER -