On solutions to a novel non-evolutionary integrable 1+1 PDE
Abstract
We investigate real solutions of a C-integrable non-evolutionary partial differential equation in the form of a scalar conservation law where the flux density depends both on the density and on its first derivatives with respect to the local variables. By performing a similarity reduction dictated by one of its local symmetry generators, a nonlinear ordinary differential equation arises that is connected to the Painlev\'e III equation. Exact solutions are secured and described provided a constraint holds among the coefficients of the original equation. In the most general case, we pinpoint the generation of additional singularities by numerical integration. Then, we discuss the evolution of given initial profiles. Finally, we mention aspects concerning rational solutions with a finite number of poles.
Cite
@article{arxiv.2310.08940,
title = {On solutions to a novel non-evolutionary integrable 1+1 PDE},
author = {Francesco Giglio and Giulio Landolfi and Luigi Martina},
journal= {arXiv preprint arXiv:2310.08940},
year = {2023}
}
Comments
Some references added. The section previously named Appendix A has been removed; the relevant equations there contained have been moved at the beginning of Section III. Subsection II.A. now appears as Subsection III.A and most of the material previously appearing as Section III has been cast as the successive Subsection III.B. Figures 3 and 4 now appears in the correct sequential order