On solutions to a novel non-evolutionary integrable 1+1 PDE

Giglio, F. , Landolfi, G. and Martina, L. (2023) On solutions to a novel non-evolutionary integrable 1+1 PDE. Journal of Physics A: Mathematical and Theoretical, 56(48), 485205. (doi: 10.1088/1751-8121/ad04a5)

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 Painlevé 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.

• Item Type: Articles
• Additional Information: This work was supported by EPSRC grant no EP/R014604/1. F G also acknowledges the hospitality of the Lecce’s division of INFN and of the Department of 22 J. Phys. A: Math. Theor. 56 (2023) 485205 F Giglio et al Mathematics and Physics ‘Ennio De Giorgi’ of the University of Salento. G L also acknowledges the hospitality of the School of Mathematics and Statistics of the University of Glasgow. G L and L M are partially supported by INFN IS-MMNLP.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Giglio, Dr Francesco
• Authors: Giglio, F., Landolfi, G., and Martina, L.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of Physics A: Mathematical and Theoretical
• Publisher: IOP Publishing
• ISSN: 1751-8113
• ISSN (Online): 1751-8121
• Published Online: 07 November 2023
• Copyright Holders: Copyright: © 2023 The Author(s)
• First Published: First published in Journal of Physics A: Mathematical and Theoretical 56(48): 485205
• Publisher Policy: Reproduced under a Creative Commons licence