On a matrix KdV6 equation

Gordoa, Pilar R.; Pickering, Andrew; Wattis, Jonathan A.D.

Abstract

The so-called KdV6 equation has, since its discovery, been the subject of much interest. In this paper we present a matrix version of this equation. On the one hand, we use the Darboux transformation to derive its Bäcklund transformation and a nonlinear superposition formula. These are then used, in the case of upper-triangular matrices, to obtain one- and two-soliton solutions. We find that wave components can combine to produce rogue waves. On the other hand, we derive a second matrix partial differential equation, for which we give auto-Bäcklund transformations of a different kind, similar to those usually given for Painlevé equations.

Citation

Gordoa, P. R., Pickering, A., & Wattis, J. A. (2025). On a matrix KdV6 equation. Communications in Nonlinear Science and Numerical Simulation, 143, Article 108605. https://doi.org/10.1016/j.cnsns.2025.108605

• Journal Article Type: Article
• Acceptance Date: Jan 6, 2025
• Online Publication Date: Jan 12, 2025
• Publication Date: Apr 1, 2025
• Deposit Date: Jan 15, 2025
• Publicly Available Date: Jan 15, 2025
• Journal: Communications in Nonlinear Science and Numerical Simulation
• Print ISSN: 1007-5704
• Electronic ISSN: 1878-7274
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 143
• Article Number: 108605
• DOI: https://doi.org/10.1016/j.cnsns.2025.108605
• Keywords: Matrix kdV6, Rogue waves, Bäcklund transformations, Multisoliton solutions
• Public URL: https://nottingham-repository.worktribe.com/output/44228693
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S1007570425000164