Investigation of breaking dynamics for Riemann waves in shallow water. (March 2020)

• Record Type: Journal Article
• Authors: Saleh, R.; Kassem, M.; Mabrouk, S.M.
• Is Part Of: Chaos, solitons and fractals. Volume 132(2020)
• Publication Date: 2020-03
• Subjects: Riemann waves -- Extended Calogero–Bogoyavlenskii–Schiff equation -- Nonlocal symmetry -- Singular manifold method -- Schwarzian derivative; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2019.109571
• Languages: English
• ISSNs: 0960-0779

Abstract

Highlights: Two extensions of Calogero–Bogoyavlenskii–Schiff equation are investigated. A sequential nonlocal symmetry reduced PDEs to ODEs. Singular manifold method is applied to solve the resulting ODEs. Solution of the first extension shows an increasing train of cuspons with amplitude decrease through time evolution. Solution of the second extension shows a double kink wave with decreasing amplitude through time increasing. Abstract: Breaking dynamics of Riemann waves, has a profound impact on studying the dynamics of shock-breaking systems. Calogero– Bogoyavlenskii– Schiff (CBS) equation describes the interaction between Riemann propagating wave along y-axis with long propagating wave along x-axis. Two extensions of the CBS (2+1)-dimensional equation are investigated. Optimization of the commutative product for the nonlocal symmetry is constructed and two successive symmetry reductions reduced the equations to ordinary differential equations (ODEs). Hidden symmetries are used in the second reduction step. The singular manifold method (SMM) is then applied to the ordinary differential equations. This results in a Schwarzian derivative whose solution leads to Cuspon and Kink waves.