A semi-discrete scheme derived from variational principles for global conservative solutions of a Camassa–Holm system. (17th February 2021)

• Record Type: Journal Article
• Authors: Galtung, Sondre Tesdal; Raynaud, Xavier
• Is Part Of: Nonlinearity. Volume 34:Number 4(2021)
• Page Start: 2220
• Page End: 2274
• Publication Date: 2021-02-17
• Subjects: Camassa–Holm equation -- two-component Camassa–Holm system -- calculus of variations -- Lagrangian coordinates -- energy-preserving discretizations -- discrete Green's functions -- discrete Sturm–Liouville operators; 35Q51 -- 35A15 -- 37K58 (Primary) -- 39A60 -- 65M80 (Secondary); Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/abc101
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: We define a kinetic and a potential energy such that the principle of stationary action from Lagrangian mechanics yields a Camassa–Holm system (2CH) as the governing equations. After discretizing these energies, we use the same variational principle to derive a semi-discrete system of equations as an approximation of the 2CH system. The discretization is only available in Lagrangian coordinates and requires the inversion of a discrete Sturm–Liouville operator with time-varying coefficients. We show the existence of fundamental solutions for this operator at initial time with appropriate decay. By propagating the fundamental solutions in time, we define an equivalent semi-discrete system for which we prove that there exists unique global solutions. Finally, we show how the solutions of the semi-discrete system can be used to construct a sequence of functions converging to the conservative solution of the 2CH system.