A semi-discrete scheme derived from variational principles for global conservative solutions of a Camassa–Holm system
Peer reviewed, Journal article
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Original versionNonlinearity. 2021, 34 (4), 2220-2274. 10.1088/1361-6544/abc101
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.