Year-end Sale % OFF 
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.
Highlights
The Camassa–Holm (CH) equation ut − utxx + 3uux − 2uxuxx − uuxxx = 0, (1.1)is first known to have appeared in [25], written in an alternative form, as a special case in a hierarchy of completely integrable partial differential equations
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
Summary
Is first known to have appeared in [25], written in an alternative form, as a special case in a hierarchy of completely integrable partial differential equations. We will not follow this approach here, but rather use the fact that (1.2) can be derived as the governing equation for a different variational problem, where the action functional includes a potential energy term and the variation is performed on the group of diffeomorphisms only. This point of view enables us to derive a discretization which mimics the variational derivation of the continuous case. In (e) we plot the distribution of the potential energy given by (ρ(t, x) − ρ∞), and observe how it grows when the peaks get closer to each other
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
