The Existence and Uniqueness of Global Admissible Conservative Weak Solution for the Periodic Single-Cycle Pulse Equation

This paper is devoted to studying the existence and uniqueness of global admissible conservative weak solution for the periodic single-cycle pulse equation without any additional assumptions. Firstly, introducing a new set of variables, we transform the single-cycle pulse equation into an equivalent semilinear system. Using the standard ordinary differential equation theory, the global solution of the semilinear system is studied. Secondly, returning to the original coordinates, we get a global admissible conservative weak solution for the periodic single-cycle pulse equation. Finally, choosing some vital test functions which are different from [Bressan (Discrete Contin. Dyn. Syst 35:25-42, 2015), Brunelli (Phys. Lett. A 353:475-478, 2006)], we find a equation to single out a unique characteristic curve through each initial point. Moreover, the uniqueness of global admissible conservative weak solution is obtained.