The existence and uniqueness of global admissible conservative weak solution for the periodic single-cycle pulse equation

TL;DR

This paper establishes the existence and uniqueness of global admissible conservative weak solutions for the periodic single-cycle pulse equation by transforming it into a semilinear system and analyzing characteristic curves.

Contribution

It introduces a novel transformation to a semilinear system and proves the global existence and uniqueness of solutions without extra assumptions.

Findings

Proved global existence of solutions.

Established uniqueness of solutions.

Provided a method to identify characteristic curves.

Abstract

This paper is devoted to the study of the existence and uniqueness of global admissible conservative weak solutions for the periodic single-cycle pulse equation. We first transform the equation into an equivalent semilinear system by introducing a new set of variables. Using the standard ordinary differential equation theory, we then obtain the global solution to the semilinear system. Next, returning to the original coordinates, we get the global admissible conservative weak solution for the periodic single-cycle pulse equation. Finally, given an admissible conservative weak solution, we find a equation to single out a unique characteristic curve through each initial point and prove the uniqueness of global admissible conservative weak solution without any additional assumptions.