The existence of solution to wave equations on Legendrian curves

Abstract

The wave on a string has basically been studied in such a way that the movement of the string is in two dimensions and the solution to wave equation is obtained by D’Alembert’s Formula. This work focuses on a wave on a curve of a string where the Legendrian base curve is a straight line x-axis. The ambient space of the curve is three dimensional space together with the standard contact structure. The Legendrian condition and the initial conditions of the vector wave equation on 1-manifold are the key to acquire the theorem which leads to the necessary conditions for the existence of nontrivial Legendrian wave. The result turns out that fixed initial position on xz-plane, zero initial position along xy-plane, zero initial velocity on xy-plane and xz-plane including zero sum of the external force along y-axis are the necessary conditions. An example of the existence of such wave has been given. The study brings the conclusion that there exists a Legendrian wave on the x-axis Legendrian base curve.

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