On the Study of the Klein-Gordon Equation in the Dunkl Setting

TL;DR

This paper investigates the Dunkl-Klein-Gordon equation, deriving an integral solution representation and analyzing its properties, including energy considerations, within the generalized Fourier analysis framework of Dunkl theory.

Contribution

It introduces an integral representation for solutions of the Dunkl-Klein-Gordon equation and explores their properties, including energy analysis, extending classical PDE results to Dunkl settings.

Findings

Derived an integral formula for the solution

Analyzed properties of the solution in Dunkl setting

Studied energy associated with the Dunkl-Klein-Gordon equation

Abstract

In Dunkl theory on $\mathbb{R}^{n}$ which generalizes classical Fourier analysis, we study the solution of the Klein-Gordon-equation defined by: \begin{eqnarray} \nonumber \partial_{t}^{2}u-\Delta_{k}u=-m^{2}u \ , \ \ \ u (x,0)=g(x) \ , \ \ \ \partial_{t}u(x,0)=f(x) \end{eqnarray} with \ $m > 0$ \ and \ $\partial_{t}^{2}u$ \ is the second derivative of the solution $u$ with respect to $t$ and $\Delta_{k}u$ is the Dunkl Laplacian with respect to $x$ where $f$ and $g$ the two functions in $\mathcal{S}(\mathbb{R}^{n})$ which surround the initial conditions. We obtain an integral representation for its solution which we gives some properties. As a specific result, we studied the associated energies to the Dunkl-Klein-Gordon equation.