The Initial Value Problem for Wave Equation and a Poisson-like Integral in Hyperbolic Plane

TL;DR

This paper explores the initial value problem for the wave equation in a hyperbolic plane, deriving a Poisson-like integral formula using a geometry aligned with the wave equation's symmetries.

Contribution

It introduces a novel approach by studying the wave equation within a geometry generated by the equation itself, leading to new solutions and integral formulas.

Findings

Derived a Poisson-like integral formula for the wave equation

Identified differences between wave and Laplace equations in hyperbolic geometry

Provided new solutions based on initial data on hyperbolic arms

Abstract

In recent time, by working in a plane with the metric associated with wave equation (the Special Relativity non-definite quadratic form), a complete formalization of space-time trigonometry and a Cauchy-like integral formula have been obtained. In this paper the concept that the solution of a mathematical problem is simplified by using a "mathematics" with the symmetries of the problem, actuates us for studying the wave equation (in particular "the initial values problem") in a plane where the geometry is the one "generated" by the wave equation itself. In this way, following a classical approach, we point out the well known differences with respect to Laplace equation notwithstanding their formal equivalence (partial differential equations of second order with constant coefficients) and also show that the same conditions stated for Laplace equation allow us to find a new solution. In particular taking as "initial data" for the wave equation an arbitrary function given on an arm of an equilateral hyperbola, a "Poisson-like" integral formula holds.