Functions of noncommuting operators in an asymptotic problem for a 2D wave equation with variable velocity and localized right-hand side

TL;DR

This paper applies noncommutative operator theory to derive explicit asymptotic solutions for a 2D wave equation with variable velocity, demonstrating practical formulas for complex PDE problems like tsunami modeling.

Contribution

It introduces a novel approach using noncommuting operators to obtain explicit asymptotic solutions for variable-coefficient wave equations.

Findings

Derived explicit formulas for asymptotic solutions

Applied the method to tsunami wave modeling

Showed the practicality of abstract operator theory in PDEs

Abstract

We use the theory of functions of noncommuting operators (noncommutative analysis) to solve an asymptotic problem for a partial differential equation and show how, starting from general constructions and operator formulas that seem to be rather abstract from the viewpoint of differential equations, one can end up with very specific, easy-to-evaluate expressions for the solution, useful, e.g., in the tsunami wave problem.