An exactly solvable problem of wave fronts and applications to the asymptotic theory

TL;DR

This paper presents an exactly solvable 4D Hamiltonian system applied to wave front problems and asymptotic analysis of PDEs, with explicit solutions for wave and Helmholtz equations in variable depth basins.

Contribution

It introduces a nontrivially solvable Hamiltonian model for wave front propagation in variable depth basins and derives explicit asymptotic solutions for related PDEs.

Findings

Explicit solutions for wave fronts in variable depth basins.

Asymptotic solutions for wave and Helmholtz equations with localized sources.

Application of the Hamiltonian system to realistic basin shapes.

Abstract

This is the full and extended version of the brief note arXiv:1908.00938. A nontrivially solvable 4-dimensional Hamiltonian system is applied to the problem of wave fronts and to the asymptotic theory of partial differential equations. The Hamilton function we consider is $H(\mathbf x,\mathbf p)=\sqrt{D(\mathbf{x})}|\mathbf{p}|$. Such Hamiltonians arise when describing the fronts of linear waves generated by a localized source in a basin with a variable depth. We consider two \emph{realistic} types of bottom shape: 1) the depth of the basin is determined, in the polar coordinates, by the function $D(\varrho,\varphi)=(\varrho^2+b)/(\varrho^2+a)$ and 2) the depth function is $D(x,y)=(x^2+b)/(x^2+a)$. As an application, we construct the asymptotic solution to the wave equation with localized initial conditions and asymptotic solutions of the Helmholtz equation with a localized right-hand side.