Periodic localized traveling waves in the two-dimensional suspension bridge equation

TL;DR

This paper develops computer-assisted proof methods to rigorously establish the existence of periodic localized traveling waves in the two-dimensional suspension bridge equation, extending prior work from one dimension.

Contribution

It introduces a novel, rigorous computational approach to prove localized traveling waves in 2D nonlinear wave equations, overcoming challenges from exponential nonlinearities.

Findings

Proved existence of 2D localized traveling waves.

Provided explicit bounds on numerical approximation errors.

Method applicable to other nonlinear wave and elliptic PDEs.

Abstract

In the dynamics generated by the suspension bridge equation, traveling waves are an essential feature. The existing literature focuses primarily on the idealized one-dimensional case, while traveling structures in two spatial dimensions have only been studied via numerical simulations. We use computer-assisted proof methods based on a Newton-Kantorovich type argument to find and prove periodic localized traveling waves in two dimensions. The main obstacle is the exponential nonlinearity in combination with the resulting large amplitude of the localized waves. Our analysis hinges on establishing computable bounds to control the aliasing error in the computed Fourier coefficients. This leads to existence proofs of different traveling wave solutions, accompanied by small, explicit, rigorous bounds on the deficiency of numerical approximations. This approach is directly extendable to other wave equation models and elliptic partial differential equations with analytic nonlinearities, in two as well as in higher dimensions.