Lower Bounds for Admissible Values of the Travelling Wave Speed in Asymmetrically Supported Beam

TL;DR

This paper investigates the range of wave speeds allowing traveling wave solutions in an asymmetrically supported beam model, revealing restrictions due to negative solution parts and connecting to Fučík spectra.

Contribution

It establishes the maximal admissible wave speed range using variational methods, links it to Dirichlet problems and Fučík spectra, and offers analytical approximations and conjectures.

Findings

Identified maximal wave speed range for existence of solutions.

Connected wave speed restrictions to Fučík spectra.

Provided analytical approximations and posed a conjecture.

Abstract

We study the admissible values of the wave speed $c$ for which the beam equation with jumping nonlinearity possesses a travelling wave solution. In contrast to previously studied problems modelling suspension bridges, the presence of the term with negative part of the solution in the equation results in restrictions of $c$. In this paper, we provide the maximal wave speed range for which the existence of the travelling wave solution can be proved using the Mountain Pass Theorem. We also introduce its close connection with related Dirichlet problems and their Fu\v{c}\'{i}k spectra. Moreover, we present several analytical approximations of the main existence result with assumptions that are easy to verify. Finally, we formulate a conjecture that the infimum of the admissible wave speed range can be described by the Fu\v{c}\'{i}k spectrum of a simple periodic problem.