Notes on number of one-troughed travelling waves in asymmetrically supported bending beam

TL;DR

This paper investigates the existence and bounds of one-troughed travelling wave solutions in a nonlinear fourth-order PDE modeling asymmetrically supported bending beams, providing new insights into solution multiplicity and parameter effects.

Contribution

It establishes the existence of multiple one-troughed travelling waves for specific parameters and offers bounds and visualizations for these solutions.

Findings

Existence of at least two different one-troughed travelling waves for certain parameters.

Upper bounds for the number of such solutions.

Visual representations of solutions and parameter effects.

Abstract

We study the boundary value problem for nonlinear fourth-order partial differential equation with jumping nonlinearity which can serve, e.g., as a model of an asymmetrically supported bending beam. We focus on a special type of solutions, the so-called one-troughed travelling waves. The main goal of this paper is to show the existence of at least two different one-troughed travelling waves for particular wave speeds and input parameters of the studied problem. We present the upper bounds for the maximal number of one-troughed solutions together with a visualisation of obtained results and corresponding solutions. Finally, we list several open questions regarding this topic.