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

TL;DR

The paper addresses the multiplicity of one-troughed travelling waves in a nonlinear fourth-order boundary value problem modeling an asymmetrically supported beam with a jumping nonlinearity. It constructs travelling-wave solutions via a reduction to a piecewise-linear ODE, encapsulated by the scalar function $L(c,p,\theta)$, and reduces the search to solving $L(c,p,\theta)=k$ for integers $k$. The main contributions include a rigorous upper bound of at most six even one-troughed travelling waves for admissible speeds, plus a visualization-driven approach suggesting the actual maximum is five; an algorithm to count and visualize the number of such waves in the $(c,p)$ plane is provided. The work substantially informs multiplicity questions for beam models with non-smooth nonlinearities and highlights open problems in characterizing solution counts and symmetry properties.

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.

Figures (6)

• Figure 1: On the left, the illustration of the border line of the area corresponding to the necessary condition $b/a < c^4/4 < 1$ (in black) and sufficient conditions $b/a < 9c^4/100$ from holubova_leva (in gray) and $b/a < \beta^{\ast}(a/b)c^4/4$ from holubova_leva_necesal (in orange) for the existence of travelling wave solution. On the right, the left picture is rescaled onto a rectangle by $b/a = p c^4/4$ with $p \in (0,1)$.
• Figure 2: On the left, the illustration of the process of finding $\theta_{min}$. On the right, the dependence of $\theta_{min}$ on $c$.
• Figure 3: On the left, the illustration of the empty interval for admissible $\theta$. The orange area corresponds to the situation where $\theta_{min}\geq \theta_2$, the black curve is determined by $\theta_{min}=\theta_1$ and the gray curve $p_0$ is given by \ref{['eq:p0']}. On the right, the illustration of strips in the $cp$ plane given by \ref{['eq:pk']} with different upper bounds for the number of solutions.
• Figure 4: Number of one-troughed travelling waves for particular pairs $c$ and $p$, with $\beta^{\ast}$ from holubova_leva_necesal. For every $p<\beta^{\ast}$ there exists a travelling wave solution.
• Figure 5: Graph of function $L$ with $c=0.61005$ and $p=0.00065$ on the left. Graph of function $L$ with $c=0.99$ and $p=0.4$ on the right.

Theorems & Definitions (10)

• Lemma 1
• proof
• Corollary 2
• Lemma 3
• proof
• Remark 4
• Lemma 5
• proof
• Theorem 6
• proof