Existence proofs of traveling wave solutions on an infinite strip for the suspension bridge equation and proof of orbital stability

TL;DR

The paper addresses the existence and orbital stability of traveling wave solutions to the 2D suspension bridge equation on the infinite strip $\Omega=\mathbb{R}\times(-d_2,d_2)$, formulated as $\Delta^2 u + c^2 \partial_{x_1}^2 u + e^{u}-1=0$ with decay and Neumann boundary conditions. It develops a computer-assisted Newton-Kantorovich framework, constructing a computable approximate solution via Fourier/cosine series on a bounded domain and an approximate inverse to prove the existence of a true solution near the approximate one, with explicit bounds to certify contraction. The stability analysis relies on enclosing the spectrum of the linearization $D\mathbb{F}(\tilde{u})$ and computing the sign of $\theta$, yielding a rigorously proven orbital stability for the main solution (stable at $c=1.2$) and orbital instability for other pattern speeds, corroborated by multiple CAP-based results. The work demonstrates a robust CAP pipeline for nonlinear PDEs with exponential nonlinearities on unbounded domains, providing explicit numerical bounds, spectral information, and reproducible proofs via interval arithmetic.

Abstract

In this paper, we present a computer-assisted approach for constructively proving the existence of traveling wave solutions of the suspension bridge equation on the infinite strip $Ω= \mathbb{R} \times (-d_2,d_2)$. Using a meticulous Fourier analysis, we derive a quantifiable approximate inverse $\mathbb{A}$ for the Jacobian $D\mathbb{F}(\bar{u})$ of the PDE at an approximate traveling wave solution $\bar{u}$. Such approximate objects are obtained thanks to Fourier coefficients sequences and operators, arising from Fourier series expansions on a rectangle $Ω_0 = (-d_1,d_1) \times (-d_2,d_2)$. In particular, the challenging exponential nonlinearity of the equation is tackled using a rigorous control of the aliasing error when computing related Fourier coefficients. This allows to establish a Newton-Kantorovich approach, from which the existence of a true traveling wave solution of the PDE can be proven in a vicinity of $\bar{u}$. We successfully apply such a methodology in the case of the suspension bridge equation and prove the existence of multiple traveling wave solutions on $Ω$. Finally, given a proven solution $\tilde{u}$, a Fourier series approximation on $Ω_0$ allows us to accurately enclose the spectrum of $D\mathbb{F}(\tilde{u})$. Such a tight control provides the number of negative eigenvalues, which in turns, allows to conclude about the orbital (in)stability of $\tilde{u}$.

Figures (2)

• Figure 1: Visualization of an approximate solution $\bar{u}$ to \ref{['eq : suspension bridge equation']} with $c = 1.2$. The approximation is truncated to a finite domain in this plot.
• Figure 2: Visualization of approximate two-peak solutions $\bar{u}$ to \ref{['eq : suspension bridge equation']}. The approximations are truncated to a finite domain in these plots.

Theorems & Definitions (36)

• Lemma 2.1
• proof
• Remark 2.2
• Theorem 3.1
• Remark 3.2
• Lemma 4.1: Bound $\mathcal{Y}_{0}$
• proof
• Remark 4.2
• Lemma 4.3: Bound $\mathcal{Z}_{2}$
• proof