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

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 $\tilde{u}$.