Existence and Stability for Traveling Waves of Fourth-Order Semilinear Wave and Schrödinger Equations

Abstract

We investigate the existence and spectral stability of traveling wave solutions for a class of fourth-order semilinear wave equations, commonly referred to as beam equations. Using variational methods based on a constrained maximization problem, we establish the existence of smooth, exponentially decaying traveling wave profiles for wavespeeds in the interval $(-\sqrt{2},\sqrt{2})$. We derive precise spectral properties of the associated linearized operators and prove a Vakhitov–Kolokolov (VK)-type stability criterion that completely characterizes spectral stability. Furthermore, we determine the sharp exponential decay rate of the traveling waves and demonstrate that it matches the decay rate of the Green's function for the linearized operator. Our analysis extends to fourth-order nonlinear Schrödinger equations, for which we establish analogous existence and stability results. The theoretical findings are complemented by numerical computations that verify the stability predictions and reveal the transition from unstable to stable regimes as the wavespeed varies. These results provide a comprehensive mathematical framework for understanding wave propagation phenomena in structural mechanics, particularly suspension bridge models.