Lindsey van der Aalst

This thesis focuses on the rigorous mathematical verification of traveling wave solutions and bifurcation structures in nonlinear partial differential equations, spanning both structural mechanics and fluid dynamics. As explicit analytical solutions to these systems are often impossible to find and standard numerical simulations lack absolute mathematical certainty, this research employs computer-assisted proofs. By framing the differential equations as zero-finding problems in appropriate Banach spaces and utilizing Newton-Kantorovich type arguments, this thesis guarantees the existence of solutions with explicit error bounds. The first part of this thesis explores structural mechanics. In particular, Chapters 2 and 3 apply computer-assisted techniques to the two-dimensional suspension bridge equation. Chapter 2 proves the existence of periodic, localized traveling waves on a large bounded domain by developing computable bounds to control the aliasing error introduced by the equation’s exponential nonlinearity. Chapter 3 extends this analysis to an unbounded domain, specifically an infinite strip defined as Ω = R × (−d2, d2) for d2 > 0. Moving beyond existence proofs, it introduces a rigorous method to determine the orbital stability of the waves. By enclosing the spectrum of a linearized operator and counting its negative eigenvalues, we establish the orbital (in)stability of the solutions. Chapters 4 and 5 transition to fluid dynamics by analyzing the three-dimensional Navier-Stokes equations. Chapter 4 applies the computer-assisted proof approach to prove the existence of a traveling wave. To manage the computational load, spatial symmetries such as reflection and rotation are imposed, reducing the size of the system. Chapter 5 shifts to the system’s bifurcation structure by focusing on the proof of a Hopf bifurcation. Moreover, this chapter outlines the method to prove the existence of a branch of periodic solutions emerging from the bifurcation point. Hence, this thesis demonstrates that computer-assisted proofs can be extended to handle nonpolynomial nonlinearities, unbounded domains and fluid systems, all in multiple spatial dimensions. It offers a robust framework for verifying dynamical phenomena that are out of reach for classical analytical methods.