The Stability of Periodic Orbits for Kawahara Equation

In the presented work, we investigate the Kawahara equation, a non-integrable partial differential equation that generalizes the classical Korteweg-de Vries (KdV) equation by including a fifth-order derivative. Our study encompasses two key contributions. First, we analyze the traveling wave solutions of the Kawahara equation, focusing on cases where the associated ordinary differential equation (ODE) exhibits complex characteristic exponents. Using Hamiltonian dynamics, we explore the geometry of invariant manifolds arising from non-real characteristic exponents. Second, we extend this analysis by introducing a control-theoretic perspective, modifying the ODE to include a control term. This approach leads to the design and experimental validation of a novel linear observer, demonstrating its stability and potential applications in systems governed by periodic coefficients.