Convective Linear Stability of Solitary Waves for Boussinesq Equations

Source abstract

Boussinesq was the first to explain the existence of Scott Russell's solitary wave mathematically. He employed a variety of asymptotically equivalent equations to describe water waves in the small‐amplitude, long‐wave regime. We study the linearized stability of solitary waves for three linearly well‐posed Boussinesq models. These are problems for which well‐developed Lyapunov methods of stability analysis appear to fail. However, we are able to analyze the eigenvalue problem for small‐amplitude solitary waves, by comparison to the equation that Boussinesq himself used to describe the solitary wave, which is now called the Korteweg–de Vries equation. With respect to a weighted norm designed to diminish as perturbations convect away from the wave profile, we prove that nonzero eigenvalues are absent in a half‐plane of the form R λ>− b for some b >0, for all three Boussinesq models. This result is used to prove the decay of solutions of the evolution equations linearized about the solitary wave, in two of the models. This “convective linear stability” property has played a central role in the proof of nonlinear asymptotic stability of solitary‐wave‐like solutions in other systems.