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Convective Linear Stability of Solitary Waves for Boussinesq Equations
Author(s) -
Pego Robert L.,
Weinstein Michael I.
Publication year - 1997
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00063
Subject(s) - eigenvalues and eigenvectors , mathematics , mathematical analysis , nonlinear system , boussinesq approximation (buoyancy) , stability theory , stability (learning theory) , linear stability , exponential stability , amplitude , norm (philosophy) , physics , convection , mechanics , law , quantum mechanics , natural convection , machine learning , computer science , rayleigh number , political science
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.