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The Lyapunov stability of the \(n\)-soliton solutions in the Lax hierarchy of the Benjamin-Ono equation. (English) Zbl 1112.37061

Summary: The Lyapunov stability is established for the \(N\)-soliton solutions in the Lax hierarchy of the Benjamin-Ono (BO) equation. We characterize the \(N\)-soliton profiles as critical points of certain Lyapunov functional. By using several results derived by the inverse scattering transform of the BO equation, we demonstrate the convexity of the Lyapunov functional when evaluated at the \(N\)-soliton profiles. From this fact, we deduce that the \(N\)-soliton solutions are energetically stable.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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