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Linear problems related to asymptotic stability of solitons of the generalized KdV equations. (English) Zbl 1126.35055

The author continues investigation of asymptotic properties of soliton solutions of the generalized KdV equation \(\psi_t + \psi_{xxx} +(\psi^p)_x = 0\) [J. Math. Pures Appl. (9) 79, 339–425 (2000; Zbl 0963.37058)] and [Arch. Ration. Mech. Anal. 157, 219–254 (2001; Zbl 0981.35073)]. One of the key ingredients, the linear Liouville property, previously established for positive integers \(p \leq 5\), is now given a new proof successful for any real \(p > 1\). The relation of the linear Liouville property to asymptotic stability and asymptotic completeness is briefly discussed. Analogously, the linear Liouville property is then proved for velocity \(c > 1\) travelling wave solutions of the Benjamin-Bona-Mahony equation.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
35Q51 Soliton equations
35B40 Asymptotic behavior of solutions to PDEs
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