## 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

### Citations:

Zbl 0963.37058; Zbl 0981.35073
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