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Asymptotic stability of solitons for the Benjamin-Ono equation. (English) Zbl 1247.35133

The authors considered the Benjamin-Ono equation of the form: \[ u_t+{\mathcal H}u_{xx}+uu_x=0\quad (t,x)\in{\mathbb R}+{\mathbb R}, \tag{1} \] where \({\mathcal H}\) denotes the Hilbert transform \[ {\mathcal H}u(x)={1\over\pi}\text{p.v.}\intop_{-\infty}^{+\infty}{u(y)\over y-x}dy= {1\over\pi}\lim_{\varepsilon\to 0}\intop_{|y-x|>\varepsilon}{u(y)\over y-x}dy \tag{2} \] with initial data \[ u(0)=u_0(x). \tag{3} \] The main results of thise paper is devoted to the proof of the asymptotic stability of the family of solutions of the problem (1)–(3) in the energy space. The proof of the main theorem is based on a Liouville property for solutions close to the solitons for this equation.
As a corollary of the proofs of the theorems, the authors obtain stability and asymptotic stability of exact multi-solitons.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
35B40 Asymptotic behavior of solutions to PDEs
35C08 Soliton solutions
35B35 Stability in context of PDEs
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References:

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