## Dynamics of KdV solitons in the presence of a slowly varying potential.(English)Zbl 1247.35132

Author’s abstract: We study the dynamics of solitons as solutions to the perturbed KdV (pKdV) equation $$\partial_tu=-\partial_x(\partial_x^2u+3u^2-bu)$$, where $$b(x,t)=b_{0}(hx, ht)$$, $$h\ll 1$$, is a slowly varying, but not small, potential. We obtain an explicit description of the trajectory of the soliton parameters of scale and position on the dynamically relevant time scale $$\delta h^{-1} \log h^{-1}$$, together with an estimate on the error of size $$h^{1/2}$$. In addition to the Lyapunov analysis commonly applied to these problems, we use a local virial estimate due to Y. Martel and F. Merle [Nonlinearity 18, No. 1, 55–80 (2005; Zbl 1064.35171)]. The results are supported by numerics. The proof does not rely on the inverse scattering machinery and is expected to carry through for the $$L^{2}$$ subcritical gKdV-$$p$$ equation, $$1<p<5$$.

### MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 35B35 Stability in context of PDEs 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems

### Keywords:

KdV equation; linear operator; soliton; spectral; virial estimate

Zbl 1064.35171
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