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, \(1<p<5\).


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


Zbl 1064.35171
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