Dynamics of Kdv Solitons in the Presence of a Slowly Varying Potential

We study the dynamics of solitons as solutions to the perturbed KdV (pKdV) equation ∂tu = −∂x(∂ xu + 3u − bu), where b(x, t) = b0(hx, ht), h 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 δh−1 log h−1, together with an estimate on the error of size h. In addition to the Lyapunov analysis commonly applied to these problems, we use a local virial estimate due to Martel-Merle [15]. 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 subcritical gKdV-p equation, 1 < p < 5. The case of p = 3, the modified Korteweg-de Vries (mKdV) equation, is structurally simpler and more precise results can be obtained by the method of Holmer-Zworski [9].