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

作者：Holmer Justin^*

来源：International Mathematics Research Notices, 2011, (23): 5367-5397.

DOI：10.1093/imrn/rnq284

We study the dynamics of solitons as solutions to the perturbed KdV (pKdV) equation partial derivative(t)u = -partial derivative(x)(partial derivative(2)(x)u+ 3u(2) - bu), where b(x, t) = b(0)(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 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 Martel and Merle "Asymptotic stability of solitons for subcritical gKdV equations revisited." Nonlinearity 18 (2005): 55-80. 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.

• 出版日期2011