Soliton dynamics for a non-Hamiltonian perturbation of mKdV

We study the dynamics of soliton solutions to the perturbed mKdV equation ${\mathrm{\partial }}_{t}u={\mathrm{\partial }}_x(-{\mathrm{\partial }}_x^{2}u-2u^3)+ϵVu$, where $V\in $\mathcal{C}$^1_b($\mathbb{R}$)$,$0<\epsilon\ll 1$. This type of perturbation is non-Hamiltonian. Nevertheless, via symplectic considerations, we show that solutions remain $O(ϵ\text{la}t{\text{ra}}^{1\mathrm{/}2})$ close to a soliton on an $O({ϵ}^{-1})$ time scale. Furthermore, we show that the soliton parameters can be chosen to evolve according to specific exact ODEs on the shorter, but still dynamically relevant, time scale $O({ϵ}^{-1\mathrm{/}2})$. Over this time scale, the perturbation can impart an O(1) influence on the soliton position.