Effective Dynamics of Double Solitons for Perturbed mKdV

Abstract

We consider the perturbed mKdV equation \({\partial_t u = -\partial_x (\partial_x^2u + 2u^3- b(x,t)u)}\) , where the potential \({b(x,t)=b_0(hx,ht), 0 < h \ll 1 }\) , is slowly varying with a double soliton initial data. On a dynamically interesting time scale the solution is \({ {\mathcal{O}}(h^2) }\) close in H2 to a double soliton whose position and scale parameters follow an effective dynamics, a simple system of ordinary differential equations. These equations are formally obtained as Hamilton’s equations for the restriction of the mKdV Hamiltonian to the submanifold of solitons. The interplay between algebraic aspects of complete integrability of the unperturbed equation and the analytic ideas related to soliton stability is central in the proof.