Perturbation Theory for Kinks of the Defocusing Modified Korteweg–de Vries Equation

Abstract

In this work, we develop an integrable perturbation theory for the defocusing modified Korteweg–de Vries kink solution based on the squared eigenfunction expansion associated with the underlying Zakharov–Shabat scattering problem. We derive the completeness relation for the squared eigenfunctions appropriate to the kink background, establish the adjoint structure needed to handle perturbations of both the continuous and discrete spectral components, and obtain explicit evolution equations for the perturbed kink parameters at leading order. The study of the first-order correction shows that perturbations generically produce a radiative shelf in front of the kink. We also apply our results to certain physically relevant perturbations and show that the predictions are consistent with direct numerical simulations.