Isoinertial operators around the KdV multi-solitons

Abstract

We consider the Korteweg–de Vries (KdV) equation, by employing the recursion operator of the KdV equation, some new operator identities for the higher order linearized Hamiltonians around one soliton are verified, it follows that the linearized operators can be diagonalized to their constant coefficient counterparts. The linearized operators around the KdV multi-solitons are isoinertial and the discrete eigenvalues of which are determined. As a direct consequence, we give a new proof of the stability of the KdV multi-solitons and extend the approaches of Maddocks and Sachs (1993) and Neves and Lopes (2006). Finally, some discussions of the KdV H−1 conservation law are presented by evaluating the inverse recursion operator of the KdV equation.