Isoinertial family of operators and convergence of KdV cnoidal waves to solitons

In this paper we show the convergence of Korteweg–de Vries cnoidal waves to the limit soliton. It is proved that the convergence is uniform and in H2-norm, as the period of the solutions tends to infinity. Families of Hill operators are also studied. We obtain a condition under which families of operators are isoinertial. This condition is satisfied for classes of Hill operators that are obtained by linearization. Our application is to the family of linearized operators at the KdV cnoidal waves. It is proved that this family is isoinertial and also the value of the inertial index is calculated.