Asymptotic stability of a nonlinear Korteweg–de Vries equation with critical lengths

We study an initial–boundary-value problem of a nonlinear Korteweg–de Vries equation posed on the finite interval (0,2kπ)(0,2kπ) where k is a positive integer. The whole system has Dirichlet boundary condition at the left end-point, and both of Dirichlet and Neumann homogeneous boundary conditions at the right end-point. It is known that the origin is not asymptotically stable for the linearized system around the origin. We prove that the origin is (locally) asymptotically stable for the nonlinear system if the integer k   is such that the kernel of the linear Korteweg–de Vries stationary equation is of dimension 1. This is for example the case if k=1k=1.