Decay for the nonlinear KdV equations at critical lengths

Abstract The nonlinear KdV equation in a bounded interval equipped with the Dirichlet boundary condition and the Neumann boundary condition on the right is considered. It is known that there is a set of critical lengths for which the solutions of the linearized system conserve the L 2 -norm if their initial data belong to a finite dimensional space M . We show that all solutions of the nonlinear system decay to 0 at least with the rate 1 / t 1 / 2 when dim ⁡ M = 1 or when dim ⁡ M is even and a specific condition is satisfied, for sufficiently small initial data. Our analysis is inspired by the power series expansion approach and involves the theory of quasi-periodic functions. Consequently, we rediscover all known results by a different approach and obtain new results. We also show that the decay rate is not slower than ln ⁡ ( t + 2 ) / t for all critical lengths.