Neumann problem for the Korteweg–de Vries equation

We consider Neumann initial-boundary value problem for the Korteweg–de Vries equation on a half-line We prove that if the initial data and the norm , where ε>0 is small enough , and . Then there exists a unique solution of the initial-boundary value problem (0.1). Moreover there exists a constant C such that the solution has the following asymptotics for t→∞ uniformly with respect to x>0, where and Ai(q) is the Airy function