The Korteweg-de Vries equation on the quarter plane with asymptotically t-periodic data via the Fokas method

We study the Korteweg-de Vries equation posed on the quarter plane with asymptotically t-periodic boundary data for large t > 0. We derive an expression for the Dirichlet to Neumann map to all orders in the perturbative expansion of a small epsilon > 0 in the case of the asymptotically periodic boundary data. More precisely, we show that if the unknown Neumann boundary data are asymptotically periodic for large t in the sense that u(x)(0, t) and u(xx)(0,t) tend to periodic functions (g) over tilde (1)(t) and (g) over tilde (2)(t) for large t, respectively, then the periodic functions (g) over tilde (1)(t) and (g) over tilde (2)(t) can be characterized in terms of the given asymptotically periodic Dirichlet boundary datum u(0, t). Moreover, we determine effectively the Fourier coefficients of the functions (g) over tilde (1)(t) and (g) over tilde (2)(t) by solving a certain recursive algebraic equations.