The modified Korteweg-de Vries equation on the quarter plane with t-periodic data

We study the modified Korteweg-de Vries equation posed on the quarter plane with asymptotically t-periodic Dirichlet boundary datum u(0,t) in the sense that u(0,t) tends to a periodic function g0 (t) with period τ as t → ∞. We consider the perturbative expansion of the solution in a small e > 0. Here we show that if the unknown boundary data ux(0,t) and uxx(0,t) are asymptotically t-periodic with period τ which tend to the functions g1 (t) and g2 (t) as t → ∞, respectively, then the periodic functions g1 (t) and g2 (t) can be uniquely determined in terms of the function g0 (t). Furthermore, we characterize the Fourier coefficients of g1 (t) and g2 (t) to all orders in the perturbative expansion by solving an infinite system of algebraic equations. As an illustrative example, we consider the case of a sine-wave as Dirichlet datum and we explicitly determine the coefficients for large t up to the third order in the perturbative expansion.