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 g?₀ (t) with period τ as t → ∞. We consider the perturbative expansion of the solution in a small ε > 0. Here we show that if the unknown boundary data u_x(0,t) and u_xx(0,t) are asymptotically t-periodic with period τ which tend to the functions g?₁ (t) and g?₂ (t) as t → ∞, respectively, then the periodic functions g?₁ (t) and g?₂ (t) can be uniquely determined in terms of the function g?₀ (t). Furthermore, we characterize the Fourier coefficients of g?₁ (t) and g?₂ (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.