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

Abstract

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 g̃0(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 ux (0, t) and uxx (0, t) are asymptotically t-periodic with period τ which tend to the functions g̃1(t) and g̃2(t) as t → ∞, respectively, then the periodic functions g̃0(t) and g̃0(t) can be uniquely determined in terms of the function g̃1(t). Furthermore, we characterize the Fourier coefficients of g̃1(t) and g̃2(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.