Riemann-Hilbert problem for the fifth-order modified Korteweg-de Vries equation with the prescribed initial and boundary values

In this paper, we investigate the fifth-order modified Korteweg-de Vries (mKdV) equation on the half-line via the Fokas unified transformation approach. We show that the solution u(x, t) of the fifth-order mKdV equation can be represented by the solution of the matrix Riemann-Hilbert problem constructed on the plane of complex spectral parameter theta. The jump matrix L(x, t,theta) has an explicit representation dependent on x, t and it can be represented exactly by the two pairs of spectral functions y(theta), z(theta) (obtained from the initial value u(0)(x)) and Y(theta), Z(theta) (obtained from the boundary conditions v(0)(t), {vk (t)}4 (1)). Furthermore, the two pairs of spectral functions y(theta), z (theta) and Y(theta), Z(theta) are not independent of each other, but are related to the compatibility condition, the so-called global relation.