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

Abstract

Abstract 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 θ. The jump matrix L(x, t, θ) has an explicit representation dependent on x, t and it can be represented exactly by the two pairs of spectral functions y(θ), z(θ) (obtained from the initial value u 0(x)) and Y(θ), Z(θ) (obtained from the boundary conditions v 0(t), { v k ( t ) } 1 4 ). Furthermore, the two pairs of spectral functions y(θ), z(θ) and Y(θ), Z(θ) are not independent of each other, but are related to the compatibility condition, the so-called global relation.