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

摘要

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 u0(x))and Y(θ),Z(θ)(obtained from the boundary conditions v0(t),{vk(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.