The three-wave equation on the half-line

• We present a Riemann–Hilbert problem formalism for the initial–boundary value problem of the three-wave equation.
• The residue conditions of matrix function M in the Riemann–Hilbert problem is obtained.
• The explicit jump matrix J in the Riemann–Hilbert problem is constructed.

The Fokas method is used to analyze the initial–boundary value problem for the three-wave equationpij,t−bi−bjai−ajpij,x+∑k(bk−bjak−aj−bi−bkai−ak)pikpkj=0,i,j,k=1,2,3, on the half-line. Assuming that the solution pij(x,t)pij(x,t) exists, we show that it can be recovered from its initial and boundary values via the solution of a Riemann–Hilbert problem formulated in the plane of the complex spectral parameter λ.