Boundary value problems for the N-wave interaction equations

We consider boundary value problems for the N  -wave interaction equations in one and two space dimensions, posed for x⩾0x⩾0 and x,y⩾0x,y⩾0, respectively. Following the recent work of Fokas, we develop an inverse scattering formalism to solve these problems by considering the simultaneous spectral analysis of the two ordinary differential equations in the associated Lax pair. The solution of the boundary value problems is obtained through the solution of a local Riemann–Hilbert problem in the one-dimensional case, and a nonlocal Riemann–Hilbert problem in the two-dimensional case.