INVERSE PROBLEM FOR N X N HYPERBOLIC SYSTEMS ON THE PLANE AND THE N-WAVE INTERACTIONS

We study rigorously the solvability of the direct and inverse problems associated with psi-x - J-psi-y = Q-psi, (x, y) is-an-element-of R2, where (i) psi is an N x N-matrix-valued function on R2 (N greater-than-or-equal-to 2), (ii) J is a constant, real, diagonal N x N matrix with entries J1 > J2 > ... > J(N), and (iii) Q is off-diagonal with rapidly decreasing (Schwartz) component functions. In particular we show that the direct problem is always solvable and give a small norm condition for the solvability of the inverse problem. In the particular case that Q is skew Hermitian the inverse problem is solvable without the small norm assumption. Furthermore we show how these results can be used to solve certain Cauchy problems for the associated nonlinear evolution equations. For concreteness we consider the N-wave interactions and show that if a certain norm of Q(x, y, 0) is small or if Q(x, y, 0) is skew Hermitian then the N-wave interactions equation has a unique global solution.