Inverse problem for a nonlinear Helmholtz equation

This paper is devoted to the uniqueness of the coefficients theta, phi is an element of L-infinity(R-3), and psi is an element of L-infinity(R-3, R-3) for the nonlinear Helmholtz equations -Deltav(x) - k(2)v(x) = theta(x)v(x)F(\v(x)\) and -Deltav(x) - k(2)v(x) = (phi(x)v(x) + ipsi(x). delv(x))\delv(x)\(r) \v(x)\(5). For small values of),, a solution v is uniquely constructed by adding a small outgoing perturbation to the plane wave x --> lambdae(ikx.d), where \d\ = 1 and lambda greater than or equal to 0. We can write v = v(x, lambda, d) = lambdae(ikx.d) + u(infinity)(s) (x/\x\, d, lambda)e(ik\x\) /\x\ + O(1/\x\(2)) for large \x\. For a fixed k > 0, weyould like to prove that theta, phi and div psi can be uniquely reconstructed from the behaviour of u(infinity)(s) (x/\x\, d, lambda) as lambda --> 0. We prove the uniqueness in this paper. (C) 2003 Elsevier SAS. All rights reserved.