The inverse problem for a nonlinear Schrödinger equation

Abstract

Using variational methods, the solution of the inverse problem of finding the refractive index of a nonlinear medium in a multidimensional Schrbdinger equation is studied. The correctness of the statement of the problem under consideration is investigated, and a necessary condition that must be satisfied by the solution of this problem is found. Bibliography: 8 titles. Finding the refractive index of a nonlinear medium is one of the fundamental problems of nonlinear optics [1]. Inverse problems of finding the refractive index of a nonlinear medium in variational statements were studied in [2, 3] for the one-dimensional Schr6dinger equation. Let D be a bounded domain of the Euclidean space E3; F the boundary of the domain D, which is assumed to be sufficiently smooth; T > 0 a given number; 0 < t < T; ~t = D • (0, t); ~ = ~T, S = F x (0, T); x = (Xl,X2, x3) an arbitrary point of the domain D; C~ T], B) the Banach space consisting of all welldefined and continuous functions on [0, iF] with values in the Banach space B; Lp(D) the Lebesgue space of functions, summable up to the power p > 1; -~-E , , where the spaces L~ ([0, T], B) and Wk(D), w~'m(~), p >_ 1, k, m ~ O, are as defined in [4, 5]; the symbol means that the property is valid for almost all values of a variable. Consider a system whose state is described by the nonlinear Schrbdinger equation .0r 3 02 r l-~ +no C Ox~. v(x,t)r162162 = f(x,t), (1) j=l where i = x/-L-f, a0, al > 0 are given numbers, and f E W21'1 (~t) is a given function. Let Eq. (1) be subject to the following initial and boundary conditions: 0r s = O, (2) r I =o = x e D, where ~(x) is a given function from W2(D), for which 0~ r Ou = O, and u is the direction of the outward normal of the boundary F. Our goal is to find the coefficient v(x, t) on the basis of additional information r Is = Y(~, t) about the solution of the Schr6dinger equation (1). The function v(x, t) is sought on the set { Ov(x,t) Ov(x,t) E o } V-v:v=v(x,t), veW#'~(fi), Iv(z,t)l<_bo, at <_bl, ~ <--52, V(x,t) efi , where bo, bt, b2 > 0 are given numbers. The variational statement of the problem at hand lies in the minimization of the functional J(v) II% = = -llL=(s) on the set V under conditions (1), (2), where y E W~/2'1/4(S) is a given function.From this point on, we assume that the above conditions on the problem are fulfilled.(3)

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The solvability of the optimal control problem for a nonlinear Schrödinger equation

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