Coefficient Determination for Nonlinear Schrödinger Equations on Manifolds

We consider an inverse problem of recovering the unknown coefficients β(t, x)and V (t, x) appearing in a time dependent nonlinear Schrödinger equation (i∂t+∆+V )u+βu2 = 0 in (0, T ) × M, on Euclidean geometry as well as on Riemannian geometry. We consider measurements in Ω ⊂ M that is a neighborhood of the boundary of M and the source-to-solution map Lβ,V that maps a source f supported in Ω×(0, T ) to the restriction of the solution u in Ω × (0, T). We show that the map Lβ,V uniquely determines the time-dependent potential and the coefficient of the non linearity, for the above non-linear Schrödinger equation and for the Gross-Pitaevskii equation, with a cubic non-linear term β|u|2 u, that is encountered in quantum physics.