Variational method for determining the complex-valued coefficients of a nonlinear nonstationary Schrödinger-type equation

This study is devoted to variational methods for solving the problem of simultaneous determination of the unknown complex-valued coefficients multiplying the lower and nonlinear terms of a nonstationary Schrödinger-type equation generalizing the well-known quantum mechanical Schrödinger equation. The sought coefficient of the lower term is a complex-valued quantum potential. Problems of this type arise in nonlinear optics, in the study of processes in quantum waveguides, and in other areas. The solvability of the variational statement of the problem under consideration is proved, a necessary condition for its solution is established, and an expression for the gradient of the cost functional based on the final observation is obtained. These results are used to develop and justify an iterative algorithm for solving the problem. An example of the instability of its solution is given, and an iterative regularizing algorithm for solving the problem is described.