Abstract

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.

中文翻译：

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确定非线性非平稳薛定ö型方程复数值系数的变分方法

摘要

本研究致力于解决变分方法，以解决同时确定未知复数值系数的问题，该方法将非平稳Schrödinger型方程的下限和非线性项相乘，从而推广了著名的量子力学Schrödinger方程。寻求的较低项的系数是复数值量子势。这种类型的问题出现在非线性光学，量子波导过程以及其他领域的研究中。证明了所考虑问题的变式陈述的可解性，确定了其解决的必要条件，并基于最终观察结果获得了成本函数梯度的表达式。这些结果用于开发和证明解决该问题的迭代算法。