About one exact solution of linear Schrӧdinger equation, describing femtosecond pulse propagation
Abstract
In current paper, a model of a femtosecond laser pulse propagation in a linear medium is considered. At the study of laser physics problems, described by the Schrödinger equations, a special role is played by the problem of an exact solution existence. The present paper proposes one of the approaches for the construction of exact solution, which consists in the transform of the Schrödinger equation to the Airy equation. An exact solution is developed, using this approach. It should be stressed that even in the case of the propagation of a laser pulse in a linear medium, no exact solutions were found early for the equation under consideration in the paper, and this problem is an urgent one. The exact solution was found for the Schrödinger equation as a product of the exponential function and the Airy functions, and linearly independent solutions of the Airy equation were written, whose properties were used under the analysis of initial laser pulse distributions. Let us note that the obtained solution includes a free parameter that corresponds to a change in the amplitude of the solution oscillations on one of the boundaries of the region considered in the problem. In addition, initial distributions, corresponding to obtained solution and depending on the physical parameters of the problem, are depicted in Figures.
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