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Publication# An implicit split-operator algorithm for the nonlinear time-dependent Schrödinger equation

Abstract

The explicit split-operator algorithm is often used for solving the linear and nonlinear time-dependent Schrödinger equations. However, when applied to certain nonlinear time-dependent Schrödinger equations, this algorithm loses time reversibility and second-order accuracy, which makes it very inefficient. Here, we propose to overcome the limitations of the explicit split-operator algorithm by abandoning its explicit nature. We describe a family of high-order implicit split-operator algorithms that are norm-conserving, time-reversible, and very efficient. The geometric properties of the integrators are proven analytically and demonstrated numerically on the local control of a two-dimensional model of retinal. Although they are only applicable to separable Hamiltonians, the implicit split-operator algorithms are, in this setting, more efficient than the recently proposed integrators based on the implicit midpoint method.

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Related concepts (3)

Geometry

Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.

Nonlinear Schrödinger equation

In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose–Einstein condensates confined to highly anisotropic, cigar-shaped traps, in the mean-field regime.

Integrator

An integrator in measurement and control applications is an element whose output signal is the time integral of its input signal. It accumulates the input quantity over a defined time to produce a representative output. Integration is an important part of many engineering and scientific applications. Mechanical integrators are the oldest type and are still used for metering water flow or electrical power. Electronic analogue integrators are the basis of analog computers and charge amplifiers.