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 midpointmethod.
Highlights
The nonlinear time-dependent Schrödinger equation (NL-TDSE) appears in the approximate treatment of many physical processes, where the approximate Hamiltonian depends on the state of the system
The best known NL-TDSE is the Gross–Pitaevskii equation,[22–26] which models the dynamics of Bose–Einstein condensates.[27,28]
To overcome the limitations of the explicit split-operator algorithm applied to general NL-TDSEs, in our previous work,[35] we developed high-order integrators by symmetrically composing the implicit midpoint method
Summary
The nonlinear time-dependent Schrödinger equation (NL-TDSE) appears in the approximate treatment of many physical processes, where the approximate Hamiltonian depends on the state of the system. The best known NL-TDSE is the Gross–Pitaevskii equation,[22–26] which models the dynamics of Bose–Einstein condensates.[27,28] To solve this NL-TDSE with cubic nonlinearity, the explicit second-order split-operator algorithm[29–32] is frequently used[33] because it is efficient and geometric.[34]. To overcome the limitations of the explicit split-operator algorithm applied to general NL-TDSEs, in our previous work,[35] we developed high-order integrators by symmetrically composing the implicit midpoint method. These integrators are applicable to the general nonlinear Schrödinger equation with both separable and nonseparable Hamiltonians and, in contrast to the explicit splitoperator algorithm, are efficient while preserving the geometric properties of the exact solution.
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