2,154 research outputs found
Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients
open access articleMotivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge-Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations
Reduced-Order Modeling based on Approximated Lax Pairs
A reduced-order model algorithm, based on approximations of Lax pairs, is
proposed to solve nonlinear evolution partial differential equations. Contrary
to other reduced-order methods, like Proper Orthogonal Decomposition, the space
where the solution is searched for evolves according to a dynamics specific to
the problem. It is therefore well-suited to solving problems with progressive
waves or front propagation. Numerical examples are shown for the KdV and FKPP
(nonlinear reaction diffusion) equations, in one and two dimensions
Krotov: A Python implementation of Krotov's method for quantum optimal control
We present a new open-source Python package, krotov, implementing the quantum optimal control method of that name. It allows to determine time-dependent external fields for a wide range of quantum control problems, including state-to-state transfer, quantum gate implementation and optimization towards an arbitrary perfect entangler. Krotov's method compares to other gradient-based optimization methods such as gradient-ascent and guarantees monotonic convergence for approximately time-continuous control fields. The user-friendly interface allows for combination with other Python packages, and thus high-level customization
Generalization of splitting methods based on modified potentials to nonlinear evolution equations of parabolic and Schr\"odinger type
The present work is concerned with the extension of modified potential
operator splitting methods to specific classes of nonlinear evolution
equations. The considered partial differential equations of Schr{\"o}dinger and
parabolic type comprise the Laplacian, a potential acting as multiplication
operator, and a cubic nonlinearity. Moreover, an invariance principle is
deduced that has a significant impact on the efficient realisation of the
resulting modified operator splitting methods for the Schr{\"o}dinger case.}
Numerical illustrations for the time-dependent Gross--Pitaevskii equation in
the physically most relevant case of three space dimensions and for its
parabolic counterpart related to ground state and excited state computations
confirm the benefits of the proposed fourth-order modified operator splitting
method in comparison with standard splitting methods.
The presented results are novel and of particular interest from both, a
theoretical perspective to inspire future investigations of modified operator
splitting methods for other classes of nonlinear evolution equations and a
practical perspective to advance the reliable and efficient simulation of
Gross--Pitaevskii systems in real and imaginary time.Comment: 30 pages, 6 figure
Kinetic Energy Estimates for the Accuracy of the Time-Dependent Hartree-Fock Approximation with Coulomb Interaction
We study the time evolution of a system of spinless fermions in
which interact through a pair potential, e.g., the Coulomb
potential. We compare the dynamics given by the solution to Schr{\"o}dinger's
equation with the time-dependent Hartree-Fock approximation, and we give an
estimate for the accuracy of this approximation in terms of the kinetic energy
of the system. This leads, in turn, to bounds in terms of the initial total
energy of the system.Comment: 35 page
Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients
Motivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge–Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with a periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations
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