# American Institute of Mathematical Sciences

December  2019, 6(2): 147-169. doi: 10.3934/jcd.2019008

## Efficient time integration methods for Gross-Pitaevskii equations with rotation term

 1 Universitat Jaume I, Departament de Matemàtiques, 12071 Castellón, Spain 2 Universitat Politècnica de València, Instituto Universitario de Matemática Multidisciplinar, 46022 Valencia, Spain 3 Universitat Jaume I, IMAC and Departament de Matemàtiques, 12071 Castellón, Spain 4 Leopold-Franzens-Universität Innsbruck, Institut für Mathematik, 6020 Innsbruck, Austria

Received  March 2019 Revised  August 2019 Published  November 2019

The objective of this work is the introduction and investigation of favourable time integration methods for the Gross-Pitaevskii equation with rotation term. Employing a reformulation in rotating Lagrangian coordinates, the equation takes the form of a nonlinear Schrödinger equation involving a space-time-dependent potential. A natural approach that combines commutator-free quasi-Magnus exponential integrators with operator splitting methods and Fourier spectral space discretisations is proposed. Furthermore, the special structure of the Hamilton operator permits the design of specifically tailored schemes. Numerical experiments confirm the good performance of the resulting exponential integrators.

Citation: Philipp Bader, Sergio Blanes, Fernando Casas, Mechthild Thalhammer. Efficient time integration methods for Gross-Pitaevskii equations with rotation term. Journal of Computational Dynamics, 2019, 6 (2) : 147-169. doi: 10.3934/jcd.2019008
##### References:

show all references

##### References:
Overview of exponential time integration methods. Commutator-free quasi-Magnus exponential integrators (CFQM, order $p$, number of compositions $J$, number of quadrature nodes $K$) or modified commutator-free quasi-Magnus exponential integrator (Scheme BBK, order $p = 6$), respectively, combined with operator splitting methods (order $p$, number of compositions $s$)
Time integration of non-autonomous linear test equation in two (left) and three (right) space dimensions by quasi-Magnus exponential integrators and operator splitting methods, see (8), (21), and Figure 1. Global errors versus time stepsizes (first row) or number of fast Fourier transforms and inverse fast Fourier transforms (second row), respectively
Time integration of non-autonomous nonlinear test equation in two (left) and three (right) space dimensions with $\vartheta = 1$ by quasi-Magnus exponential integrators and operator splitting methods, see (8), (21), and Figure 1. Global errors versus time stepsizes (first row) or number of fast Fourier transforms and inverse fast Fourier transforms (second row), respectively
Time integration of non-autonomous nonlinear test equation in two (left) and three (right) space dimensions with $\vartheta = 10$ by quasi-Magnus exponential integrators and operator splitting methods, see (8), (21), and Figure 1. Global errors versus time stepsizes (first row) or number of fast Fourier transforms and inverse fast Fourier transforms (second row), respectively
Time integration of two-dimensional rotational Gross-Pitaevskii equation (22) by sixth-order modified CFQM exponential integrator (20) and optimised sixth-order splitting method proposed in [18]. Solution profile $|{\psi(x, t)}|^2$ displayed for spatial section $x \in [- 5, 5]^2$ and time $t = 15$. Consistent results obtained by time-splitting generalised-Laguerre-Fourier spectral methods [22,Eq. (7)-(8),Fig. 2]
 [1] Jeremy L. Marzuola, Michael I. Weinstein. Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1505-1554. doi: 10.3934/dcds.2010.28.1505 [2] A. Alamo, J. M. Sanz-Serna. Word combinatorics for stochastic differential equations: Splitting integrators. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2163-2195. doi: 10.3934/cpaa.2019097 [3] Roy H. Goodman, Jeremy L. Marzuola, Michael I. Weinstein. Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 225-246. doi: 10.3934/dcds.2015.35.225 [4] Michael Entov, Leonid Polterovich, Daniel Rosen. Poisson brackets, quasi-states and symplectic integrators. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1455-1468. doi: 10.3934/dcds.2010.28.1455 [5] Norman E. Dancer. On the converse problem for the Gross-Pitaevskii equations with a large parameter. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2481-2493. doi: 10.3934/dcds.2014.34.2481 [6] Mats Vermeeren. Modified equations for variational integrators applied to Lagrangians linear in velocities. Journal of Geometric Mechanics, 2019, 11 (1) : 1-22. doi: 10.3934/jgm.2019001 [7] Jorge Cortés. Energy conserving nonholonomic integrators. Conference Publications, 2003, 2003 (Special) : 189-199. doi: 10.3934/proc.2003.2003.189 [8] Yujin Guo, Xiaoyu Zeng, Huan-Song Zhou. Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3749-3786. doi: 10.3934/dcds.2017159 [9] Dong Deng, Ruikuan Liu. Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3175-3193. doi: 10.3934/dcdsb.2018306 [10] Pedro L. García, Antonio Fernández, César Rodrigo. Variational integrators for discrete Lagrange problems. Journal of Geometric Mechanics, 2010, 2 (4) : 343-374. doi: 10.3934/jgm.2010.2.343 [11] Oscar E. Fernandez, Anthony M. Bloch, P. J. Olver. Variational Integrators for Hamiltonizable Nonholonomic Systems. Journal of Geometric Mechanics, 2012, 4 (2) : 137-163. doi: 10.3934/jgm.2012.4.137 [12] Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337 [13] Claude Le Bris, Frédéric Legoll. Integrators for highly oscillatory Hamiltonian systems: An homogenization approach. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 347-373. doi: 10.3934/dcdsb.2010.13.347 [14] Fasma Diele, Carmela Marangi. Positive symplectic integrators for predator-prey dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2661-2678. doi: 10.3934/dcdsb.2017185 [15] Leonardo Colombo, Fernando Jiménez, David Martín de Diego. Variational integrators for mechanical control systems with symmetries. Journal of Computational Dynamics, 2015, 2 (2) : 193-225. doi: 10.3934/jcd.2015003 [16] Werner Bauer, François Gay-Balmaz. Variational integrators for anelastic and pseudo-incompressible flows. Journal of Geometric Mechanics, 2019, 11 (4) : 511-537. doi: 10.3934/jgm.2019025 [17] Benedict Leimkuhler, Charles Matthews, Tiffany Vlaar. Partitioned integrators for thermodynamic parameterization of neural networks. Foundations of Data Science, 2019, 1 (4) : 457-489. doi: 10.3934/fods.2019019 [18] Shalva Amiranashvili, Raimondas  Čiegis, Mindaugas Radziunas. Numerical methods for a class of generalized nonlinear Schrödinger equations. Kinetic & Related Models, 2015, 8 (2) : 215-234. doi: 10.3934/krm.2015.8.215 [19] Suna Ma, Huiyuan Li, Zhimin Zhang. Novel spectral methods for Schrödinger equations with an inverse square potential on the whole space. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1589-1615. doi: 10.3934/dcdsb.2018221 [20] Ko-Shin Chen, Peter Sternberg. Dynamics of Ginzburg-Landau and Gross-Pitaevskii vortices on manifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1905-1931. doi: 10.3934/dcds.2014.34.1905

Impact Factor:

## Tools

Article outline

Figures and Tables