Multi-symplectic methods for generalized Schrödinger equations
Introduction
Hamiltonian systems possess a variety of geometric properties, the most notable of which is a symplectic structure. When numerically simulating Hamiltonian ODEs it has proven advantageous to preserve the underlying symplectic structure. In particular, let be canonical coordinates and let be a smooth function. A Hamiltonian system in , together with its associated variational system of equations, are of the form where is the identity matrix and a symmetric matrix. The symplectic form is conserved by the phase flow, i.e. ωt=0. Symplectic integrators, which are approximations to (1) that preserve a discretization of the symplectic conservation law, have demonstrated a remarkable ability to preserve the phase space geometry for very long time [4].
In an effort to generalize these ideas to Hamiltonian PDEs, the concepts of multi-symplecticity and multi-symplectic integrators have recently been introduced [1], [2], [3], [8], [9]. A Hamiltonian PDE with N space dimensions is said to be multi-symplectic if it can be written as where are skew-symmetric matrices and a smooth function. The variational system of equations is given by and associated with it are the N+1 two-forms which define a symplectic structure associated with time and the space directions xi, respectively. Multi-symplectic integrators are schemes which preserve the multi-symplectic structure of (2) and can be constructed, for example, by concatenating uni-directional ODE symplectic integrators [3], [9].
This approach was used to develop a multi-symplectic centered cell (MS-CC) discretization for the nonlinear Schrödinger (NLS) equation [6]. The multi-symplectic integrator was found to be simple, fast and more efficient than standard integrators in reproducing the qualitative features of the wave profile and possessed remarkable conservation properties for local as well as global invariants [6]. However, the fidelity of the MS-CC integrator diminished as the complexity of the waveform increased.
The NLS equation in two space dimensions with an external potential is used to model the mean-field dynamics of a dilute-gas Bose–Einstein condenstate (BEC) [5]. The equation is referred to as the Gross–Pitaevskii (GP) equation and, although it has the additional feature of variable coefficients, has a multi-symplectic formulation [7]. Systems with spatial dimension D≥2 are particularly important as they provide an important test for applicability and scalability of the properties of multi-symplectic integrators.
In this paper, new multi-symplectic spectral (MS-S) integrators are developed for the NLS and GP equations. For the one-dimensional NLS equation we find that the MS-S discretization outperforms the MS-CC discretization. The MS-S integrator more faithfully captures the properties of the complicated solutions that pose problems for the MS-CC integrator. Further, the MS-S integrator better resolves the local conservation laws. The MS-S method is found to scale well to several spatial dimensions and is, in fact, the method of choice for the variable coefficient GP equation.
Section snippets
Local and global conservation laws
For Hamiltonian PDEs, an important feature of a multi-symplectic structure is the multi-symplectic conservation law which can be obtained via the variational equation (3). Conservation of multi-symplecticity (5) is analogous to preservation of the two-form, ωt=0, for Hamiltonian ODEs.
One consequence of multi-symplecticity is that when the Hamiltonian is independent of t and xi, the PDE has local energy and momentum conservation laws (MCLs) [3]
A multi-symplectic finite difference scheme
Let the discretization of the multi-symplectic PDE (2) and the conservation law of multi-symplecticity be written schematically as using the notation , and ∂ti,j and ∂xi,j are discretizations of the corresponding derivatives ∂t and ∂x. Definition 1 The numerical scheme (8) is said to be multi-symplectic if (9) is a discrete conservation law for (8).
The NLS equation
The focusing one-dimensional NLS equation: can be written in multi-symplectic form by letting u=p+iq and introducing the conjugate momenta v=px,w=qx [6]. Separating (15) into real and imaginary parts, the multi-symplectic form (Eq. (2) with N=1) for the NLS equation is obtained with and Hamiltonian . The multi-symplectic conservation law for the NLS equation is given by
The GP equation
The two-dimensional GP equation is used to model a repulsive dilute-gas BEC in a lattice potential. After rescaling the physical variables, the GP equation is given by where u(x,y,t) is the macroscopic wave function of the condensate and an experimentally generated macroscopic potential. The GP equation has the form of a generalized NLS equation with a variable coefficient. The parameter α determines whether (24) is repulsive (α=1, defocusing nonlinearity),
Numerical experiments
We are interested in simulating multi-phase quasi-periodic (in time) solutions to the NLS and GP equations under periodic boundary conditions.
Acknowledgements
Support for this research was provided by the National Science Foundation through grant NSF-0204714.
A.L. Islas received his Masters degree in Mathematics from New York University and is currently a PhD student at Old Dominion University. His research interests are numerical partial differential equations.
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Cited by (30)
Two classes of linearly implicit local energy-preserving approach for general multi-symplectic Hamiltonian PDEs
2020, Journal of Computational PhysicsCitation Excerpt :In the last decade, many systematic methodologies (also called multi-symplectic integrators) holding the discrete version of the MSCL have been proposed for the system (1) in one dimension, such as the box/Preissmann scheme [26], the Euler-box scheme [30], the Fourier pseudo-spectral collocation scheme [14], the wavelet collocation scheme [43] and the diamond scheme [29]. These multi-symplectic integrators have been successfully applied to the Schrödinger-type equations [14,21], KdV equation [2], RLW equation [7], the Camassa-Holm equation [16], Maxwell's equations [24,38] and so on. For more details, one can refer to the review paper [26] and references therein.
On the multi-symplectic structure of Boussinesq-type systems. II: Geometric discretization
2019, Physica D: Nonlinear PhenomenaCitation Excerpt :The PRK semi-discretization avoids the singularity in cases like the nonlinear wave equation, the NLS equation or the ‘good’ Boussinesq equation [26], and the dispersion relation does not contain spurious waves, although only covers a discontinuous part of the continuous frequency. A second approach in constructing MS integrators may be represented e.g. by the references [27–30], where a discrete multi-symplectic property of a discretization based on Fourier pseudospectral approximation in space and a symplectic time integration is analysed in different PDEs with MS structure. This approach pays attention to the boundary conditions (of periodic type in the cases treated in the previous references), showing how they force to treat time and space variables in a different way.
Almost structure-preserving analysis for weakly linear damping nonlinear Schrödinger equation with periodic perturbation
2017, Communications in Nonlinear Science and Numerical SimulationDispersion-managed solitons in fibre systems and lasers
2012, Physics ReportsSymplectic structure-preserving integrators for the two-dimensional GrossPitaevskii equation for BEC
2011, Journal of Computational and Applied MathematicsSymplectic and multi-symplectic wavelet collocation methods for two-dimensional Schrödinger equations
2011, Applied Numerical Mathematics
A.L. Islas received his Masters degree in Mathematics from New York University and is currently a PhD student at Old Dominion University. His research interests are numerical partial differential equations.
C.M. Schober received her PhD in Applied Mathematics from the University of Arizona and is an Associate Professor in the Department of Mathematics at Old Dominion University (on leave) and at the University of Central Florida. Her main research interests are nonlinear wave equations and their applications in nonlinear optics and fluid dynamics as well as geometric integrators for nonlinear partial differential equations.