Multisymplectic Structures Research Articles

Geometry of bundle-valued multisymplectic structures with Lie algebroids

We study multisymplectic structures taking values in vector bundles with connections from the viewpoint of the Hamiltonian symmetry. We introduce the notion of bundle-valued n-plectic structures and exhibit some properties of them. In addition, we define bundle-valued homotopy momentum sections for bundle-valued n-plectic manifolds with Lie algebroids to discuss momentum map theories in both cases of quaternionic Kähler manifolds and hyper-Kähler manifolds. Furthermore, we generalize the Marsden-Weinstein-Meyer reduction theorem for symplectic manifolds and construct two kinds of reductions of vector-valued 1-plectic manifolds.

Multi-symplectic simulations of W/M-shape-peaks solitons and cuspons for FORQ equation

The FORQ equation, which is bi-Hamiltonian, integrable, and hosts of a range of soliton solutions, is viewed afresh from the viewpoint of multi-symplectic structures. A multi-symplectic formulation is derived and used as a guide in the construction of numerical methods for simulation; in particular, a multi-symplectic Preissmann box scheme is constructed for the FORQ equation, which is second-order accurate in both space and time. Simulations show that the cuspons and the W/M-shape-peaks solitons for the FORQ equation are reproduced accurately. To verify the precision and the validity of the numerical results, the relative errors between the numerical results and the theoretical solutions, and the maximum values of the errors of the local conservation laws are presented.

On the multi-symplectic structure of Boussinesq-type systems. II: Geometric discretization

In this paper we consider the numerical approximation of systems of Boussinesq-type to model surface wave propagation. Some theoretical properties of these systems (multi-symplectic and Hamiltonian formulations, well-posedness and existence of solitary-wave solutions) were previously analyzed by the authors in Part I. As a second part of the study, considered here is the construction of geometric schemes for the numerical integration. By using the method of lines, the geometric properties, based on the multi-symplectic and Hamiltonian structures, of different strategies for the spatial and time discretizations are discussed and illustrated.

Multisymplectic structures and invariant tensors for Lie systems

A Lie system is a non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot–Guldberg Lie algebra. This work pioneers the analysis of Lie systems admitting a Vessiot–Guldberg Lie algebra of Hamiltonian vector fields relative to a multisymplectic structure: the multisymplectic Lie systems. Geometric methods are developed to consider a Lie system as a multisymplectic one. By attaching a multisymplectic Lie system via its multisymplectic structure with a tensor coalgebra, we find multisymplectic and coalgebra methods to derive superposition rules, constants of motion, and invariant tensor fields relative to its evolution. Our results are illustrated with examples occurring in physics and mathematics such as Schwarz equations, control systems, or diffusion-type equations. This also shows the interest of multisymplectic techniques in ordinary differential equations, which represents a not much investigated field of application of multisymplectic geometry.

On nonparaxial nonlinear Schrödinger-type equations

In this paper the one-dimensional nonparaxial nonlinear Schrödinger equation is considered. This was proposed as an alternative to the classical nonlinear Schrödinger equation in those situations where the assumption of paraxiality may fail. The paper contributes to the mathematical properties of the equation in a two-fold way. First, some theoretical results on linear well-posedness, Hamiltonian and multi-symplectic formulations are derived. Then we propose to take into account these properties in order to deal with the numerical approximation. In this sense, different numerical procedures that preserve the Hamiltonian and multi-symplectic structures are discussed and illustrated with numerical experiments.

Multisymplectic structures induced by symplectic structures

A symplectic structure on a smooth manifold M induces multisymplectic structures on M, defined by the powers of the symplectic form. We study the relationship between the symplectic structure and these multisymplectic structures.

Dissipation-preserving spectral element method for damped seismic wave equations

This article describes the extension of the conformal symplectic method to solve the damped acoustic wave equation and the elastic wave equations in the framework of the spectral element method. The conformal symplectic method is a variation of conventional symplectic methods to treat non-conservative time evolution problems, which has superior behaviors in long-time stability and dissipation preservation. To reveal the intrinsic dissipative properties of the model equations, we first reformulate the original systems in their equivalent conformal multi-symplectic structures and derive the corresponding conformal symplectic conservation laws. We thereafter separate each system into a conservative Hamiltonian system and a purely dissipative ordinary differential equation system. Based on the splitting methodology, we solve the two subsystems respectively. The dissipative one is cheaply solved by its analytic solution. While for the conservative system, we combine a fourth-order symplectic Nyström method in time and the spectral element method in space to cover the circumstances in realistic geological structures involving complex free-surface topography. The Strang composition method is adopted thereby to concatenate the corresponding two parts of solutions and generate the completed conformal symplectic method. A relative larger Courant number than that of the traditional Newmark scheme is found in the numerical experiments in conjunction with a spatial sampling of approximately 5 points per wavelength. A benchmark test for the damped acoustic wave equation validates the effectiveness of our proposed method in precisely capturing dissipation rate. The classical Lamb problem is used to demonstrate the ability of modeling Rayleigh wave in elastic wave propagation. More comprehensive numerical experiments are presented to investigate the long-time simulation, low dispersion and energy conservation properties of the conformal symplectic methods in both the attenuating homogeneous and heterogeneous media.

On Higher Dirac Structures

We study higher-order analogues of Dirac structures, extending the multisymplectic structures that arise in field theory. We define higher Dirac structures as involutive subbundles of $TM+\wedge^k TM^*$ satisfying a weak version of the usual lagrangian condition (which agrees with it only when $k=1$). Higher Dirac structures transversal to $TM$ recover the higher Poisson structures introduced in [8] as the infinitesimal counterparts of multisymplectic groupoids. We describe the leafwise geometry underlying an involutive isotropic subbundle in terms of a distinguished 1-cocycle in a natural differential complex, generalizing the presymplectic foliation of a Dirac structure. We also identify the global objects integrating higher Dirac structures.

Stochastic symplectic and multi-symplectic methods for nonlinear Schrödinger equation with white noise dispersion

We indicate that the nonlinear Schrödinger equation with white noise dispersion possesses stochastic symplectic and multi-symplectic structures. Based on these structures, we propose the stochastic symplectic and multi-symplectic methods, which preserve the continuous and discrete charge conservation laws, respectively. Moreover, we show that the proposed methods are convergent with temporal order one in probability. Numerical experiments are presented to verify our theoretical results.

Vector bundle valued differential forms on ℕQ-manifolds

Geometric structures on $\mathbb N Q$-manifolds, i.e.~non-negatively graded manifolds with an homological vector field, encode non-graded geometric data on Lie algebroids and their higher analogues. A particularly relevant class of structures consists of vector bundle valued differential forms. Symplectic forms, contact structures and, more generally, distributions are in this class. We describe vector bundle valued differential forms on non-negatively graded manifolds in terms of non-graded geometric data. Moreover, we use this description to present, in a unified way, novel proofs of known results, and new results about degree one $\mathbb N Q$-manifolds equipped with certain geometric structures, namely symplectic structures, contact structures, involutive distributions (already present in literature) and locally conformal symplectic structures, and generic vector bundle valued higher order forms, in particular presymplectic and multisymplectic structures (not yet present in literature).

Numerical study of long-time Camassa–Holm solution behavior for soliton transport

In this paper a three-step solution scheme is employed to numerically explore the long-time solution behavior of the Camassa–Holm equation. In the present u−P−α formation, we conduct modified equation analysis to eliminate several leading discretization error terms and perform Fourier analysis for minimizing the wave-like type of error. A three-point seventh-order spatially accurate combined compact upwind scheme is developed for the approximation of first-order derivative term. For the purpose of retaining Hamiltonian and multi-symplectic geometric structures in the non-dissipative Camassa–Holm equation, the adopted time integrator conserves symplecticity. Another main emphasis of this study is to numerically shed light on the scenario of the soliton transport.

Exact combined traveling wave solutions and multi-symplectic structure of the variant Boussinesq–Whitham–Broer–Kaup type equations

The homogeneous balance of undetermined coefficients method (HBUCM) is firstly proposed to construct not only the exact traveling wave solutions, three-wave solutions, homoclinic solutions, N-soliton solutions, but also multi-symplectic structures of some nonlinear partial differential equations (NLPDEs). By applying the proposed method to the variant Boussinesq equations (VBEs), the exact combined traveling wave solutions and a multi-symplectic structure of the VBEs are obtained directly. Then, the definition and a multi-symplectic structure of the variant Boussinesq–Whitham–Broer–Kaup type equations (VBWBKTEs) which can degenerate to the VBEs, the Whitham–Broer–Kaup equations (WBKEs) and the Broer–Kaup equations (BKEs) are given in the multi-symplectic sense. The HBUCM is also a standard and computable method, which can be generalized to obtain the exact solutions and multi-symplectic structures for some types of NLPDEs.

Variational discretizations for the generalized Rosenau-type equations

The generalized Rosenau-type equations include the Rosenau–RLW equation and the Rosenau–KdV equation, which both admit the third-order Lagrangians. In the Lagrangian framework, this paper presents the variational formulations of the generalized Rosenau-type equations as well as their multisymplectic structures. Based on the discrete variational principle, we construct the variational discretizations for solving the evolutions of solitary solutions of this class of equations. We simulate the motion of the single solitary wave, and also observe the different kind of collisions for the generalized Rosenau-type equations with various coefficients.

Numerical study of the generalised Klein–Gordon equations

In this study, we discuss an approximate set of equations describing water wave propagating in deep water. These generalised Klein–Gordon (gKG) equations possess a variational formulation, as well as a canonical Hamiltonian and multi-symplectic structures. Periodic travelling wave solutions are constructed numerically to high accuracy and compared to a seventh-order Stokes expansion of the full Euler equations. Then, we propose an efficient pseudo-spectral discretisation, which allows to assess the stability of travelling waves and localised wave packets.

New explicit multi-symplectic scheme for nonlinear wave equation

Based on splitting multi-symplectic structures, a new multi-symplectic scheme is proposed and applied to a nonlinear wave equation. The explicit multi-symplectic scheme of the nonlinear wave equation is obtained, and the corresponding multi-symplectic conservation property is proved. The backward error analysis shows that the explicit multi-symplectic scheme has good accuracy. The sine-Gordon equation and the Klein-Gordon equation are simulated by an explicit multi-symplectic scheme. The numerical results show that the new explicit multi-symplectic scheme can well simulate the solitary wave behaviors of the nonlinear wave equation and approximately preserve the relative energy error of the equation.

Remarks on Hamiltonian structures in G2-geometry

In this article, we treat G2-geometry as a special case of multisymplectic geometry and make a number of remarks regarding Hamiltonian multivector fields and Hamiltonian differential forms on manifolds with an integrable G2-structure; in particular, we discuss existence and make a number of identifications of the spaces of Hamiltonian structures associated to the two multisymplectic structures associated to an integrable G2-structure. Along the way, we prove some results in multisymplectic geometry that are generalizations of results from symplectic geometry.

Lagrangian distributions and connections in multisymplectic and polysymplectic geometry

We discuss the interplay between lagrangian distributions and connections in (generalized) symplectic geometry, beginning with the traditional case of symplectic manifolds and then passing to the more general context of poly- and multisymplectic structures on fiber bundles, which is relevant for the covariant hamiltonian formulation of classical field theory. In particular, we generalize Weinsteinʼs tubular neighborhood theorem for symplectic manifolds carrying a (simple) lagrangian foliation to this situation. In all cases, the Bott connection, or an appropriately extended version thereof, plays a central role.

Nonstandard finite difference variational integrators for nonlinear Schrödinger equation with variable coefficients

In this paper, the idea of nonstandard finite difference discretization is employed to develop two variational integrators for the nonlinear Schrödinger equation with variable coefficients. These integrators are naturally multi-symplectic, and their multi-symplectic structures are presented by the multi-symplectic form formulas. Local truncation errors and convergences of the integrators are briefly discussed. The effectiveness and efficiency of the proposed schemes, such as the convergence order, numerical stability, and the capability in preserving the norm conservation, are verified in the numerical experiments.

Numerical dispersion analysis of a multi-symplectic scheme for the three dimensional Maxwell’s equations

In this paper, we study a multi-symplectic scheme for three dimensional Maxwell’s equations in a simple medium. This is a system of PDEs with multi-symplectic structures. We prove that this multi-symplectic scheme preserves the discrete version of local and global energy conservation law and the discrete divergence. Furthermore, we extend the discussion to several dispersion properties of the multi-symplectic scheme including the numerical dispersion relation, the numerical group velocity, the effect of large time steps and the CFL condition.

Nonstandard Finite Difference Variational Integrators for Multisymplectic PDEs

We use the idea of nonstandard finite difference methods to derive the discrete variational integrators for multisymplectic PDEs. We obtain a nonstandard finite difference variational integrator for linear wave equation with a triangle discretization and two nonstandard finite difference variational integrators for the nonlinear Klein‐Gordon equation with a triangle discretization and a square discretization, respectively. These methods are naturally multisymplectic. Their discrete multisymplectic structures are presented by the multisymplectic form formulas. The convergence of the discretization schemes is discussed. The effectiveness and efficiency of the proposed methods are verified by the numerical experiments.