Multisymplectic Formulation Research Articles

Multi-Symplectic Simulation on Soliton-Collision for Nonlinear Perturbed Schrödinger Equation

Seeking solitary wave solutions and revealing their interactional characteristics for nonlinear evolution equations help us lot to comprehend the motion laws of the microparticles. As a local nonlinear dynamic behavior, the soliton-collision is difficult to be reproduced numerically. In this paper, the soliton-collision process in the nonlinear perturbed Schrödinger equation is simulated employing the multi-symplectic method. The multi-symplectic formulations are derived including the multi-symplectic form and three local conservation laws of the nonlinear perturbed Schrödinger equation. Employing the implicit midpoint rule, we construct a multi-symplectic scheme, which is equivalent to the Preissmann box scheme, for the nonlinear perturbed Schrödinger equation. The elegant structure-preserving properties of the multi-symplectic scheme are illustrated by the tiny maximum absolute residual of the discrete multi-symplectic structure at each time step in the numerical simulations. The effects of the perturbation strength on the soliton-collision in the nonlinear perturbed Schrödinger equation are reported in the numerical results in detail.

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.

Discrete Lagrange problems with constraints valued in a Lie group

The Lagrange problem is established in the discrete field theory subject to constraints with values in a Lie group. For the admissible sections that satisfy a certain regularity condition, we prove that the critical sections of such problems are the solutions of a canonically unconstrained variational problem associated with the Lagrange problem (discrete Lagrange multiplier rule). This variational problem has a discrete Cartan 1-form, from which a Noether theory of symmetries and a multisymplectic form formula are established. The whole theory is applied to the Euler-Poincaré reduction in the discrete field theory, concluding as an illustration with the remarkable example of the harmonic maps of the discrete plane in the Lie group SO(n).

Global energy preserving model reduction for multi-symplectic PDEs

Many Hamiltonian systems can be recast in multi-symplectic form. We develop a reduced-order model (ROM) for multi-symplectic Hamiltonian partial differential equations (PDEs) that preserves the global energy. The full-order solutions are obtained by finite difference discretization in space and the global energy preserving average vector field (AVF) method. The ROM is constructed in the same way as the full-order model (FOM) applying proper orthogonal decomposition (POD) with the Galerkin projection. The reduced-order system has the same structure as the FOM, and preserves the discrete reduced global energy. Applying the discrete empirical interpolation method (DEIM), the reduced-order solutions are computed efficiently in the online stage. A priori error bound is derived for the DEIM approximation to the nonlinear Hamiltonian. The accuracy and computational efficiency of the ROMs are demonstrated for the Korteweg de Vries (KdV) equation, Zakharov-Kuznetzov (ZK) equation, and nonlinear Schrödinger (NLS) equation in multi-symplectic form. Preservation of the reduced energies shows that the reduced-order solutions ensure the long-term stability of the solutions.

Efficient linearized local energy-preserving method for the Kadomtsev-Petviashvili equation

<p style='text-indent:20px;'>A linearized implicit local energy-preserving (LEP) scheme is proposed for the KPI equation by discretizing its multi-symplectic Hamiltonian form with the Kahan's method in time and symplectic Euler-box rule in space. It can be implemented easily, and also it is less storage-consuming and more efficient than the fully implicit methods. Several numerical experiments, including simulations of evolution of the line-soliton and lump-type soliton and interaction of the two lumps, are carried out to show the good performance of the scheme.</p>

Cauchy data space and multisymplectic formulation of conformal classical field theories

In this article we inspect the dynamics of classical field theories with a local conformal behavior. Our interest in the multisymplectic setting comes from its suitable description of field theories, and the conformal character has been added to account for field theories that are scale invariant, flat spaces, and because some conformal fields can be exactly solved or classified. In particular, we will solve the example of a conformal scalar field using the geometric Hamilton–Jacobi theory that is explicitly proposed for conformal fields on a multisymplectic manifold. To complete the geometric approach to study field theories, we propose the Hamilton–Jacobi theory for conformal fields in a Cauchy data space, in which space and time are split separately and the dynamics is depicted in an infinite-dimensional manifold.

Discrete conservation laws for finite element discretisations of multisymplectic PDEs

In this work we propose a new, arbitrary order space-time finite element discretisation for Hamiltonian PDEs in multisymplectic formulation. We show that the new method which is obtained by using both continuous and discontinuous discretisations in space, admits a local and global conservation law of energy. We also show existence and uniqueness of solutions of the discrete equations. Further, we illustrate the error behaviour and the conservation properties of the proposed discretisation in extensive numerical experiments on the linear and nonlinear wave equation and the nonlinear Schrödinger equation.

A review on geometric formulations for classical field theory: the Bonzom–Livine model for gravity

Motivated by the study of physical models associated with general relativity, we review some finite-dimensional, geometric and covariant formulations that allow us to characterize in a simple manner the symmetries for classical field theory by implementing an appropriate fibre-bundle structure, either at the Lagrangian, the Hamiltonian multisymplectic or the polysymplectic levels. In particular, we are able to formulate Noether’s theorems by means of the covariant momentum maps and to systematically introduce a covariant Poisson–Hamiltonian framework. Also, by focusing on the space plus time decomposition for a generic classical field theory and its relation to these geometric formulations, we are able to successfully recover the gauge content and the true local degrees of freedom for the theory. In order to illustrate the relevance of these geometric frameworks, we center our attention to the analysis of a model for three-dimensional theory of general relativity that involves an arbitrary Immirzi-like parameter. At the Lagrangian level, we reproduce the field equations of the system which for this model turn out to be equivalent to the vanishing torsion condition and the Einstein equations. We also concentrate on the analysis of the gauge symmetries of the system in order to obtain the Lagrangian covariant momentum map associated with the theory and, consequently, its corresponding Noether currents. Next, within the Hamiltonian multisymplectic approach, we aim our attention to describing how the gauge symmetries of the model yield covariant canonical transformations on the covariant multimomenta phase-space, thus giving rise to the existence of a covariant momentum map. Besides, we analyze the physical system under consideration within the De Donder–Weyl canonical theory implemented at the polysymplectic level, thus establishing a relation from the covariant momentum map to the conserved currents of the theory within this covariant Hamiltonian approach. Finally, after performing the space plus time decomposition of the space-time manifold, and taking as a starting point the Hamiltonian multisymplectic formulation, we are able to recover both the extended Hamiltonian and the gauge structure content that characterize the gravity model of our interest within the instantaneous Dirac–Hamiltonian formulation.

An Overview of the Hamilton–Jacobi Theory: the Classical and Geometrical Approaches and Some Extensions and Applications

This work is devoted to review the modern geometric description of the Lagrangian and Hamiltonian formalisms of the Hamilton–Jacobi theory. The relation with the “classical” Hamiltonian approach using canonical transformations is also analyzed. Furthermore, a more general framework for the theory is also briefly explained. It is also shown how, from this generic framework, the Lagrangian and Hamiltonian cases of the theory for dynamical systems are recovered, as well as how the model can be extended to other types of physical systems, such as higher-order dynamical systems and (first-order) classical field theories in their multisymplectic formulation.

Hamiltonian multiform description of an integrable hierarchy

Motivated by the notion of Lagrangian multiforms, which provide a Lagrangian formulation of integrability, and by the results of the authors on the role of covariant Hamiltonian formalism for integrable field theories, we propose the notion of Hamiltonian multiforms for integrable 1 + 1-dimensional field theories. They provide the Hamiltonian counterpart of Lagrangian multiforms and encapsulate in a single object an arbitrary number of flows within an integrable hierarchy. For a given hierarchy, taking a Lagrangian multiform as the starting point, we provide a systematic construction of a Hamiltonian multiform based on a generalization of techniques of covariant Hamiltonian field theory. This also produces two other important objects: a symplectic multiform and the related multi-time Poisson bracket. They reduce to a multisymplectic form and the related covariant Poisson bracket if we restrict our attention to a single flow in the hierarchy. Our framework offers an alternative approach to define and derive conservation laws for a hierarchy. We illustrate our results on three examples: the potential Korteweg–de Vries hierarchy, the sine-Gordon hierarchy (in light-cone coordinates), and the Ablowitz–Kaup–Newell–Segur hierarchy.

Covariant momentum map thermodynamics for parametrized field theories

A general-covariant statistical framework capable of describing classical fluctuations of the gravitational field is a thorny open problem in theoretical physics, yet ultimately necessary to understand the nature of the gravitational interaction, and a key to quantum gravity. Inspired by Souriau’s symplectic generalization of the Maxwell–Boltzmann–Gibbs equilibrium in Lie group thermodynamics, we investigate a space–time-covariant formulation of statistical mechanics for parametrized first-order field theories, as a simplified model sharing essential general covariant features with canonical general relativity. Starting from a covariant multi-symplectic phase space formulation, we define a general-covariant notion of Gibbs state in terms of the covariant momentum map associated with the lifted action of the diffeomorphisms group on the extended phase space. We show how such a covariant notion of equilibrium encodes the whole information about symmetry, gauge and dynamics carried by the theory, associated with a canonical spacetime foliation, where the covariant choice of a reference frame reflects in a Lie algebra-valued notion of local temperature. We investigate how physical equilibrium, hence time evolution, emerges from such a state and the role of the gauge symmetry in the thermodynamic description.

Generalized Multi-Symplectic Numerical Implementation of Dynamic Responses for Saturated Poroelastic Timoshenko Beam

Based on the porous media theory and Timoshenko beam theory, properties of dynamic responses in fluid-solid coupled incompressible saturated poroelastic Timoshenko beam are investigated by generalized multi-symplectic method. Dynamic response equation set of incompressible saturated poroelastic Timoshenko beam is presented at first. Then a first order generalized multi-symplectic form of this dynamic response equation set is constructed, and errors of generalized multi-symplectic conservation law, generalized multi-symplectic local momentum and generalized multi-symplectic local energy are also derived. A Preissmann Box generalized multi-symplectic scheme of the dynamic response equation set is presented, the discrete errors of generalized multi-symplectic conservation law, generalized multi-symplectic local momentum conservation law and generalized multi-symplectic local energy conservation law are also obtained. In view of the dynamic responses of incompressible saturated poroelastic Timoshenko cantilever beam with two ends permeable and free end subjected to the step load, the transverse dynamic response process of the solid skeleton is simulated numerically, the evolution processes of solid effective stress and the equivalent moment of the pore fluid pressure over time are also presented numerically. The effects of fluid-solid coupled interaction parameter and slenderness ratio of the beam on the solid dynamic response process are revealed, as well as the effects on all generalized multi-symplectic numerical errors are checked simultaneously. From results obtained, the processes for solid deflection, solid effective stress and the equivalent moment of the pore fluid pressure approaching to their steady response values are all shortened with increasing of fluid-solid coupled interaction parameter, while the response process of solid deflection and the pore fluid equivalent moment are lengthened with increasing of slenderness ratio of the beam. Moreover, the steady value of solid deflection is much closer to the static deflection value of classic single phase elastic Euler-Bernoulli beam with increasing of the slenderness ratio. As time goes on, the solid skeleton of the beam will support all outside load, so equivalent moment of the pore fluid pressure becomes zero at last. In addition, it is presented all generalized multi-symplectic numerical errors decrease with the decreasing of parameters representing the dissipation effect for the dynamic system.

Covariant Variational Evolution and Jacobi brackets: Fields

The analysis of the covariant brackets on the space of functions on the solutions to a variational problem in the framework of contact geometry initiated in the companion letter[Formula: see text] is extended to the case of the multisymplectic formulation of the free Klein–Gordon theory and of the free Schrödinger equation.

Multi-Symplectic Method for the Logarithmic-KdV Equation

The multi-symplectic integrator, as a numerical integration approach with symmetry, is known to have the characteristic of preserving the qualitative features and geometric properties of certain systems. Using the multi-symplectic integrator, the numerical simulation of the Gaussian solitary wave propagation of the logarithmic Korteweg–de Vries (logarithmic-KdV) equation was investigated. The multi-symplectic formulation of the logarithmic-KdV equation was explored by introducing some intermediate variables. A fully implicit version of the centered box scheme was used to discretize the multi-symplectic equations. In addition, numerical experiments were carried out to show the conservative properties of the proposed scheme.

The problem of Lagrange in discrete field theory

A discrete Lagrange problem in a fibered manifold on a cellular complex is defined as a Lagrangian density and a constraint submanifold in the 1-jets space of the manifold. After defining the concepts of admissible section and infinitesimal admissible variation, the aim of these problems is to find admissible sections that are critical for the action functional of the Lagrangian density with respect to the infinitesimal admissible variations. For admissible sections satisfying a certain regularity condition, we prove that such critical sections are the solutions of an extended unconstrained discrete variational problem canonically associated to the problem of Lagrange (discrete Lagrange multiplier rule). This variational problem has a discrete Cartan 1-form, from which a Noether theory of symmetries and a multisymplectic form formula are established for the discrete Lagrange problems considered. Finally, the whole theory is illustrated with an example of geometric interest: the discrete harmonic 1-forms on the elliptic or hyperbolic discrete plane, for which we obtain two explicit variational integrators: one with initial conditions (Cauchy problem) and other with boundary conditions (Dirichlet problem).

New multisymplectic approach to the Metric-Affine (Einstein-Palatini) action for gravity

We present a covariant multisymplectic formulation for the Einstein-Palatini (or Metric-Affine) model of General Relativity (without energy-matter sources). As it is described by a first-order affine Lagrangian (in the derivatives of the fields), it is singular and, hence, this is a gauge field theory with constraints. These constraints are obtained after applying a constraint algorithm to the field equations, both in the Lagrangian and the Hamiltonian formalisms. In order to do this, the covariant field equations must be written in a suitable geometrical way, using integrable distributions which are represented by multivector fields of a certain type. We obtain and explain the geometrical and physical meaning of the Lagrangian constraints and we construct the multimomentum (covariant) Hamiltonian formalism. The gauge symmetries of the model are discussed in both formalisms and, from them, the equivalence with the Einstein-Hilbert model is established.

Local structure-preserving algorithms for general multi-symplectic Hamiltonian PDEs

Many PDEs can be recast into the general multi-symplectic formulation possessing three local conservation laws. We devote the present paper to some systematic methods, which hold the discrete versions of the local conservation laws respectively, for the general multi-symplectic PDEs. For the original problem subjected to appropriate boundary conditions, the proposed methods are globally conservative. The proposed methods are successfully applied to many one-dimensional and multi-dimensional Hamiltonian PDEs, such as KdV equation, G–P equation, Maxwell’s equations and so on. Numerical experiments are carried out to verify the theoretical analysis.

Structure-preserving algorithm for thermal conduction of local thermal nonequilibrium saturated porous medium

Based on the two-energy equation model and constitutive equations of local thermal nonequilibrium saturated porous media, one-dimensional thermal conduction equations of solid skeleton and pore fluid are established. Introducing orthogonal variables, a first order generalized multi-symplectic structure-preserving form of thermal conduction equations is derived as well as the errors of generalized multi-symplectic conservation law and local momentum. The temperature profiles of two phases are obtained and the effect of the heat exchange coefficient between two phases on the process from local thermal nonequilibrium to local thermal equilibrium is revealed numerically. According to the relative error between numerical solution and analytical solution derived from the separating variables method, it can be concluded that this structure-preserving scheme is a valid scheme with high accuracy. Numerical errors of generalized multi-symplectic conservation law and local momentum are presented for two typical types of the thermal conduction in one-dimensional saturated porous media. According to these numerical results, it can be concluded that this structure-preserving algorithm has long-time numerical stability and good conservation properties.

New variational and multisymplectic formulations of the Euler-Poincaré equation on the Virasoro-Bott group using the inverse map.

We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler-Poincaré equations defined on the Virasoro-Bott group, by using the inverse map (also called 'back-to-labels' map). This family contains as special cases the well-known Korteweg-de Vries, Camassa-Holm and Hunter-Saxton soliton equations. In the conclusion section, we sketch opportunities for future work that would apply the new Clebsch momentum map with 2-cocycles derived here to investigate a new type of interplay among nonlinearity, dispersion and noise.

Multisymplectic unified formalism for Einstein-Hilbert gravity

We present a covariant multisymplectic formulation for the Einstein-Hilbert model of general relativity. As it is described by a second-order singular Lagrangian, this is a gauge field theory with constraints. The use of the unified Lagrangian-Hamiltonian formalism is particularly interesting when it is applied to these kinds of theories, since it simplifies the treatment of them, in particular, the implementation of the constraint algorithm, the retrieval of the Lagrangian description, and the construction of the covariant Hamiltonian formalism. In order to apply this algorithm to the covariant field equations, they must be written in a suitable geometrical way, which consists of using integrable distributions, represented by multivector fields of a certain type. We apply all these tools to the Einstein-Hilbert model without and with energy-matter sources. We obtain and explain the geometrical and physical meaning of the Lagrangian constraints and we construct the multimomentum (covariant) Hamiltonian formalisms in both cases. As a consequence of the gauge freedom and the constraint algorithm, we see how this model is equivalent to a first-order regular theory, without gauge freedom. In the case of the presence of energy-matter sources, we show how some relevant geometrical and physical characteristics of the theory depend on the type of source. In all the cases, we obtain explicitly multivector fields which are solutions to the gravitational field equations. Finally, a brief study of symmetries and conservation laws is done in this context.