Hamiltonian Partial Differential Equations Research Articles

The multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs

In this article we consider partitioned Runge-Kutta (PRK) methods for Hamiltonian partial differential equations (PDEs) and present some sufficient conditions for multi-symplecticity of PRK methods of Hamiltonian PDEs.

Transient radiative behavior of Hamiltonian systems in finite domains

We show that Hamiltonian partial differential equations on compact spatial domains can display transient radiative behavior, usually associated with infinite domains. This is done by considering a model of a single oscillator coupled to a wave field, which damps due to the resonant coupling of the oscillator with a discrete frequency with the continuous spectrum of the field. The analysis carried out illustrates that despite the “discretization” of the continuous spectrum due to the finiteness of the domain, a remnant of the resonance mechanism persists. In particular, this explains how numerical computations on bounded domains accurately simulate, on large but finite time scales, phenomena associated with infinite spatial domains. Numerical simulations in the present model show good agreement with our theoretical predictions.

Diffusion of Power in Randomly Perturbed Hamiltonian Partial Differential Equations

We study the evolution of the energy (mode-power) distribution for a class of randomly perturbed Hamiltonian partial differential equations and derive master equations for the dynamics of the expected power in the discrete modes. In the case where the unperturbed dynamics has only discrete frequencies (finitely or infinitely many) the mode-power distribution is governed by an equation of discrete diffusion type for times of order (ɛ−2). Here ɛ denotes the size of the random perturbation. If the unperturbed system has discrete and continuous spectrum the mode-power distribution is governed by an equation of discrete diffusion-damping type for times of order (ɛ−2). The methods involve an extension of the authors’ work on deterministic periodic and almost periodic perturbations, and yield new results which complement results of others, derived by probabilistic methods.

Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations

This is a partial survey of various developments of the classical KAM theory in dynamical systems to infinite-dimensional lattice models and Hamiltonian PDEs. Particular emphasis is given on Nekhoroshev stability and growth of higher Sobolev norms. Some new results are also presented.

On a class of discretizations of Hamiltonian nonlinear partial differential equations

We present a new class of discretizations of partial differential equations (PDEs) that preserve a (momentum-like) integral of the motion. This results in an “effective” translational invariance for the dynamical problem and the absence of a Peierls–Nabarro barrier that is usually present in discretizations of Hamiltonian PDEs. A general method to construct such discretizations for any nonlinearity is given and the properties of the resulting differential–difference equations are analyzed in a number of different cases for nonlinear Klein–Gordon as well for nonlinear Schrödinger type systems. While for the former nonintegrability of the dynamical problem is evident, in the latter case numerical evidence suggests that the behavior is close to the one of integrable systems. We also show how static solutions of the equations can be constructed for these discretizations and discuss the similarities and differences of the method with previously reported ones.

Constructing the symplectic Evans matrix using maximally analytic individual vectors

For linear systems with a multi-symplectic structure, arising from the linearization of Hamiltonian partial differential equations about a solitary wave, the Evans function can be characterized as the determinant of a matrix, and each entry of this matrix is a restricted symplectic form. This variant of the Evans function is useful for a geometric analysis of the linear stability problem. But, in general, this matrix of two-forms may have branch points at isolated points, shrinking the natural region of analyticity. In this paper, a new construction of the symplectic Evans matrix is presented, which is based on individual vectors but is analytic at the branch points—indeed, maximally analytic. In fact, this result has greater generality than just the symplectic case; it solves the following open problem in the literature: can the Evans function be constructed in a maximally analytic way when individual vectors are used? Although the non-symplectic case will be discussed in passing, the paper will concentrate on the symplectic case, where there are geometric reasons for evaluating the Evans function on individual vectors. This result simplifies and generalizes the multi-symplectic framework for the stability analysis of solitary waves, and some of the implications are discussed.

Exponential averaging for Hamiltonian evolution equations

We derive estimates on the magnitude of non-adiabatic interaction between a Hamiltonian partial differential equation and a high-frequency nonlinear oscillator. Assuming spatial analyticity of the initial conditions, we show that the dynamics can be transformed to the uncoupled dynamics of an infinite-dimensional Hamiltonian system and an anharmonic oscillator, up to coupling terms which are exponentially small in a certain power of the frequency of the oscillator. The result is derived from an abstract averaging theorem for infinite-dimensional analytic evolution equations in Gevrey spaces. Refining upon a similar result by Neishtadt for analytic ordinary differential equations, the temporal estimate crucially depends on the spatial regularity of the initial condition. The result shows to what extent the strong resonances between rapid forcing and highly oscillatory spatial modes can be suppressed by the choice of sufficiently smooth initial data. An application is provided by a system of nonlinear Schrödinger equations, coupled to a rapidly forcing single mode, representing small-scale oscillations. We provide an example showing that the estimates for partial differential equations we derive here are necessarily different from those in the context of ordinary differential equations.

Conservation laws of semidiscrete Hamiltonian equations

Many evolution partial differential equations (PDEs) can be cast into Hamiltonian form. Conservation laws of these equations are related to one-parameter Hamiltonian symmetries admitted by the PDEs [P. J. Olver, Applications of Lie Groups to Differential Equations (Springer, New York, 1986)]. In this paper we consider symmetries and Noether’s theorem for semidiscrete Hamiltonian equations which are obtained by space discretization of Hamiltonian PDEs. Using symmetries, one can find conservation laws of these equations. Several applications including a transfer equation and the Korteweg–de Vries equation are presented.

The Symplectic Evans Matrix,¶and the Instability of Solitary Waves and Fronts

Hamiltonian evolution equations which are equivariant with respect to the action of a Lie group are models for physical phenomena such as oceanographic flows, optical fibres and atmospheric flows, and such systems often have a wide variety of solitary wave or front solutions. In this paper, we present a new symplectic framework for analyzing the spectral problem associated with the linearization about such solitary waves and fronts. At the heart of the analysis is a multi-symplectic formulation of Hamiltonian partial differential equations where a distinct symplectic structure is assigned for the time and space directions, with a third symplectic structure -- with two-form denoted by Omega - associated with a coordinate frame moving at the speed of the wave. This leads to a geometric decomposition and symplectification of the Evans function formulation for the linearization about solitary waves and fronts. We introduce the concept of the "symplectic Evans matrix", a matrix consisting of restricted "Omega-symplectic" forms. By applying Hodge duality to the exterior algebra formulation of the Evans function, we find that the zeros of the Evans function correspond to zeros of the determinant of the symplectic Evans matrix. Based on this formulation, we prove several new properties of the Evans function. Restricting the spectral parameter lambda to the real axis, we obtain rigorous results on the derivatives of the Evans function near the origin, based solely on the abstract geometry of the equations, and results for the large $|\lambda|$ behaviour which use primarily the symplectic structure, but also extend to the non-symplectic case. The Lie group symmetry affects the Evans function by generating zero eigenvalues of large multiplicity in the so-called systems at infinity. We present a new geometric theory which describes precisely how these zero eigenvalues behave under perturbation. By combining all these results, a new rigorous sufficient condition for instability of solitary waves and fronts is obtained. The theory applies to a large class of solitary waves and fronts including waves which are biasymptotic to a nonconstant manifold of states as $|x|$ tends to infinity. To illustrate the theory, it is applied to three examples: a Boussinesq model from oceanography, a class of nonlinear Schrodinger equations from optics and a nonlinear Klein-Gordon equation from atmospheric dynamics.

Parametrically Excited Hamiltonian Partial Differential Equations

Consider a linear autonomous Hamiltonian system with a time-periodic bound state solution. In this paper we study the structural instability of this bound state relative to time almost periodic perturbations which are small, localized, and Hamiltonian. This class of perturbations includes those whose time dependence is periodic but encompasses a large class of those with finite (quasi-periodic) or infinitely many noncommensurate frequencies. Problems of the type considered arise in many areas of applications including ionization physics and the propagation of light in optical fibers in the presence of defects. The mechanism of instability is radiation damping due to resonant coupling of the bound state to the continuum modes by the time-dependent perturbation. This results in a transfer of energy from the discrete modes to the continuum. The rate of decay of solutions is slow and hence the decaying bound states can be viewed as metastable. These results generalize those of A. Soffer and M. I. Weinstein, who treated localized time-periodic perturbations of a particular form. In the present work, new analytical issues need to be addressed in view of (i) the presence of infinitely many frequencies which may resonate with the continuum as well as (ii) the possible accumulation of such resonances in the continuous spectrum. The theory is applied to a general class of Schrodinger operators.

Optimal Prediction for Hamiltonian Partial Differential Equations

Optimal prediction methods compensate for a lack of resolution in the numerical solution of time-dependent differential equations through the use of prior statistical information. We present a new derivation of the basic methodology, show that field-theoretical perturbation theory provides a useful device for dealing with quasi-linear problems, and provide a nonlinear example that illuminates the difference between a pseudo-spectral method and an optimal prediction method with Fourier kernels. Along the way, we explain the differences and similarities between optimal prediction, the representer method in data assimilation, and duality methods for finding weak solutions. We also discuss the conditions under which a simple implementation of the optimal prediction method can be expected to perform well.

Homoclinic Tubes in Nonlinear Schr&ödinger Equation under Hamiltonian Perturbations

The concept of a homoclinic tube was introduced by Silnikov 14) in a study on the structure of the neighborhood of a homoclinic tube asymptotic to an invariant torus σ under a diffeomorphism F in a finite dimensional phase space. The asymptotic torus is of the saddle type. The homoclinic tube consists of a doubly infinite sequence of tori {σj , j = 0,±1,±2, · · ·} in the transversal intersection of the stable and unstable manifolds of σ, such that σj+1 = F ◦ σj for any j. This is a generalization of the concept of a transversal homoclinic orbit in which the points are replaced by tori. Silnikov obtained a theorem on the symbolic dynamics structures in the neighborhood of the homoclinic tube that is similar to Smale’s theorem for a transversal homoclinic orbit. 13), 12) We are interested in homoclinic tubes for several reasons: 1. Especially in high dimensions, it is often the case that the dynamics inside each invariant tube in the neighborhood of a homoclinic tube is also chaotic. We refer to such chaotic dynamics as “chaos in the small” and the symbolic dynamics of the invariant tubes as “chaos in the large”. Such cascade structures are more important than the structures in a neighborhood of a homoclinic orbit when high or infinite dimensional dynamical systems are studied. 2. Symbolic dynamics structures in the neighborhoods of homoclinic tubes are more observable than in the neighborhoods of homoclinic orbits in numerical and physical experiments. 3. When studying a high or infinite dimensional Hamiltonian system (for example, the cubic nonlinear Schrodinger equation under Hamiltonian perturbations), each invariant tube contains both KAM tori and stochastic layers (chaos in the small). Thus, not only is it the case that the dynamics inside each stochastic layer is chaotic, all these stochastic layers also move chaotically under Poincare maps. In this paper, we are interested in studying Hamiltonian partial differential equations. Denote by W (c) a normally hyperbolic center manifold, by W (cu) and W (cs)

On Numerical Methods for Hamiltonian PDEs and a Collocation Method for the Vlasov–Maxwell Equations

Hamiltonian partial differential equations often have implicit conservation laws—constants of the motion—embedded within them. It is not, in general, possible to preserve these conservation laws simply by discretization in conservative form because there is frequently only one explicit conservation law. However, by using weighted residual methods and exploiting the Hamiltonian structure of the equations it is shown that at least some of the conservation laws are preserved in a method of lines (continuous in time). In particular, the Hamiltonian can always be exactly preserved as a constant of the motion. Other conservation laws, in particular linear and quadratic Casimirs and momenta, can sometimes be conserved too, depending on the details of the equations under consideration and the form of discretization employed. Collocation methods also offer automatic conservation of linear and quadratic Casimirs. Some standard discretization methods, when applied to Hamiltonian problems are shown to be derived from a numerical approximation to the exact Poisson bracket of the system. A method for the Vlasov–Maxwell equations based on Legendre–Gauss–Lobatto collocation is presented as an example of these ideas.

Smoothed dynamics of highly oscillatory Hamiltonian systems

We consider the numerical treatment of Hamiltonian systems that contain a potential which grows large when the system deviates from the equilibrium value of the potential. Such systems arise, e.g., in molecular dynamics simulations and the spatial discretization of Hamiltonian partial differential equations. Since the presence of highly oscillatory terms in the solutions forces any explicit integrator to use very small step size, the numerical integration of such systems provides a challenging task. It has been suggested before to replace the strong potential by a holonomic constraint that forces the solutions to stay at the equilibrium value of the potential. This approach has, e.g., been successfully applied to the bond stretching in molecular dynamics simulations. In other cases, such as the bond-angle bending, this methods fails due to the introduced rigidity. Here we give a careful analysis of the analytical problem by means of a smoothing operator. This will lead us to the notion of the smoothed dynamics of a highly oscillatory Hamiltonian system. Based on our analysis, we suggest a new constrained formulation that maintains the flexibility of the system while at the same time suppressing the high-frequency components in the solutions and thus allowing for larger time steps. The new constrained formulation is Hamiltonian and can be discretized by the well-known SHAKE method.

A new operator splitting method for the numerical solution of partial differential equations

We present a new method of operator splitting for the numerical solution of a class of partial differential equations, which we call the odd-even splitting. This method extends the applicability of higher-order composition methods that have appeared recently in the literature. These composition methods can be considerably more efficient than conventional methods and additionally require at least a factor of two less storage capacity. In the case of Hamiltonian partial differential equations the schemes we present have the additional advantage that they are symplectic. We illustrate our scheme for three nonlinear wave equations in one dimension: the shallow water equations, the Boussinesq equation, and the KdV equation. The odd-even splitting is also applicable also to PDE's in higher spatial dimension.

Spectral implementation of a new operator splitting method for solving partial differential equations

We extend and review a method of operator splitting for the numerical solution of a class of partial differential equations introduced by us recently, which we call the odd–even splitting. This method extends the applicability of higher order composition methods that have appeared recently in the literature. These composition methods can be considerably more efficient than conventional methods and additionally require at least a factor of 2 less storage capacity. In addition, in some cases, it is possible to use an explicit treatment of certain terms that need to be treated implicitly in other time stepping schemes. In the case of canonical Hamiltonian partial differential equations the schemes we present have the additional advantage that they are symplectic. First we will illustrate the method with an example given previously, namely the nonlinear shallow water wave equation in one dimension. In this case spatial derivatives are approximated by finite differences. The bulk of the paper is devoted to the nontrivial examples of the Euler and Navier–Stokes equations in two dimensions. Here we demonstrate how our method is extended to higher dimensional systems and also how spectral methods for computation of spatial derivatives can be incorporated into the odd–even splitting scheme. We also present a method by which explicit treatment of diffusion terms becomes possible, while retaining the advantages of implicit methods, such as stability and the ability to take larger time steps. © 1995 American Institute of Physics.

Symplectic integration of Hamiltonian wave equations

The numerical integration of a wide class of Hamiltonian partial differential equations by standard symplectic schemes is discussed, with a consistent, Hamiltonian approach. We discretize the Hamiltonian and the Poisson structure separately, then form the the resulting ODE's. The stability, accuracy, and dispersion of different explicit splitting methods are analyzed, and we give the circumstances under which the best results can be obtained; in particular, when the Hamiltonian can be split into linear and nonlinear terms. Many different treatments and examples are compared.

Canonical transformations and generating functionals

It is shown that canonical transformations for field variables in hamiltonian partial differential equations can be obtained from generating functionals in the same way as classical canonical transformations from generating functions. A simple proof of the relation between infinitesimal invariant transformations and constants of the motion is obtained. The formalism is extended to cover finite and nonlocal transformations of the spatial variables.