Hamiltonian Evolution Equations Research Articles

SHORT-TIME EXISTENCE FOR SCALE-INVARIANT HAMILTONIAN WAVES

We prove short-time existence of smooth solutions for a class of nonlinear, and in general spatially nonlocal, Hamiltonian evolution equations that describe the self-interaction of weakly nonlinear scale-invariant waves. These equations include ones that describe weakly nonlinear hyperbolic surface waves, such as nonlinear Rayleigh waves in elasticity.

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.

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.

Complex Hamiltonian evolution equations and field theory

The form of the complex Hamiltonian evolution equations (linear and nonlinear) is deduced here from that of the classical Hamiltonian systems. Their properties and those of the associated continuous Poisson bracket are studied. It is shown that if any space-localized solution of the above type equations is expanded in any Hilbert space basis the expansion coefficients are canonically conjugate variables and obey Hamiltonian systems of complex equations. The last is proved also for sets of “canonically conjugate” functionals of general type. As an application, commutation relations which are equivalent in form to Dirac’s quantization rules (the constant in place of ℏ is undetermined) are obtained. The question of how a nonlinear field theory may contain quantum mechanics as a special case is discussed in view of this paper’s results.

SYMMETRY REDUCED EINSTEIN GRAVITY AND GENERALIZED σ AND CHIRAL MODELS

Certain features associated with the symmetry reduction of the vacuum Einstein equations by two commuting, spacelike Killing vector fields are studied. In particular, the discussion encompasses the equations for the Gowdy T3 cosmology and cylindrical gravitational waves. We first point out a relation between the SL(2,R) (or SO(3)) σ and principal chiral models, and then show that the reduced Einstein equations can be obtained from a dimensional reduction of the standard SL(2,R) σ-model in three dimensions. The reduced equations can also be derived from the action of a 'generalized' two dimensional SL(2,R) σ-model with a time dependent constraint. We give a Hamiltonian formulation of this action, and show that the Hamiltonian evolution equations for certain phase space variables are those of a certain generalization of the principal chiral model. Using these Hamiltonian equations, we give a prescription for obtaining an infinite set of constants of motion explicitly as functionals of the spacetime metric variables.

De Broglie-type relations from nonlinear evolution equations

De Broglie-type relations for the energy and momentum of the space-localized solutions to a class of nonlinear complex Hamiltonian evolution equations are derived without any “external assumptions.” It is found that the Hermitian norm of the same solutions plays a more fundamental role than a mere normalization constant. The quantum commutator is obtained from the infinite-dimensional complex Poisson bracket, and it is in complete agreement with the above find. The significance that such relationships should exist, while being entirely independent of the concepts of pointlike particle and/or point charge, is briefly discussed.

Einstein's equations and the chiral model.

The vacuum Einstein equations for spacetimes with two commuting spacelike Killing field symmetries are studied using the Ashtekar variables. The case of compact spacelike hypersurfaces which are three-tori is considered, and the determinant of the Killing two-torus metric is chosen as the time gauge. The Hamiltonian evolution equations in this gauge may be rewritten as those of a modified SL(2) principal chiral model with a time-dependent ``coupling constant,'' or equivalently, with time-dependent SL(2) structure constants. The evolution equations have a generalized zero-curvature formulation. Using this form, the explicit time dependence of an infinite number of spatial-diffeomorphism-invariant phase-space functionals is extracted, and it is shown that these are observables in the sense that they Poisson commute with the reduced Hamiltonian. An infinite set of observables that have SL(2) indices is also found. This determination of the explicit time dependence of an infinite set of spatial-diffeomorphism-invariant observables amounts to the solutions of the Hamiltonian Einstein equations for these observables. \textcopyright{} 1996 The American Physical Society.

Aspects of long time behaviour of solutions of nonlinear Hamiltonian evolution equations

Aspects of long time behaviour of solutions of nonlinear Hamiltonian evolution equations

Hamiltonian spatial structure for three-dimensional water waves in a moving frame of reference

The governing equations for three-dimensional time-dependent water waves in a moving frame of reference are reformulated in terms of the energy and momentum flux. The novelty of this approach is that time-independent motions of the system—that is, motions that are steady in a moving frame of reference—satisfy a partial differential equation, which is shown to be Hamiltonian. The theory of Hamiltonian evolution equations (canonical variables, Poisson brackets, symplectic form, conservation laws) is applied to the spatial Hamiltonian system derived for pure gravity waves. The addition of surface tension changes the spatial Hamiltonian structure in such a way that the symplectic operator becomes degenerate, and the properties of this generalized Hamiltonian system are also studied. Hamiltonian bifurcation theory is applied to the linear spatial Hamiltonian system for capillary-gravity waves, showing how new waves can be found in this framework.

Observables for spacetimes with two Killing field symmetries.

The Einstein equations for spacetimes with two commuting spacelike Killing field symmetries are studied from a Hamiltonian point of view. The complexified Ashtekar canonical variables are used, and the symmetry reduction is performed directly in the Hamiltonian theory. The reduced system corresponds to the field equations of the SL(2,R) chiral model with additional constraints. On the classical phase space, a method of obtaining an infinite number of constants of the motion, or observables, is given. The procedure involves writing the Hamiltonian evolution equations as a single `zero curvature' equation, and then employing techniques used in the study of two dimensional integrable models. Two infinite sets of observables are obtained explicitly as functionals of the phase space variables. One set carries sl(2,R) Lie algebra indices and forms an infinite dimensional Poisson algebra, while the other is formed from traces of SL(2,R) holonomies that commute with one another. The restriction of the (complex) observables to the Euclidean and Lorentzian sectors is discussed. It is also shown that the sl(2,R) observables can be associated with a solution generating technique which is linked to that given by Geroch.

Regge calculus model of Bianchi ix inflation

The continuous time 3+1 Regge calculus of Piran-Williams is applied to analyze a homogeneous anisotropic closed cosmological model. Piran-Williams' 5cell model is extended to 16 cell model, and the action for the lattice is constructed. The Hamiltonian constraint and evolution equations derived from the action are solved numerically. It is shown that this method works very well for the analysis of inflation of a homogeneous anisotropic universe. Inflation takes place for most of intial conditions, but in case that spatial curvature dominates other energy density initially, some universes recollapse before onset of inflation. The effect of the intial anisotropy helps the onset of inflation. The numerical solutions are compared with Wald's cosmic no hair theorem and difference between pure cosmological constant and effective one is discussed.

Spatial hamiltonian structure, energy flux and the water-wave problem

Travelling waves and relative equilibrium states of infinite dimensional hamiltonian systems are considered with particular attention to the water-wave problem. Relative equilibria correspond to solutions that are stationary in a moving frame of reference. The governing equations for relative equilibria are recast as a hamiltonian evolution equation in space. The mass, momentum and energy flux densities play a fundamental role in the spatial hamiltonian structure: the energy flux generates the basic symplectic operator, the momentum flux is the spatial hamiltonian and the mass flux is a second independent (spatial) integral. A complete theory is given for the water-wave problem with surface tension. The hamiltonian structure is used to show the existence of dual variational principles for travelling waves and to give an analysis of a singularity in the dispersion relation for capillary-gravity waves from a spatial hamiltonian systems viewpoint.

Bi-Hamiltonian formulation of the Hénon-Heiles system and its multidimensional extensions

We prove that a (slightly) generalized Hénon-Heiles system is equivalent to a fifth order Hamiltonian evolution equation with a third order Hamiltonian operator. This equivalence makes it possible to use the machinery of restricted flows of soliton hierarchies in order to (a) find natural extensions of integrable cases of the Hénon-Heiles system, and (b) determine (in the KdV case) a bi-Hamiltonian formulation for the (extended) Hénon-Heiles system and prove its complete integrability.

Quantum gravity and the collapse of the wavefunction

We derive a general form of non-unitary modification to the hamiltonian evolution equation caused by wormholes interacting with a microscopic system. We show that it suppresses elements of the density matrix which are off-diagonal in the configuration space representation, thus causing the wavefunction to collapse. An order-of-magnitude estimate indicates that the collapse time is very long ( t≈10 37s) for a single elementary particle. However, the effect is enhanced for the centre-of-mass motion of a large aggregate of particles. In particular, the wavefunction of a body containing Avogadro's number of particles may be localized within a Bohr radius in t≈10 −7s.

The Hamiltonian structure of a complex version of the Burgers hierarchy

In this paper we construct a class of integrable Hamiltonian nonlinear evolution equations generated by a purely differential recursion operator. It turns out that this hierarchy is a complex version of the Burgers hierarchy and can be linearized through a generalization of the Cole–Hopf transformation.

A hierarchy of Hamiltonian evolution equations associated with a generalized Schrödinger spectral problem

A hierarchy of nonlinear evolution equations associated with a generalization of the Schrödinger spectral problem is derived. It is shown that each equation is Hamiltonian and that their flows commute. The spectral equation is examined and certain difficulties in the inverse problem are pointed out.

Special families of Hamiltonian structures and integrable nonlinear evolution equations

The N-fold Backlund transformation for the Davey–Stewartson Equation III is constructed following the idea of Neugebauer and Meinel. The formula yields the N-soliton solution in a single step, for this particular nonlinear equation in (2+1) dimensions. It is shown that N=1 leads to the standard one soliton solution.

Infinite-dimensional Hamiltonian systems associated with matrix Schrödinger operators

The Gel’fand-Dikii method is used to derive families of nonlinear Hamiltonian evolution equations from matrix Schrodinger operators. These Hamiltonian systems have an infinite set of constants of motion such that the Poisson brackets of any two of them vanish identically and they admit also a formulation in terms of Lax pairs.

Schrödinger spectral problems with energy–dependent potentials as sources of nonlinear Hamiltonian evolution equations

We develop a method to derive infinite families of completely integrable nonlinear Hamiltonian evolution equations associated with Schrödinger spectral problems whose potential functions depend on the spectral parameter.

On global solutions for non-linear Hamiltonian evolution equations

It is shown that partial differential equations of Hamiltonian type admit global solutions in time if (a) the initial data is near equilibrium (or the coupling constant is small) (b) the linear terms have positive energy and (c) the non-linear terms are smooth functions in the topology of the linearized energy norm. The non-linear terms need not have positive energy. The result is applied to non-linear wave equations in which the interaction energy is not necessarily positive.