Hamiltonian Chains Research Articles

Exact enumeration of Hamiltonian circuits, walks and chains in two and three dimensions

We present an algorithm for enumerating exactly the number of Hamiltonian chains on regular lattices in low dimensions. By definition, these are sets of k disjoint paths whose union visits each lattice vertex exactly once. The well-known Hamiltonian circuits and walks appear as the special cases k = 0 and k = 1, respectively. In two dimensions, we enumerate chains on L × L square lattices up to L = 12, walks up to L = 17 and circuits up to L = 20. Some results for three dimensions are also given. Using our data we extract several quantities of physical interest.

Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

The integrability of an m-component system of hydrodynamic type, u t = V(u)u x , by the generalized hodograph method requires the diagonalizability of the m × m matrix V(u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalize this approach to hydrodynamic chains—infinite-component systems of hydrodynamic type for which the ∞ × ∞ matrix V(u) is ‘sufficiently sparse’. For such systems the Haantjes tensor is well-defined, and the calculation of its components involves finite summations only. We illustrate our approach by classifying broad classes of conservative and Hamiltonian hydrodynamic chains with the zero Haantjes tensor. We prove that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability.

Classification of integrable Hamiltonian hydrodynamic chains associated with Kupershmidt’s brackets

We characterize a class of integrable Hamiltonian hydrodynamic chains, based on the necessary condition for the integrability provided by the vanishing of the Haantjes tensor. We prove that the vanishing of the first few components of the Haantjes tensor is already sufficiently restrictive, and allows a complete description of the corresponding Hamiltonian densities. In each of the cases we were able to explicitly construct a generating function for conservation laws, thus establishing the integrability.

Hamiltonian Chains in Hypergraphs

Hamiltionian chain is a generalisation of hamiltonian cycles for hypergraphs. Among the several possible ways of generalisations this is probably the most strong one, it requires the strongest structure. Since there are many interesting questions about hamiltonian cycles in graphs, we can try to answer these questions for hypergraphs, too. In the present article we give a survey on results about such questions.

Combining constraint Propagation and meta-heuristics for searching a Maximum Weight Hamiltonian Chain

This paper presents the approach that we developed to solve the ROADEF 2003 challenge problem. This work is part of a research program whose aim is to study the benefits and the computer-aided generation of hybrid solutions that mix constraint programming and meta-heuristics, such as large neighborhood search (LNS). This paper focuses on three contributions that were obtained during this project: an improved method for propagating Hamiltonian chain constraints, a fresh look at limited discrepancy search and the introduction of randomization and de-randomization within our combination algebra. This algebra is made of terms that represent optimization algorithms, following the approach of SALSA[1], which can be generated or tuned automatically using a learning meta-strategy [2]. In this paper, the hybrid combination that is investigated mixes constraint propagation, a special form of limited discrepancy search and large neighborhood search.

Recouvrements Hamiltoniens de certains graphes

Nous étudions la possibilité de recouvrir certains graphes par des chaînes Hamiltoniennes. Nous appliquons les résultats obtenus au cas particulier du graphe divisoriel pour obtenir des réponses nouvelles à un problème posé par P. Erdös. We study the possibility of covering some graphs with Hamiltonian chains. The results are applied to the particular case of the divisorial graph to give some new answers to a question asked by P. Erdös.

Hamiltonian Chains in Hypergraphs

Hamiltionian chain is a generalisation of hamiltonian cycles for hypergraphs. Among the several possible ways of generalisations this is probably the most strong one, it requires the strongest structure. Since there are many interesting questions about hamiltonian cycles in graphs, we can try to answer these questions for hypergraphs, too. In the present article we give a survey on results about such questions.

Localization and coherence in nonintegrable systems

We study the irreversible dynamics of nonlinear, nonintegrable Hamiltonian oscillator chains approaching their statistical asymptotic states. In systems constrained by more than one conserved quantity, the partitioning of the conserved quantities leads naturally to localized and coherent structures. If the phase space is compact, the final equilibrium state is governed by entropy maximization and the coherent structures are stable lumps. In systems where the phase space is not compact, the coherent structures can be collapsed, represented in phase space by a heteroclinic connection of some unstable saddle to infinity.

On the Perron roots of principal submatrices of co-order one of irreducible nonnegative matrices

Let A be an irreducible nonnegative matrix and λ( A) be the Perron root (spectral radius) of A. Denote by λ min( A) the minimum of the Perron roots of all the principal submatrices of co-order one. It is well known that the interval ( λ min( A), λ( A)) does not contain any eigenvalues of A. Consider any principal submatrix A− v of co-order one whose Perron root is equal to λ min( A). We show that the Jordan structure of λ min( A) as an eigenvalue of A is obtained from that of the Perron root of A− v as follows: one largest Jordan block disappears and the others remain the same. So, if only one Jordan block corresponds to the Perron root of the submatrix, then λ min( A) is not an eigenvalue of A. By Schneider’s theorem, this holds if and only if there is a Hamiltonian chain in the singular digraph of A− v. In the general case the Jordan structure for the Perron root of the submatrix A− v and therefore that for the eigenvalue λ min( A) of A can be arbitrary. But if the Perron root λ( A− w) of a principal submatrix A− w of co-order one is strictly greater than λ min( A), then λ( A− w) is a simple eigenvalue of A− w. We also obtain different representations for the generalized eigenvectors corresponding to the eigenvalues of A contained in the annulus { λ: λ min( A)<| λ|< λ( A)}.

Effective Hamiltonian for travelling discrete breathers

Hamiltonian chains of oscillators in general probably do not sustain exact travelling discrete breathers. However solutions which look like moving discrete breathers for some time are not difficult to observe in numerics. In this paper we propose an abstract framework for the description of approximate travelling discrete breathers in Hamiltonian chains of oscillators. The method is based on the construction of an effective Hamiltonian enabling one to describe the dynamics of the translation degree of freedom of moving breathers. Error estimate on the approximate dynamics is also studied. The concept of the Peierls–Nabarro barrier can be made clear in this framework. We illustrate the method with two simple examples, namely the Salerno model which interpolates between the Ablowitz–Ladik lattice and the discrete nonlinear Schrodinger system, and the Fermi–Pasta–Ulam chain.

Modified Hamiltonian chain: A graph theoretic approach to group technology

Graph theory can be effectively applied to the group technology configuration problem (GTCP). Earlier attempts were made to use graph theoretic algorithms, e.g. minimal spanning tree (MST), tree search, and branch & bound to solve the group technology (GT) problem. The proposed algorithm is based on modified Hamiltonian chain (MHC) and consists of two stages. Stage I forms the graph from the machine part incidence matrix. Stage II generates a modified Hamiltonian chain which is a subgraph of the main graph developed in Stage I, and it gives machine sequence and part sequence directly. Dummy edges are considered in MHC for better accessibility in order to arrive at a block diagonal solution to the problem. This paper presents a simple approach by designing a MHC in the graph theoretic method to solve the group technology configuration problem. Results obtained from testing the method are compared with the other well-known methods and found to be satisfactory.

Hamiltonian chains in hypergraphs

Hamiltonian chains in hypergraphs

Hamiltonian chains in hypergraphs

Hamiltonian chains in hypergraphs

Relaxation of classical many-body Hamiltonians in one dimension

The relaxation of Fourier modes of Hamiltonian chains close to equilibrium is studied in the framework of a simple mode-coupling theory. Explicit estimates of the dependence of relevant time scales on the energy density (or temperature) and on the wave number of the initial excitation are given. They are in agreement with previous numerical findings on the approach to equilibrium and turn out to be also useful in the qualitative interpretation of them. The theory is compared with molecular dynamics results in the case of the quartic Fermi-Pasta-Ulam potential.

Some stability properties of breathers in hamiltonian networks of oscillators

A one-dimensional hamiltonian chain of coupled anharmonic oscillators is considered. The averaging method is used to prove that it has breathers which are stable in ℓ 2 norm over times exponentially long with the breather's frequency. It is also shown that in a suitable approximation the breather is asymptotically stable in the sense that initial data close enough to the breather lead to solutions in which the motion of each single particle tends (as t → ∞) to the breathing oscillation. It is conjectured that also breathers of the original model are asymptotically stable.

Localised modes on localised equilibria

We show that spatially localised equilibria for networks of dynamical systems can have localised eigenmodes on them, and study their spectral properties and onset thresholds. The theory is illustrated with numerics for a class of one-dimensional one-degree-of-freedom Hamiltonian oscillator chains with nearest neighbour coupling, which have a discrete kink in the uncoupled limit. We discuss implications of localised eigenmodes on spatially localised equilibria on many aspects of localised dynamics of networks.

Phonon scattering by localized equilibria of nonlinear nearest-neighbor chains

We study scattering of phonons by localized equilibria, for example, localized defects on nonlinear chains. We show that perfect transmission occurs at $k=0$ at the threshold for creation of localized modes and there exists a characteristic transition involving perfect transmission of long-wavelength phonons near the threshold. The theory is illustrated for the stationary case of a discrete kink on a translationally invariant Hamiltonian nearest neighbor chain, which is then generalized to any symmetric localized defects. The implications for discrete breathers are also discussed.

Thermodynamic limit beyond the stochasticity threshold in nonlinear chains

We present numerical experiments on Hamiltonian nonlinear chains at fixed specific energy in the stochastic domain with a growing number of degrees of freedom, up to N = 2048. Previous results on the rates of changes of action variables, with reference to translational invariance, are confirmed and specified. Furthermore, for models with other conserved quantities beside energy, an increasing number of degrees of freedom shows an increasing deviation from equipartition. Globally, the approach to equilibrium seems to be slower at larger N. This could cast a shadow over the effectiveness of stochasticity for systems with bounded spectrum and infinite degrees of freedom.

Spatio-temporal evolution of local temperature excitations in disordered harmonic systems

In strongly disordered condensed systems the Peierls-Boltzmann type of energy transport theory is not applicable. We present an alternative approach based on a local temperature excitation. The formalism is exemplified analytically and numerically for the Hamiltonian chain and for a chain with a random array of disturbed springs. Unexpected features are found, e.g. superdiffusion. Localization is verified also.

Spatial evolution of energy in the Hamiltonian chain

The spread of an energy packet in a “Hamiltonian chain” (monatomic, nearest neighbour springs) is considered, which at t = 0 is created via local momentum (ME) or via local displacement (DE) excitation, respectively. In both cases the 2nd moment M 2 displays wavelike behaviour ( M 2∼ t 2) in contrast to a diffusive one ( M 2∼ t). But otherwise quite unexpected features are found. (a) The packet never disintegrates into two wing parts. Its maximum always remains in the central region. (b) For intermediate time regimes the shape of the packet is remarkably different for both initial conditions. (c) The wings of the packets exhibit strong fluctuations which calm down in the central region. (d) In the neighbourhood of the original excitation center the shape of the packet turns independent of the position (quasi-thermalization). (e) The most surprising feature is a difference in the average spread velocities of the energy packet in the 2 prototype cases by a factor of √2. Thus, the spread strongly depends on the nature of the initial excitation even in the long time regime.