Discrete breathers are time-periodic, spatially localized solutions of equations of motion for classical degrees of freedom interacting on a lattice. They come in one-parameter families. We report on studies of energy properties of breather families in one-, two-and three-dimensional lattices. We show that breather energies have a positive lower bound if the lattice dimension of a given nonlinear lattice is greater than or equal to a certain critical value. These findings could be important for the experimental detection of discrete breathers. 03.20.+i, 03.65.-w, 03.65.Sq Typeset using REVT E X

We analyze the classical and quantum properties of the integrable dimer problem. The classical version exhibits exactly one bifurcation in phase space, which gives birth to permutational symmetry broken trajectories and a separatrix. The quantum analysis yields all tunneling rates (splittings) in leading order of perturbation. In the semiclassical regime the eigenvalue spectrum obtained by numerically exact diagonalization allows one to conclude about the presence of a separatrix and a bifurcation in the corresponding classical model. PACS numbers: 05.45.+b, 03.20.+i, 03.65.Sq The problem of correspondence between classical and quantum-mechanical properties of nonlinear systems is currently an object of wide interest [1]. One interesting topic concerns Hamiltonian systems with a given symmetry (e.g., some permutational symmetry), where classical trajectories exist which are not invariant under the corresponding symmetry operation. This topic appears in analyzing selective bond excitation in chemistry and in the quantization of discrete breathers [2].We consider an integrable system with two degrees of freedom (TDF), whose classical version exhibits exactly one bifurcation (of periodic orbits) and separatrix manifold. This manifold cuts the phase space into three parts-one with invariant trajectories, and two with noninvariant trajectories, where the corresponding symmetry is the permutational one. By varying a single parameter it is possible to "switch" between these phase space parts by crossing the separatrix. It appears natural to expect in the quantum case a drastic change in the splittings of energy levels (which should be zero in the classical limit for the noninvariant phase space parts). However, the splittings are nonzero for any given value of the control parameter. The only way to avoid contradiction between the classical and quantum cases is to assume that the quantum level splittings tend to a steplike function (of, e.g., the level pair number) in the classical limit. The step should occur at the position of the classical separatrix. This problem can be coined also dynamical tunneling through a separatrix. There exist studies of the influence of classical chaos on dynamical tunneling [3]. This paper is an extension of previous studies on classical and quantum properties of the dimer system [4][5][6].We are able to trace the splittings of the level pairs using quantum perturbation methods. We consider the quasiclassical regime and show that the step indeed occurs. Therefore we are able to extract information about the classical separatrix and bifurcation. Further, we show that the quantum density of states (the second integral of motion is fixed) exhibits a sharp maximum at the separatrix energy. By calculating the corresponding classical quantity (with the help of Weyl's formula) we find that this singularity appears due to the integration over a part of the separatrix manifold which includes a hyperbolic isolated orbit.Let us consider the integrable dimer model with Hamiltonian [4]

We develop a general mapping from given kink or pulse shaped traveling-wave solutions including their velocity to the equations of motion on one-dimensional lattices which support these solutions. We apply this mapping-by definition an inverse method-to acoustic solitons in chains with nonlinear intersite interactions, nonlinear Klein-Gordon chains, reaction-diffusion equations, and discrete nonlinear Schrödinger systems. Potential functions can be found in a unique way provided the pulse shape is reflection symmetric and pulse and kink shapes are at least C2 functions. For kinks we discuss the relation of our results to the problem of a Peierls-Nabarro potential and continuous symmetries. We then generalize our method to higher dimensional lattices for reaction-diffusion systems. We find that increasing also the number of components easily allows for moving solutions.

Cumulants represent a natural language for expressing macroscopic properties of a solid. We show that cumulants are subject to a nontrivial geometry. This geometry provides an intuitive understanding of a number of cumulant relations which have been obtained so far by using algebraic considerations. We give general expressions for their infinitesimal and finite transformations and represent a cumulant wave operator through an integration over a path in the Hilbert space. Cases are investigated where this integration can be done exactly. An expression of the ground-state wave function in terms of the cumulant wave operator is derived. In the second part of the article, we derive the cumulant counterpart of Faddeev's equations and show its connection to the method of increments.

We give definitions for different types of moving spatially localized objects in discrete nonlinear lattices. We derive general analytical relations connecting frequency, velocity and localization length of moving discrete breathers and kinks in nonlinear one-dimensional lattices. Then we propose numerical algorithms to find these solutions. Finally we discuss generalizations to higher dimensional lattices.Comment: 18 Pages, 5 figures,REVTEX. Citations and figures are added in this versio

We study the dynamics of waves in a system of diffusively coupled discrete nonlinear sources. We show that the system exhibits burst waves which are periodic in a traveling-wave reference frame. We demonstrate that the burst waves are pinned if the diffusive coupling is below a critical value. When the coupling crosses the critical value the system undergoes a depinning instability via a saddle-node bifurcation, and the wave begins to move. We obtain the universal scaling for the mean wave velocity just above threshold. PACS: 47.54.+r, 87.22.As, 82.40.Ck, 47.20.Ky The effect of discrete source distribution for spatially extended systems is a problem of interest in such disparate fields as the biophysics of the calcium release waves in living cells [1,2], pinning in the dislocation motion in crystals [3], breathers in nonlinear crystal lattices [4], Josephson junction arrays [5], and charge density waves in one-dimensional strongly correlated electron systems [5][6][7][8]. In order to elucidate the effects of discreteness, complex models of calcium release have been solved numerically [9][10][11]. The simple "fire-diffuse-fire" model constructed in [12] displays burst (or saltatory) wave fronts. These fronts either propagate or they do not exist: they cannot undergo pinning. If a system with wave pinning consists of diffusively coupled discrete sources [6], the standard approach to its analysis has been to completely discretize the dynamics. This is done by replacing the diffusion term with a difference scheme for the field at the source sites [5,7]. This simplification neglects the field structure between sites. The latter is crucial for an understanding of the system dynamics and universal behavior near the pinning/depinning transition.In this Letter we consider a discrete array of nonlinear reaction sites embedded in a continuum in which the reactant diffuses. We study both analytically and numerically the propagation of burst waves and their pinning, as well as the universal behavior of the system near the pinning threshold. Our model is represented by the following diffusion equation with discrete nonlinear sources (in N -dimensional space):where u is a dimensionless concentration, D is the diffusion coefficient, α is the production rate of the reactant, and d is the distance between neighboring sites (channels). These sites are located at x i 's. The reaction dynamics are specified by a nonlinear function f (u). To describe the waves of one stable phase propagating into another stable phase, we choose the bistable reaction dynamics [13]. The simplest example of bistability is given by f (u) = −u(u−u 0 )(u−1) , with two stable fixed points u = 0, 1 , and one unstable fixed point u = u 0 .In the present Letter we study one-dimensional (1D) wave propagation. Then, after rescaling x =xd, t = tα −1 , we obtain from Eq. (1) (after dropping tildes)with the effective dimensionless diffusion coefficient β = D/αd 2 . The system dynamics are determined by the balance between the dissipation and the local nonlinear ...

We study the onset of the propagation failure of wave fronts in systems of coupled cells. We introduce a new method to analyze the scaling of the critical external field at which fronts cease to propagate, as a function of intercellular coupling. We find the universal scaling of the field throughout the range of couplings, and show that the field becomes exponentially small for large couplings. Our method is generic and applicable to a wide class of cellular dynamics in chemical, biological, and engineering systems. We confirm our results by direct numerical simulations. PACS: 87.18.Pj, 82.40.Bj, 87.19.Hh, The impact of discreteness on the propagation of phase fronts in biophysical, chemical, and engineering systems has been intensively studied during the last decade. Among the diverse examples are calcium release waves in living cells [1], reaction fronts in chains of coupled chemical reactors [2,3], arrays of coupled diode resonators [4,5], and discontinuous propagation of action potential in cardiac tissue [6][7][8]. All these disparate systems share a common phenomenon of wave front propagation failure, independently of specific details of each system. Recently this effect has drawn considerable attention (see, e.g., Refs. [2,9]). Numerous experimental evidences show that the propagation failure occurs at finite values of the coupling strength (a critical coupling) [2,4,6]. This is contrary to continuous systems, where wave fronts propagate for arbitrary couplings [10]. A challenging problem is to establish the universal properties of the critical coupling; this is crucial for making predictions of qualitatively different regimes of system dynamics.In this Letter we consider the universal behavior of phase separation fronts in one-dimensional nonlinear discrete systems in an external field. We study the propagation failure transition for a class of simple dynamical models describing experimental observations in arrays of coupled nonlinear cells, such as chains of bistable chemical reactors [2,3], systems of cardiac cells [8], etc. A new analytical method is presented to study generic properties of the critical external field of the transition. We find, using this method, how the critical field scales with the intrachain coupling. This method is applicable for a wide range of the couplings. We confirm our analytical predictions by direct numerical simulations of the full system. Our model in general is given by the following set of coupled nonlinear equations:Here u n is the order parameter at the n-th site, γ is the damping coefficient, C is the coupling constant, and G(u n , E) is the onsite potential, where E is the applied field. The potential has at least two minima separated by a barrier u B . The external field E is responsible for the energy difference between the minima. This provides for one globally stable and one metastable minimum, u = u + and u = u − , respectively. Phase fronts connect the two minima and tend to propagate, to increase the size of the energetically more favorable phase, u + .

We analyze the origin and features of localized excitations in a discrete two-dimensional Hamiltonian lattice. The lattice obeys discrete translational symmetry, and the localized excitations exist because of the presence of nonlinearities. We connect the presence of these excitations with the existence of local integrability of the original N degree of freedom system. On the basis of this explanation we make several predictions about the existence and stability of these excitations. This work is an extension of previously published results on vibrational localization in one-dimensional nonlinear Hamiltonian lattices (Phys.Rev.E.49(1994)836). Thus we confirm earlier suggestions about the generic property of Hamiltonian lattices to exhibit localized excitations independent on the dimensionality of the lattice. PACS number(s): 03.20.+i ; 63.20.Pw ; 63.20.Ry