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.
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