Breakup threshold of solitons in systems with nonconvex interactions (original) (raw)
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Solitons in combined linear and nonlinear lattice potentials
Physical Review A, 2010
We study ordinary solitons and gap solitons (GSs) in the framework of the onedimensional Gross-Pitaevskii equation (GPE) with a combination of linear and nonlinear lattice potentials. The main points of the analysis are effects of (in)commensurability between the lattices, development of analytical methods, viz., the variational approximation (VA) for narrow ordinary solitons, and various forms of the averaging method for broad solitons of both types, and also the study of mobility of the solitons. Under the direct commensurability (equal periods of the lattices, L lin = L nonlin ), the family of ordinary solitons is similar to its counterpart in the GPE without external potentials. In the case of the subharmonic commensurability, with L lin = (1/2)L nonlin , or incommensurability, there is an existence threshold for the ordinary solitons, and the scaling relation between their amplitude and width is different from that in the absence of the potentials. GS families demonstrate a bistability, unless the direct commensurability takes place. Specific scaling relations are found for them too. Ordinary solitons can be readily set in motion by kicking. GSs are mobile too, featuring inelastic collisions. The analytical approximations are shown to be quite accurate, predicting correct scaling relations for the soliton families in different cases. The stability of the ordinary solitons is fully determined by the VK (Vakhitov-Kolokolov) criterion, i.e., a negative slope in the dependence between the solitons's chemical potential µ and norm N . The stability of GS families obeys an inverted ("anti-VK") criterion, dµ/dN > 0, which is explained by the approximation based on the averaging method. The present system provides for a unique possibility to check the anti-VK criterion, as µ(N ) dependences for GSs feature turning points, except for the case of the direct commensurability.
Solitons and Deformed Lattices I
Journal of Nonlinear Mathematical Physics, 2005
We study a model describing some aspects of the dynamics of biopolymers. The models involve either one or two finite chains with a number N of sites that represent the "units" of a biophysical system. The mechanical degrees of freedom of these chains are coupled to the internal degrees of freedom through position dependent excitation transfer functions. We reconsider the case of the one chain model discussed by Mingaleev et al. and present new results concerning the soliton sector of this model. We also give new (preliminary) results in the two chain model in which case we have introduced an interaction potential inspired by the Morse potential. J ik = (e β − 1) exp(−β| r i − r k |).
Solitons in atomic chains with long-range interactions
Physics Letters A, 1994
Nonlinear solitary excitations are studied in an anharmonic chain with cubic or Toda nearest-neighbor interaction and exponentially decaying harmonic long-range interactions. Analytic expressions are obtained for the cubic interatomic potential, and the results are in qualitative agreement with numerical simulations. The relation of amplitude and velocity is quite different to a lattice without long-range forces especially near the speed of sound with the possibility of multiple solutions for a given velocity.
Generalized Frenkel-Kontorova model: A diatomic chain in a sinusoidal potential
Physical Review B, 1998
To understand modulated structures and commensurate-incommensurate transitions, we generalize the Frenkel-Kontorova model to a diatomic chain in the presence of an external sinusoidal potential. For the classical ground states, the diatomic effects are reflected by the phase diagram and the phonon spectrum. For the quantum ground states, the diatomic effects are reflected by the distribution of atoms on the external potential, the phase diagram, correspondences between the ground states and the orbits of the area-preserving maps, the phonon spectrum, and the occurrence of a second critical point K c Ј besides K c at which a transition by breaking analyticity occurs.
Dynamics on the Double Morse Potential: A Paradigm for Roaming Reactions with no Saddle Points
Regular and Chaotic Dynamics, 2018
In this paper we analyze a two degree of freedom Hamiltonian system constructed from two planar Morse potentials. The resulting potential energy surface has two potential wells surrounded by an unbounded flat region containing no critical points. In addition, the model has an index one saddle between the potential wells. We study the dynamical mechanisms underlying transport between the two potential wells, with emphasis on the role of the flat region surrounding the wells. The model allows us to probe many of the features of the "roaming mechanism" whose reaction dynamics are of current interest in the chemistry community. Introduction: In the 1978 English version of his famous book Mathematical Methods of Classical Mechanics V. I. Arnold made the provocative statement: "Analyzing a general potential system with two degrees of freedom is beyond the capability of modern science." Despite the great progress in our understanding on nonlinear dynamical systems theory since that time, this statement is still mostly true. In this paper, we describe a new class of two-dimensional (2d) potential energy surface (PES) constructed from two planar Morse potentials whose rich dynamics certainly bears out Arnold's assessment of the situation. The PES has two potential wells, separated by an index one saddle point, and surrounded by an unbounded flat region containing no critical points. We are interested in the fate of trajectories that leave one of the potential wells. It is observed that such trajectories have three possible fates. They can 1) exit a potential well and re-enter the same potential well at a later time; 2) they can exit a potential well and enter the other potential well, or 3) they can exit a potential well, enter the flat region, and become unbounded. The collective dynamical behavior concerning inter-well transport embodied in this system has many of the features of the "roaming mechanism" for reaction dynamics that is of current interest in the chemistry community [1,2]. Roaming is a recently discovered mechanism for chemical reaction, i.e. breaking and/or forming chemical bonds between atoms [1]. Many studies of the roaming phenomenon in a
The quasi-equilibrium phase of nonlinear chains
Pramana, 2005
We show that time evolution initiated via kinetic energy perturbations in conservative, discrete, spring-mass chains with purely non-linear, non-integrable, algebraic potentials of the form V (xi − xi+1) ∼ (xi − xi+1) 2n , n ≥ 2 and an integer, occurs via discrete solitary waves (DSWs) and discrete antisolitary waves (DASWs). Presence of reflecting and periodic boundaries in the system leads to collisions between the DSWs and DASWs. Such collisions lead to the breakage and subsequent reformation of (different) DSWs and DASWs. Our calculations show that the system eventually reaches a stable 'quasi-equilibrium' phase that appears to be independent of initial conditions, possesses Gaussian velocity distribution, and has a higher mean kinetic energy and larger range of kinetic energy fluctuations as compared to the pure harmonic system with n = 1; the latter indicates possible violation of equipartition.
Nonlinear excitations and electric transport in dissipative Morse-Toda lattices
The European Physical Journal B, 2006
We investigate the onset and maintenance of nonlinear soliton-like excitations in chains of atoms with Morse interactions at rather high densities, where the exponential repulsion dominates. First we discuss the atomic interactions and approximate the Morse potential by an effective Toda potential with adapted density-dependent parameters. Then we study several mechanisms to generate and stabilize the soliton-like excitations: (i) External forcing: we shake the masses periodically, mimicking a piezoelectriclike excitation, and delay subsequent damping by thermal excitation; (ii) heating, quenching and active friction: we heat up the system to a relatively high temperature Gaussian distribution, then quench to a low temperature, and subsequently stabilize by active friction. Finally, we assume that the atoms in the chain are ionized with free electrons able to move along the lattice. We show that the nonlinear soliton-like excitations running on the chain interact with the electrons. They influence their motion in the presence of an external field creating dynamic bound states ("solectrons", etc.). We show that these bound states can move very fast and create extra current. The soliton-induced contribution to the current is constant, field-independent for a significant range of values when approaching the zero-field value.
Russian journal of nonlinear dynamics, 2022
The oscillatory motion in nonlinear nanolattices having different interatomic potential energy functions is investigated. Potential energies such as the classical Morse, Biswas-Hamann and modified Lennard-Jones potentials are considered as interaction potentials between atoms in one-dimensional nanolattices. Noteworthy phenomena are obtained with a nonlinear chain, for each of the potential functions considered. The generalized governing system of equations for the interaction potentials are formulated using the well-known Euler-Lagrange equation with Rayleigh's modification. Linearized damping terms are introduced into the nonlinear chain. The nanochain has statistical attachments of 40 atoms, which are perturbed to analyze the resulting nonlinearities in the nanolattices. The range of initial points for the initial value problem (presented as second-order ordinary differential equations) largely varies, depending on the interaction potential. The nanolattices are broken at some initial point(s), with one atom falling off the slender chain or more than one atom falling off. The broken nanochain is characterized by an amplitude of vibration growing to infinity. In general, it is observed that the nonlinear effects in the interaction potentials cause growing amplitudes of vibration, accompanied by disruptions of the nanolattice or by the translation of chaotic motion into regular motion (after the introduction of linear damping). This study provides a computationally efficient approach for understanding atomic interactions in long nanostructural components from a theoretical perspective.