Exact kink solutions in a new non-linear hyperbolic double-well potential

Abstract

We propose amodel of a kink bearing Hamiltonianwith a new non-linear potential V ( q , p ) whose double-well shape can be vaned continuously as a function of the parameter p and which has the q 4 potential as a particular case. Exact classical kink solutions that dependonp areobtained. The rest masses and restenergiesof the kinksare also determined, In recent years non-linear monatomic chain models have been extensively used in condensed-matter physics because they provide a non-perturbation approach to strongly anharmonic systems. The governing equations frequently admit large-amplitude localized field profiles that are physically distinct from those obtainable by superposition of small-amplitude or linearized profiles. These localized large-amplitude excitations canpropagate through thesystem without distortionofshaFeandarecommonly referred to as solitary waves. They exhibit remarkable stability and other particle-like properties.Because of their localized nature, they have found widespread use as one-dimensional models of extended particles in non-linear quantum-field theories, dislocations in crystals, planar domain walls in ferromagnets and ferroelectrics, propagatingflux quanta in Josephson transmission lines, disgyration planes in superfluid He, charge carriers in weakly pinned charge-density-wavecondensates, and charged dislocations in superionic conductors, to mention only a few examples.Various kinds of non-linear potential have been proposed to explain these phenomena. As examples, we may retain the Toda potential (1967), the Lennard-Jones potential, the Morse potential, the sineGordon and double sine-Gordon potentials, the q4, @and @fields, adouble quadratic potential, the Schmidt potential (1979), the Magyari potential (1981), the Behera and Khare potential (1981) and the Remoissenet-Peyrardpotential(l981).Consider a general class of non-linear solitary wave bearing one-dimensional lattice Hamiltonians (see Currie er al1980).where pi is a one-component dimensionless field defined on a one-dimensional lattice of points with lattice constant 1. The first term represents the kinetic energy carried by the field, the second represents harmonic coupling between field values at neighbouring

Oscillons in hyperbolic models

“…1a for the potential V 1 , where the barrier height is constant. The static kink solution is given by [30] φ K (x) = 1 arcsinh(µ) arctanh µ…”

Analytical calculation of the Peierls–Nabarro pinning barrier for one-dimensional parametric double-well models

“…At this step it is instructive to remark that according to the two model parameters, only the first will give rise to kink family whose widths effectively vary(decrease) as function of (with increasing)µ, whereas all kinks of the second family have their widths bounded to the constant d k whatever µ. However, their "kink shape" is deeply affected by the deformability parameter as noticed elsewhere [17]. As our primary goal is the kink-lattice interactions, proceeding with we appeal to the PNP approach which assumes the existence of an effective potential field provided by the lattice substrate and via which kinks get pinned due to lattice discreteness.…”

“…For this last context, it has even been argued [15] that the DM model is to date the best candidate giving relatively good account of the so-called "geometric effect" . In addition to the DM, several other parametrized DWP models are currently present in the literature [16,17]. By their essential virtues they allow theoretical manipulations at one wish.…”

Stochastic resonance in periodically driven bistable systems subjected to anomalous diffusion

“…A possible extension of the present study would be to look at the effects of deformability of the bistable potential, such as the change of confinement of the potential well (as done in ref. [22]), or of positions of the doubledegenerate potential minima [14], on profiles of the stochastic resonance quantifiers. Indeed the bistable potential U(x) considered in the present work is the so-called 4 [11][12][13] whose rigid profile, reflected in its fixed minima positions and the fixed barrier height, limit their applicability to systems with soft profiles as it is common in polymers and biophysical systems.…”

“…As typical example, thermally activated processes such as noise-induced escape from metastable states [1][2][3][4][5] and the phenomenon of stochastic resonance (SR) [3,6,7] have attracted a great deal of attention because of their fundamental role in several areas of physics, chemistry, biophysics, social as well as financial sciences [5,[8][9][10]. In particular, there has been an unprecedented interest in noise-driven phenomena pertaining to the so-called Kramers escape rate problem [5], due to their intimate connection with second-order phase transitions [11] in physical contexts where dynamical instabilities are governed by double-well potentials [12][13][14]. For these systems, noise-driven transport phenomena are well known to be dominated by standard diffusion processes, whereby oscillators exhibit normal Brownian motions and the escape rate has the standard Arrhenius law [3].…”