Nonlinear dynamics of the Frenkel-Kontorova model with impurities

Influence of elastic property on the friction between atomic force microscope tips and 2D materials

The relationship between the elastic property of solid materials and friction has been discussed and studied by theoretical calculation and analysis. In the present work, we perform an experimental study concerning this relationship. Atomic force microscope (AFM) scanning of four different transition metal dichalcogenides is conducted under different experimental conditions. It is found that materials with smaller vertical interlayer force constant, which also means smaller elasticity modulus, have larger friction. We attribute this phenomenon to larger elastic deformation in softer materials, which results in a larger obstacle to the motion of AFM tips.

Input-Independent Energy Harvesting in Bistable Lattices from Transition Waves

We demonstrate the utilisation of transition waves for realising input-invariant, frequency-independent energy harvesting in 1D lattices of bistable elements. We propose a metamaterial-inspired design with an integrated electromechanical transduction mechanism to the unit cell, rendering the power conversion capability an intrinsic property of the lattice. Moreover, focusing of transmitted energy to desired locations is demonstrated numerically and experimentally by introducing engineered defects in the form of perturbation in mass or inter-element forcing. We achieve further localisation of energy and numerically observe a breather-like mode for the first time in this type of lattice, improving the harvesting performance by an order of magnitude. Our approach considers generic bistable unit cells and thus provides a universal mechanism to harvest energy and realise metamaterials effectively behaving as a capacitor and power delivery system.

Dissipation-driven behavior of nonpropagating hydrodynamic solitons under confinement

We have identified a physical mechanism that rules the confinement of nonpropagating hydrodynamic solitons. We show that thin boundary layers arising on walls are responsible for a jump in the local damping. The outcome is a weak dissipation-driven repulsion that determines decisively the solitons' long-time behavior. Numerical simulations of our model are consistent with experiments. Our results uncover how confinement can generate a localized distribution of dissipation in out-of-equilibrium systems. Moreover, they show the preponderance of such a subtle effect in the behavior of localized structures. The reported results should explain the dynamic behavior of other confined dissipative systems.

Internal nonlinear dynamics of a short lattice DNA model in terms of propagating kink-antikink solitons

We study the internal nonlinear dynamics of an inhomogeneous short lattice DNA model by solving numerically the governing discrete perturbed sine-Gordon equations under the limits of a uniform and a nonuniform angular rotation of bases. The internal dynamics is expressed in terms of open-state configurations represented by kink and antikink solitons with fluctuations. The inhomogeneity in the strands and hydrogen bonds as well as nonuniformity in the rotation of bases introduce fluctuations in the profile of the solitons without affecting their robust nature and the propagation. These fluctuations spread into the tail regions of the soliton in the case of periodic inhomogeneity. However, the localized form of inhomogeneity generates amplified fluctuations in the profile of the soliton. The fluctuations are expected to enhance the denaturation process in the DNA molecule.

Influence of impurities on solitons in the nonlinear LC transmission line

This paper studies the propagation of solitons in the nonlinear LC transmission line (NLCTL) with capacitor impurity. Based on Kirchhoff's laws, the numerical simulation shows that the amplitude of the soliton will be increased or decreased when it is close to the positive or negative impurity. Then, it will be split into reflected and transmitted waves by both the positive and negative impurities. Furthermore, their final amplitude and propagating speeds are almost independent of the impurity polarity. The observations near the impurity can be understood in the physical picture of the linear uncoupled energy absorption. By these results, we find that the impurity-soliton interactions (ISIs) in NLCTLs for both inductance and capacitance impurities, which have been seen to be different before, actually can be unified. They also indicate that the ISIs in NLCTLs essentially can be integrated with those in many other soliton systems, such as the Frenkel-Kontorova model and hydrodynamics. Moreover, the impurity-induced influence on the NLCTL solitons can also be well interpreted in the framework of the nonlinear Schrödinger model with an impurity term derived from the discrete voltage propagation equations by means of the perturbation method.

Strain waves, earthquakes, slow earthquakes, and afterslip in the framework of the Frenkel-Kontorova model

The one-dimensional Frenkel-Kontorova (FK) model, well known from the theory of dislocations in crystal materials, is applied to the simulation of the process of nonelastic stress propagation along transform faults. Dynamic parameters of plate boundary earthquakes as well as slow earthquakes and afterslip are quantitatively described, including propagation velocity along the strike, plate boundary velocity during and after the strike, stress drop, displacement, extent of the rupture zone, and spatiotemporal distribution of stress and strain. The three fundamental speeds of plate movement, earthquake migration, and seismic waves are shown to be connected in framework of the continuum FK model. The magnitude of the strain wave velocity is a strong (almost exponential) function of accumulated stress or strain. It changes from a few km/s during earthquakes to a few dozen km per day, month, or year during afterslip and interearthquake periods. Results of the earthquake parameter calculation based on real data are in reasonable agreement with measured values. The distributions of aftershocks in this model are consistent with the Omori law for temporal distribution and a 1/r for the spatial distributions.

Discrete nonlinear Schrödinger equations with arbitrarily high-order nonlinearities

A class of discrete nonlinear Schrödinger equations with arbitrarily high-order nonlinearities is introduced. These equations are derived from the same Hamiltonian using different Poisson brackets and include as particular cases the saturable discrete nonlinear Schrödinger equation and the Ablowitz-Ladik equation. As a common property, these equations possess three kinds of exact analytical stationary solutions for which the Peierls-Nabarro barrier is zero. Several properties of these solutions, including stability, discrete breathers, and moving solutions, are investigated.

Exact solitary wave solutions for a discrete lambdaphi4 field theory in 1+1 dimensions

We have found exact, periodic, time-dependent solitary wave solutions of a discrete phi4 field theory model. For finite lattices, depending on whether one is considering a repulsive or attractive case, the solutions are Jacobi elliptic functions, either sn (x,m) [which reduce to the kink function tanh (x) for m-->1 ], or they are dn (x,m) and cn (x,m) [which reduce to the pulse function sech (x) for m-->1 ]. We have studied the stability of these solutions numerically, and we find that our solutions are linearly stable in most cases. We show that this model is a Hamiltonian system, and that the effective Peierls-Nabarro barrier due to discreteness is zero not only for the two localized modes but even for all three periodic solutions. We also present results of numerical simulations of scattering of kink-antikink and pulse-antipulse solitary wave solutions.

Influence of impurities on hydrodynamic solitons

The interaction between impurities and solitary waves has been experimentally observed on the surface of a defective water layer subjected to vertical vibration. A slightly rugged surface on one sidewall of the water layer serves as the impurity, making the layer breadth at the defect slightly different elsewhere. A wide-breadth impurity will attract or pin not only the hydrodynamic breather at lower driving frequency, but also the kink at higher driving frequency, while a narrow-breadth one will repel them. Using a multiple scale expansion method, a nonlinear Schro dinger equation with an impurity term (NLSI) was derived from the basic hydrodynamic equation. Furthermore, we present numerical calculations that show good agreement between the NLSI-based theoretical model and the experimental results.

Breather statics and dynamics in Klein-Gordon chains with a bend

In this paper, we examine a nonlinear model with an impurity emulating a bend. We justify the geometric interpretation of the model and connect it with earlier work on models including geometric effects. We focus on both the bifurcation and stability analysis of the modes that emerge as a function of the strength of the bend angle, but we also examine dynamical effects including the scattering of mobile localized modes (discrete breathers) off of such a geometric structure. The potential outcomes of such numerical experiments (including transmission, trapping within the bend as well as reflection) are highlighted and qualitatively explained. Such models are of interest both theoretically in understanding the interplay of breathers with curvature, but also practically in simple models of photonic crystals or of bent chains of DNA.

Interaction of a discrete breather with a lattice junction

We study the scattering of a moving discrete breather (DB) on a junction in a Fermi-Pasta-Ulam chain consisting of two segments with different masses of the particles. We consider four distinct cases: (i) a light-heavy (abrupt) junction in which the DB impinges on the junction from the segment with lighter mass, (ii) a heavy-light junction, (iii) an up mass ramp in which the mass in the heavier segment increases continuously as one moves away from the junction point, and (iv) a down mass ramp. Depending on the mass difference and DB characteristics (frequency and velocity), the DB can either reflect from, or transmit through, or get trapped at the junction or on the ramp. For the heavy-light junction, the DB can even split at the junction into a reflected and a transmitted DB. The latter is found to subsequently split into two or more DBs. For the down mass ramp the DB gets accelerated in several stages, with accompanying radiation (phonons). These results are rationalized by calculating the Peierls-Nabarro barrier for the various cases. We also point out implications of our results in realistic situations such as electron-phonon coupled chains.

Tuning friction with noise and disorder

We present numerical and experimental evidence which demonstrates that under certain conditions friction can be reduced by spatial disorder and/or thermal noise. We discuss possible mechanisms for this behavior.