Kink Antikink Research Articles

Energy transfer in the Holstein approach for the interplay between periodic on-site and linear acoustic potentials

We study the problem of a transferring electron along a lattice of phonons, in the continuous long wave limit, holding periodic on-site and linear longitudinal interactions in Holstein’s approach. We thus find that the continuum limit of our modeling produces an effective coupling between the linear Schrödinger and sine–Gordon equations. Then, we take advantage of the existence of trapped kink–anti kink solutions in the sine–Gordon equation to variationally describe traveling localized coupled solutions. We validate our variational findings by solving numerically the full coupled system. Very reasonable agreement is found between the variational and full numerical solutions for the amplitude evolution of both profiles; the wave function and the trapped kink–anti kink. Our results show the significance of permitting longitudinal interactions in the Holstein’s approach to hold trapped localized solutions. It is actually found a critical ratio between longitudinal and on-site interactions, as depending on the velocity of propagation, from where coupled localized solutions exist.

A microscopic traffic flow model for explaining nonlinear traffic phenomena: Modeling, stability analysis and validation

As the core of reproducing the real nonlinear phenomena of traffic congestion, the investigation of car-following models, which take into account human factors, has consistently assumed a central role within the domain of transportation science. Nevertheless, it is noteworthy that prior studies have invariably employed a fixed value for driver reaction time to stimuli, whereas it is imperative to recognize the dynamic nature of this parameter, closely associated with the instantaneous vehicle speed. To address this issue, this paper constructs a dynamic sensitivity coefficient (DSC), and further develops a nonlinear human factoring car-following model to reveal the relation between the reaction time and the current speed. Firstly, the linear critical stability condition and the mKdV equation of the model are derived by the linear stability analysis and kink–antikink solution analytic method, respectively. Then, the numerical experiments are conducted to demonstrate that the proposed model is more practical and can reproduce the real driving behavior and phenomena. Finally, calibration and validation results exhibit the actual vehicle trajectory can be simulated well.

Nonlinear Analysis of an Extended Heterogeneous Lattice Hydrodynamic Model Considering on/off-Ramps

To study the impact of on/off-ramps on traffic stability in heterogeneous traffic flow, a novel lattice hydrodynamic model was presented. The new model’s stability condition was determined using the linear stability analysis method. The theoretical results reveal that traffic flow stability is influenced by the proportion of vehicles with different maximum speeds and safe headway, as well as the presence of on-ramps and off-ramps to a certain degree. Through the approximate perturbation method, the mKdV equation and the kink–antikink solution of the traffic density at the jam area are obtained. In order to verify the effectiveness and feasibility of the new model, numerical simulations were conducted to demonstrate on/off ramp effect and different proportions of vehicles which possess bigger maximum velocity or safe space headway affect the traffic stability. The numerical results indicate that in heterogeneous traffic flow scenarios, increasing the ratio of vehicles which possess bigger maximum velocity or bigger safe space headway can lead to a deterioration of traffic stability. The effect of on-ramp could cause traffic instability, while the effect of off-ramp is beneficial for easing traffic congestion.

Novel superposed kinklike and pulselike solutions for several nonlocal nonlinear equations

We show that a number of nonlocal nonlinear equations, including the Ablowitz–Musslimani and Yang variant of the nonlocal nonlinear Schrödinger (NLS) equation, the nonlocal modified Korteweg de Vries (mKdV) equation, and the nonlocal Hirota equation, admit novel kinklike and pulselike superposed periodic solutions. Furthermore, we show that the nonlocal mKdV equation also admits the superposed (hyperbolic) kink–antikink solution. In addition, we show that while the nonlocal Ablowitz–Musslimani variant of the NLS admits complex parity-time reversal-invariant kink and pulse solutions, neither the local NLS nor the Yang variant of the nonlocal NLS admits such solutions. Finally, except for the Yang variant of the nonlocal NLS, we show that the other three nonlocal equations admit both the kink and pulse solutions in the same model.

Bifurcation analysis and dynamics of abundant traveling waves of the modified Novikov equation

This paper concerns traveling waves of the modified Novikov equation by dynamical systems approach. We show dynamics of abundant traveling waves for the modified Novikov equation, including solitary waves, periodic waves, compactons, kink (antikink), and kink‐like (antikink‐like) waves. The kink waves are newly founded in the Novikov equation. Additionally, we also derive exact explicit solitary waves under special parameter conditions.

Scattering of the asymmetric $$\phi ^6$$ kinks from a $${\mathcal{PT}\mathcal{}}$$-symmetric perturbation: creating multiple kink–antikink pairs from phonons

Interaction of asymmetric $\phi^6$ kinks with a spatially localized $\mathcal{PT}$-symmetric perturbation is investigated numerically. It has been shown that when the kink (antikink) hits the defect from the gain side, a final velocity of the kink decreases (increases), while for the kink and antikink coming from the opposite direction their final velocities remain unchanged. It is also found that when the kink interacts with the defect from the gain side multiple pair of the kink-antikink are formed from small amplitude waves (phonons) in the final states depending on the initial velocity of the initial kink and parameter of the perturbation.

Bifurcation Analysis and Exact Wave Solutions for the Double‐Chain Model of DNA

This work aims to study analytically the nonlinear model for deoxyribonucleic acid (DNA). Based on the complete discrimination and direct method, some new wave solutions are introduced. These solutions are sorted into solitary, periodic, kink (antikink), and singular solutions. Moreover, a part of them is illustrated graphically. Based on Hamilton concepts, we study the bifurcation and phase portrait for the Hamilton system corresponding to the model under consideration.

Sphalerons and resonance phenomenon in kink-antikink collisions

We show that in some kink-antikink (KAK) collisions sphalerons, i.e., unstable static solutions - rather than the asymptotic free soliton states - can be the source of the internal degrees of freedom (normal modes) which trigger the resonance phenomenon responsible for the fractal structure in the final state formation.

Dipole–dipole interactions effects on mobility and conductivity of ionic and bonded defects in hydrogen-bonded chains

Defect formation and transport in a hydrogen-bonded system is studied via a two-sublattice soliton-bearing one-dimensional model. Our two component model completes and extends our previous study (Tchakoutio Nguetcho and Kofane, 2007; Tchakoutio Nguetcho et al., 2008; Li et al., 2015) made on the dynamics of defects in a quasi-one-dimensional nonlinear model of complex hydrogen bond systems. The inclusion of dipole–dipole interactions provides a new class of differential equations possessing several key parameters and many singular straight lines. In order to evaluate and measure the variations effects on each interaction parameter, we use dynamic system methods to first say what are really the key parameters of the system, and then discuss on phase portraits bifurcations to derive a variety of exotic solutions corresponding to phase trajectories under different parameter conditions. Among the rich variety of soliton patterns obtained, we focus our attention to kink (antikink) solutions. The response of these solitonic defects to an externally applied DC electric field makes this system an excellent model to describe the protonic conductivity in complex hydrogen-bonded networks. Therefore, we evaluate analytically the effect of dipole–dipole interactions on the mobility of the kink–antikink pair and the specific electrical-conductivity of the protons transfer.

Painlevé analysis, auto-Bäcklund transformation and analytic solutions for modified KdV equation with variable coefficients describing dust acoustic solitary structures in magnetized dusty plasmas

In this paper, variable coefficients mKdV equation is examined by using Painlevé analysis and auto-Bäcklund transformation method. The proposed equation is an important equation in magnetized dusty plasmas. The Painlevé analysis is used to determine the integrability whereas an auto-Bäcklund transformation technique is being explored to derive unique family of analytical solutions for variable coefficients mKdV equation. New kink–antikink and periodic-kink- type soliton solutions have been determined successfully for the considered equation. This paper shows that auto-Bäcklund transformation method is effective, direct and easy to use, and used to determine the analytic soliton solutions of various nonlinear evolution equations in the field of science and engineering. The results are plotted graphically to signify the potency and applicability of this proposed scheme for solving the above considered equation. The obtained results are in the form of soliton-like solutions, solitary wave solutions, exponential and trigonometric function solutions. Therefore, these solutions help us to understand the potential and physical behaviors of the proposed equation.

A Theoretical Model for Release Dynamics of an Antifungal Agent Covalently Bonded to the Chitosan

The aim of the study was to create a mathematical model useful for monitoring the release of bioactive aldehydes covalently bonded to the chitosan by reversible imine linkage, considered as a polymer–drug system. For this purpose, two hydrogels were prepared by the acid condensation reaction of chitosan with the antifungal 2-formyl-phenyl-boronic acid and their particularities; influencing the release of the antifungal aldehyde by shifting the imination equilibrium to the reagents was considered, i.e., the supramolecular nature of the hydrogels was highlighted by polarized light microscopy, while scanning electron microscopy showed their microporous morphology. Furthermore, the in vitro fungicidal activity was investigated on two fungal strains and the in vitro release curves of the antifungal aldehyde triggered by the pH stimulus were drawn. The theoretical model was developed starting from the hypothesis that the imine-chitosan system, both structurally and functionally, can be assimilated, from a mathematical point of view, with a multifractal object, and its dynamics were analyzed in the framework of the Scale Relativity Theory. Thus, through Riccati-type gauges, two synchronous dynamics, one in the scale space, associated with the fungicidal activity, and the other in the usual space, associated with the antifungal aldehyde release, become operational. Their synchronicity, reducible to the isomorphism of two SL(2R)-type groups, implies, by means of its joint invariant functions, bioactive aldehyde compound release dynamics in the form of “kink–antikink pairs” dynamics of a multifractal type. Finally, the theoretical model was validated through the experimental data.

Kink–antikink scattering-induced breathing bound states and oscillons in a parametrized ϕ4 model

Recent studies have emphasized the important role that a shape deformability of scalar-field models pertaining to the same class with the standard [Formula: see text] field, can play in controlling the production of a specific type of breathing bound states so-called oscillons. In the context of cosmology, the built-in mechanism of oscillons suggests that they can affect the standard picture of scalar ultra-light dark matter. In this paper, kink scatterings are investigated in a parametrized model of bistable system admitting the classical [Formula: see text] field as an asymptotic limit, with focus on the formation of long-lived low-amplitude almost harmonic oscillations of the scalar field around a vacuum. The parametrized model is characterized by a double-well potential with a shape-deformation parameter that changes only the steepness of the potential walls, and hence the flatness of the hump of the potential barrier, leaving unaffected the two degenerate minima and the barrier height. It is found that the variation of the deformability parameter promotes several additional vibrational modes in the kink-phonon scattering potential, leading to suppression of the two-bounce windows in kink–antikink scatterings and the production of oscillons. Numerical results suggest that the anharmonicity of the potential barrier, characterized by a flat barrier hump, is the main determinant factor for the production of oscillons in double-well systems.

Kink-antikink collisions in the periodic [formula omitted] model

We borrow the form of potential of the well-known kink-bearing φ4 system in the range between its two vacua and paste it repeatedly into the other ranges to introduce the periodic φ4 system. The paper is devoted to providing a comparative numerical study of the properties of the two systems. Although the two systems are quite similar for a kink (antikink) solution, they usually exhibit different behaviors throughout collisions. For instance, they have different critical velocities, different results during collisions, and a different rule in their quasi-fractal structures. Their quasi-fractal structures will be studied in the disturbed kink-antikink collisions as well. Hence, three types of scattering windows will be introduced with respect to the incoming speed, the amplitude, and initial phase of the internal mode, respectively. Moreover, a detailed comparative study of the collisions between two kinks and one antikink will be done at the end.

Nonlinear analysis of the car-following model considering headway changes with memory and backward looking effect

In order to research the effects of headway changes with memory and backward looking effect on traffic flow, in the light of the full velocity difference model (FVDM), a novel car-following model that considers backward looking effect and headway changes with memory is proposed. The use of linear stability theory is to obtain the stability criterion that can judge traffic flow stability. In addition, on the basis of nonlinear theory, the time-dependent Ginzburg–Lan (TDGL) equation and the modified Korteweg–de Vries (mKdV) equation near the critical stability point are inferred. The kink–antikink solutions of the above two equations can be used to describe the traffic jam. In the end, numerical simulation is carried out, and the results further demonstrate that the novel model can stabilize traffic flow and eliminate traffic jam better.

New feedback control for a novel two-dimensional lattice hydrodynamic model considering driver’s memory effect

Considering the driver’s memory effect and new feedback control signal, a novel two-dimensional lattice hydrodynamic model is proposed. The linear stability condition of the new model is derived by exploiting control method. Via nonlinear analysis, the kink–antikink solution of modified Korteweg–de Vries (mKdV) equation is derived, which can be used to give a description of the density waves near the critical points. Then numerical simulation is conducted to verify the theoretical analysis. The results of theoretical analysis and numerical simulation both show that the new control signal availably stabilizes the traffic flow. On the contrary, as the driver’s memory time increases, the traffic flow becomes more unstable.

Kink–antikink interaction forces and bound states in a biharmonic ϕ4 model

We consider the interaction of solitons in a biharmonic, beam model analogue of the well-studied ϕ4 Klein–Gordon theory. Specifically, we calculate the force between a well separated kink and antikink. Knowing their accelerations as a function of separation, we can determine their motion using a simple ordinary differential equations. There is good agreement between this asymptotic analysis and numerical computation. Importantly, we find the force has an exponentially-decaying oscillatory behaviour (unlike the monotonically attractive interaction in the Klein–Gordon case). Corresponding to the zeros of the force, we predict the existence of an infinite set of field theory equilibria, i.e., kink–antikink bound states. We confirm the first few of these at the partial differential equations level, and verify their anticipated stability or instability. We also explore the implications of this interaction force in the collision between a kink and an oppositely moving antikink.

Analysis of an extended two-lane lattice hydrodynamic model considering mixed traffic flow and self-stabilization effect

PurposeThe purpose of this paper is to explore the impact of the mixed traffic flow, self-stabilization effect and the lane changing behavior on traffic flow stability.Design/methodology/approachAn extended two-lane lattice hydrodynamic model considering mixed traffic flow and self-stabilization effect is proposed in this paper. Through linear analysis, the stability conditions of the extended model are derived. Then, the nonlinear analysis of the model is carried out by using the perturbation theory, the modified Kortweg–de Vries equation of the density of the blocking area is derived and the kink–antikink solution about the density is obtained. Furthermore, the results of theoretical analysis are verified by numerical simulation.FindingsThe results of numerical simulation show that the increase of the proportion of vehicles with larger maximum speed or larger safe headway in the mix flow are not conducive to the stability of traffic flow, while the self-stabilization effect and lane changing behavior is positive to the alleviation of traffic congestion.Research limitations/implicationsThis paper does not take into account the factors such as curve and slope in the actual road environment, which will have more or less influence on the stability of traffic flow, so there is still a certain gap with the real traffic environment.Originality/valueThe existing two-lane lattice hydrodynamic models are rarely discussed in the case of mixed traffic flow. The improved model proposed in this paper can better reflect the actual traffic, which can also provide a theoretical reference for the actual traffic governance.

Padé schemes with Richardson extrapolation for the sine-Gordon equation

Four novel implicit finite difference methods with (q+s)-th order in space based on (q, s)-Padé approximations have been analyzed and developed for the sine-Gordon equation. Specifically, (4,0)-, (2,2)-, (4,2)-, and (4,4)-Padé methods. All of them share the treatment for the nonlinearity and integration in time, specifically, the one that results in an energy-conserving (2,0)-Padé scheme. The five methods have been developed with and without Richardson extrapolation in time. All the methods are linearly, unconditionally stable. A comparison among them for both the kink–antikink and breather solutions in terms of global error, computational cost and energy conservation is presented. Our results indicate that the (4,0)- and (4,4)-Padé methods without Richardson extrapolation are the most cost-effective ones for small and large global error, respectively; and the (4,4)-Padé methods in all the cases when Richardson extrapolation is used.

Exact explicit nonlinear wave solutions to a modified cKdV equation

In this paper, we study nonlinear wave solutions to a modified cKdV equation by exploiting Bifurcation method of Hamiltonian systems. We identify all possible bifurcation conditions and obtain the phase portraits of the system in different regions of the parametric space, through which, we obtain exact explicit nonlinear wave solutions, including solitary wave solutions, singular wave solutions, periodic singular wave solutions, and kink (antikink) wave solutions. Of particular interest is the appearance of the so-called V-shaped kink (antikink) wave solutions, W-shaped solitary wave solutions, and W-shaped periodic wave solutions, which were not found in previous studies.

Kink scattering in hyperbolic models

In this work, we study kink–antikink and antikink–kink collisions in hyperbolic models of fourth and sixth orders. We compared the patterns of scattering with known results from polynomial models of the same order. The hyperbolic models considered here tend to the polynomial [Formula: see text] and [Formula: see text] models in the limit of small values of the scalar field. We show that kinks and antikinks that interact hyperbolically with the fourth order differ sensibly from those governed by the polynomial [Formula: see text] model. The increasing of the order of interaction to the sixth order shows that the hyperbolic and polynomial models give intricate structures of scattering that differ only slightly. The dependence on the order of interaction is related to some characteristics of the models such as the potential of perturbations and the number of vibrational modes.