Kink Antikink Research Articles

Kink–Antikink Scattering in φ4 and ϕ6 Models

For kink–antikink scattering within the φ4 non-linear field theory in one space and one time dimension resonance type configurations emerge when the relative velocity between kink and antikink falls below a critical value. It has been conjectured that the vibrational excitation of the kink would be the source for these resonances because (simplified) collective coordinate calculations, that emphasized on this excitation, qualitatively reproduced those resonances. Surprisingly a numerical study in the ϕ6 field theory also exhibited such resonances even though it does not contain the vibrational excitation. To explore this contradiction we start from the working hypothesis that in either model any collective coordinate ansatz which includes a degree of freedom similar to the vibrational excitation leads to resonances, regardless of whether or not this mode emerges as a solution to the (linearized) field equations. To this end we compare numerical results in the φ4 and ϕ6 models that arise from the full set of partial differential equations to those from the ordinary differential equations for a collective coordinate ansatz. An inaccuracy in literature formulas for the collective coordinate approach in the φ4 model requires to revisit those calculations.

Bifurcation of Travelling Wave Solutions of the Generalized Zakharov Equation

By using bifurcation theory of planar ordinary differential equations all different bounded travelling wave solutions of the generalized Zakharov equation are classified in to different parametric regions. In each of these parametric regions the exact explicit parametric representation of all solitary, kink (antikink), and periodic wave solutions as well as their numerical simulation and their corresponding phase portraits are obtained.

Traveling-wave pulse on a superconductive active transmission line using resonant tunneling diodes

Analytic study and computer experiment investigations on a superconductive active transmission line using resonant tunneling diodes (RTDs) are discussed. It is shown, based on nonlinear wave propagation effects, that the line supports pulse propagation appearing as pairs of kink–antikink profiles. This behavior is due to compensation between the effects of amplification and dissipation along the network.

Flow difference effect in the two-lane lattice hydrodynamic model

By introducing a flow difference effect, a modified lattice two-lane traffic flow model is proposed, which is proved to be capable of improving the stability of traffic flow. Both the linear stability condition and the kink—antikink solution derived from the modified Korteweg—de Vries (mKdV) equation are analyzed. Numerical simulations verify the theoretical analysis. Furthermore, the evolution laws under different disturbances in the metastable region are studied.

Friction coefficient and radius of curvature effects upon traffic flow on a curved Road

We investigate the friction coefficient and radius of curvature effects upon traffic flow analytically and numerically. A new model is proposed to describe the motion of vehicles running on a curved road. The stability condition is obtained by the use of control theories. The nonlinear analysis method is taken to derive the modified KdV (Korteweg–de Vries) equation and the kink–antikink solution is obtained near the critical point. Simulations are carried out to verify the double effects upon traffic flow. The flux increases but the stability decreases with an increase of the two parameters. Numerical results are in good agreement with the analytical results.

Multiple flux difference effect in the lattice hydrodynamic model

Considering the effect of multiple flux difference, an extended lattice model is proposed to improve the stability of traffic flow. The stability condition of the new model is obtained by using linear stability theory. The theoretical analysis result shows that considering the flux difference effect ahead can stabilize traffic flow. The nonlinear analysis is also conducted by using a reductive perturbation method. The modified KdV (mKdV) equation near the critical point is derived and the kink—antikink solution is obtained from the mKdV equation. Numerical simulation results show that the multiple flux difference effect can suppress the traffic jam considerably, which is in line with the analytical result.

Extension on Bifurcations of Traveling Wave Solutions for a Two‐Component Fornberg‐Whitham Equation

Fan et al. studied the bifurcations of traveling wave solutions for a two‐component Fornberg‐Whitham equation. They gave a part of possible phase portraits and obtained some uncertain parametric conditions for solitons and kink (antikink) solutions. However, the exact explicit parametric conditions have not been given for the existence of solitons and kink (antikink) solutions. In this paper, we study the bifurcations for the two‐component Fornberg‐Whitham equation in detalis, present all possible phase portraits, and give the exact explicit parametric conditions for various solutions. In addition, not only solitons and kink (antikink) solutions, but also peakons and periodic cusp waves are obtained. Our results extend the previous study.

Bifurcation of Traveling Wave Solutions for a Two‐Component Generalized θ‐Equation

We study the bifurcation of traveling wave solutions for a two‐component generalized θ‐equation. We show all the explicit bifurcation parametric conditions and all possible phase portraits of the system. Especially, the explicit conditions, under which there exist kink (or antikink) solutions, are given. Additionally, not only solitons and kink (antikink) solutions, but also peakons and periodic cusp waves with explicit expressions, are obtained.

The time-dependent Ginzburg—Landau equation for the two-velocity difference model

A thermodynamic theory is formulated to describe the phase transition and critical phenomenon in traffic flow. Based on the two-velocity difference model, the time-dependent Ginzburg—Landau (TDGL) equation under certain condition is derived to describe the traffic flow near the critical point through the nonlinear analytical method. The corresponding two solutions, the uniform and the kink solutions, are given. The coexisting curve, spinodal line and critical point are obtained by the first and second derivatives of the thermodynamic potential. The modified Korteweg de Vries (mKdV) equation around the critical point is derived by using the reductive perturbation method and its kink—antikink solution is also obtained. The relation between the TDGL equation and the mKdV equation is shown. The simulation result is consistent with the nonlinear analytical result.

A traffic flow lattice model considering relative current influence and its numerical simulation

Based on Xue's lattice model, an extended lattice model is proposed by considering the relative current information about next-nearest-neighbour sites ahead. The linear stability condition of the presented model is obtained by employing the linear stability theory. The density wave is investigated analytically with the perturbation method. The results show that the occurrence of traffic jamming transitions can be described by the kink–antikink solution of the modified Korteweg-de Vries (mKdV) equation. The simulation results are in good agreement with the analytical results, showing that the stability of traffic flow can be enhanced when the relative current of next-nearest-neighbour sites ahead is considered.

Relaxation of terrace-width distributions: Physical information from Fokker–Planck time

Recently some of us have constructed a Fokker–Planck formalism to describe the equilibration of the terrace-width distribution of a vicinal surface from an arbitrary initial configuration. However, the meaning of the associated relaxation time, related to the strength of the random noise in the underlying Langevin equation, was rather unclear. Here we present a set of careful kinetic Monte Carlo simulations that demonstrate convincingly that the time constant shows activated behavior with a barrier that has a physically plausible dependence on the energies of the governing microscopic model. Remarkably, the rate-limiting step for relaxation in the far-from-equilibrium regime is the generation of kink–antikink pairs, involving the breaking of three lateral bonds on a cubic {0 0 1} surface, in contrast to the processes breaking two bonds that dominate equilibrium fluctuations. After an initial regime, the Fokker–Planck time at least semiquantitatively tracks the actual physical time.

Numerical simulation of soliton and kink density waves in traffic flow with periodic boundaries

The soliton and kink–antikink density waves are simulated with periodic boundaries, by adding perturbation in the initial condition on single-lane road based on a car-following model. They are reproduced in the form of the space–time evolution of headway, both of which propagate backwards. It is found that the solitons appear only near the neutral stability line regardless of the boundary conditions, and they exhibit upward form when the initial headway is smaller than the safety distance, otherwise they exhibit downward form. Comparison is made between the numerical and analytical results about the amplitude of kink–antikink wave, and the underlying mechanism is analyzed. Besides, it is indicated that the maximal current of traffic flow increases with decreasing safety distance. The numerical simulation shows a good agreement with the analytical results.

Superconducting Transitions in Wire Network under Spatially Modulated Magnetic Field

Superconducting wire network subjected to a uniform and a checkerboard-patterned magnetic field is investigated. The checkerboard field is created by an array of ferromagnetic dots placed on top of the superconducting network. Measurements of the current–voltage ( I – V ) characteristics have revealed the canonical Kosterlitz–Thouless–Berezinskii (KTB) transition at zero magnetic field. The superconducting transition at α=1/2 (α being the frustration parameter) is studied in detail, and compared with the theoretical predictions on the fully frustrated XY model which has a double degeneracy in the ground state vortex configuration. The I – V characteristics for α=1/2 is consistent with the scenario of kink–antikink unbinding at antiphase domain boundaries first proposed by Korshunov [Phys. Rev. Lett. 88 (2002) 167007]. Introduction of the checkerboard field with amplitude β to the α=1/2 case lifts the double degeneracy. The effect of non-zero β on the nature of superconducting transition at α=1/2 is consis...

Kink–antikink density wave of an extended car-following model in a cooperative driving system

We propose an extended optimal velocity model applicable to cooperative driving control system by considering the headway of arbitrary number of cars that precede and the relative velocity. The stability condition of the extended model is obtained by using the linear stability theory. The modified Korteweg-de Vries (mKdV) equation is derived to describe the traffic behavior near the critical point by applying the nonlinear analysis. Thus the traffic jams can be described by the kink–antikink density wave which is the solution of the mKdV equation. The simulation results confirm the analytical results and show that the traffic jams are suppressed more efficiently with considering not only the headway of more vehicles ahead but also the relative velocity.

The return of the kink

We present a simple theory for the statistics of subsequent passages of kinks at a particular position along a fluctuating step on a crystal surface. Two situations are treated, namely one where the waiting time until the next passage is measured starting from the previous passage of a kink at the observation point, and one where the waiting time is measured starting from a random instant in time. In both cases the waiting time distribution is shown not to obey simple Poisson statistics. Monte Carlo simulations of kink motion show that Poisson statistics emerges for longer waiting times only when the creation and annihilation of kink–antikink pairs are explicitly included. Together, the theory and simulations provide an alternative interpretation of experimental results obtained by Giesen et al. with Scanning Tunneling Microscopy [M. Giesen-Seibert, H. Ibach, Surf. Sci. 316 (1994) 205].

Phase compactons

We study the phase dynamics of a chain of autonomous, self-sustained, dispersively coupled oscillators. In the quasicontinuum limit the basic discrete model reduces to a Korteveg–de Vries-like equation, but with a nonlinear dispersion. The system supports compactons–solitary waves with a compact support–and kovatons–compact formations of glued together kink–antikink pairs that propagate with a unique speed, but may assume an arbitrary width. We demonstrate that lattice solitary waves, though not exactly compact, have tails which decay at a superexponential rate. They are robust and collide nearly elastically and together with wave sources are the building blocks of the dynamics that emerges from typical initial conditions. In finite lattices, after a long time, the dynamics becomes chaotic. Numerical studies of the complex Ginzburg–Landau lattice show that the non-dispersive coupling causes a damping and deceleration, or growth and acceleration, of compactons. A simple perturbation method is applied to study these effects.

Kinks in dipole chains

It is shown that the topological discrete sine-Gordon system introduced by Speight and Ward models the dynamics of an infinite uniform chain of electric dipoles constrained to rotate in a plane containing the chain. Such a chain admits a novel type of static kink solution which may occupy any position relative to the spatial lattice and experiences no Peierls–Nabarro barrier. Consequently the dynamics of a single kink is highly continuum-like, despite the strongly discrete nature of the model. Static multikinks and kink–antikink pairs are constructed, and it is shown that all such static solutions are unstable. Exact propagating kinks are sought numerically using the pseudo-spectral method, but it is found that none exist, except, perhaps, at very low speed.

Full phase diagram of the massive Gross–Neveu model

The massive Gross–Neveu model is solved in the large- N limit at finite temperature and chemical potential. The scalar potential is given in terms of Jacobi elliptic functions. It contains three parameters which are determined by transcendental equations. Self-consistency of the scalar potential is proved. The phase diagram for non-zero bare quark mass is found to contain a kink–antikink crystal phase as well as a massive fermion gas phase featuring a cross-over from light to heavy effective fermion mass. For zero bare quark mass, we recover the three known phases kink–antikink crystal, massless fermion gas, and massive fermion gas. All phase transitions are shown to be of second order. Equations for the phase boundaries are given and solved numerically. Implications on condensed matter physics are indicated where our results generalize the bipolaron lattice in non-degenerate conducting polymers to finite temperature.

Kink shape solutions of the Maxwell-Lorentz system

In the limit of high amplitude oscillating electromagnetic fields, a sequence of kink antikink shaped optical waves has been found in the Maxwell's equations coupled to a single Lorentz oscillator and with Kerr nonlinearity. The individual kinks and antikinks result from a traveling wave assumption and their stability has been assessed by numerical simulations. For typical physical parameter values the kink width is of the order of tens of femtoseconds.

REGULAR AND PERIODIC TACHYON KINKS

We search for regular tachyon kinks in an extended model, which includes the tachyon action recently proposed to describe the tachyon field. The extended model that we propose adds a new contribution to the tachyon action, which allows one to obtain stable tachyon kinks of regular profile, which may appropriately lead to the singular kink found by Sen sometime ago. Also, under specific conditions we may find periodic array of kink–antikink configurations.