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We describe a physical mechanism for creating multi-soliton bound states where solitons are glued together by attraction between the beams they guide, `solitonic gluons'. We verify the concept of the solitonic gluons experimentally, observing a suppression of the repulsion between dark solitons due to an attractive force between bright guided beams.
We find power-law behaviors of grain velocity in both propagation and backscattering in a gravitationally compacted granular chain with nonlinear contact force. We focus on the leading peak of the velocity signal which decreases in a power-law $d^{-α}$, where $d$ is the location of the peak, as the signal goes down. The ratio of backscattered to incident leading velocity also follows a power-law $d_i^{-β}$, where $d_i$ is the depth of impurity. The up-going backscattered signal is nearly solitary. Therefore, the overall change of the leading velocity peak is given by the power-law $d_i^{-(α+β)}$. We find $α=0.250$ and $β=0.167$ for the Hertzian contact force.
The response of the living tissue to the effects of strong heating or cooling can cause the blood flow rate to vary by an order of magnitude. A mathematical model for the freezing of living tissue is formulated which takes into account the nonlocal temperature dependence of the blood flow rate when the temperature distribution in the tissue is substantially nonuniform, as in cryosurgery.
Withdrawn by the author. Further research is needed.
The paper was withdrawn by the author. Further research is needed.
We consider the damped hyperbolic equation in one space dimension $εu_{tt} + u_t = u_{xx} + F(u)$, where $ε$ is a positive, not necessarily small parameter. We assume that $F(0)=F(1)=0$ and that $F$ is concave on the interval $[0,1]$. Under these assumptions, our equation has a continuous family of monotone propagating fronts (or travelling waves) indexed by the speed parameter $c \ge c_*$. Using energy estimates, we first show that the travelling waves are locally stable with respect to perturbations in a weighted Sobolev space. Then, under additional assumptions on the non-linearity, we obtain global stability results using a suitable version of the hyperbolic Maximum Principle. Finally, in the critical case $c = c_*$, we use self-similar variables to compute the exact asymptotic behavior of the perturbations as $t \to +\infty$. In particular, setting $ε= 0$, we recover several stability results for the travelling waves of the corresponding parabolic equation.
A number of mechanisms that lead to the confinement of patterns to a small part of a translationally symmetric pattern-forming system are reviewed: nonadiabatic locking of fronts, global coupling and conservation laws, dispersion, and coupling to additional slow modes $via$ gradients. Various connections with experimental results are made.
The dynamics of hexagon patterns in rotating systems are investigated within the framework of modified Swift-Hohenberg equations that can be considered as simple models for rotating convection with broken up-down symmetry, e.g. non-Boussinesq Rayleigh-Benard or Marangoni convection. In the weakly nonlinear regime a linear stability analysis of the hexagons reveals long- and short-wave instabilities, which can be steady or oscillatory. The oscillatory short-wave instabilities can lead to stable hexagon patterns that are periodically modulated in space and time, or to a state of spatio-temporal chaos with a Fourier spectrum that precesses on average in time. The chaotic state can exhibit bistability with the steady hexagon pattern. There exist regimes in which the steady hexagon patterns are unstable at all wavenumbers.
Patterns in reaction-diffusion systems near primary bifurcations can be studied locally and classified by means of amplitude equations. This is not possible for excitable reaction-diffusion systems. In this Letter we propose a global classification of two variable excitable reaction-diffusion systems. In particular, we claim that the topology of the underlying two-dimensional homogeneous dynamics can be used to organize the system's behavior.
How can a system in a macroscopically stable state explore energetically more favorable states, which are far away from the current equilibrium state? Based on continuum mechanical considerations we derive a Boussinesq-type equation which models the dynamics of martensitic phase transformations. The solutions of the system, which we refer to as microkinetically regularized wave equation exhibit strong oscillations after short times, thermalization can be confirmed. That means that macroscopic fluctuations of the solutions decay at the benefit of microscopic fluctuations. First analytical and numerical results on the propagation of phase boundaries and thermalization effects are presented. Despite the fact that model is conservative, it exhibits the hysteretic behavior. Such a behavior is usually interpreted in macroscopic models in terms of dissipative threshold which the driving force has to overcome to ensure that the phase transformation proceeds. The threshold value depends on the amount of the transformed phase as it is observed in known experiments. Secondly we investigate the dynamics of oscillatory solutions. We present a formalism based on Young measures, which allows us to describe the effective dynamics of rapidly fluctuating solutions. The new method enables us to derive a numerical scheme for oscillatory solutions based on particle methods.
An impact of kink-type solitons on infrared lattice vibrations is studied for incommensurate Frenkel-Kontorova model. It is shown that the vibration of particles involved into the kink formation is very similar to that in a gap mode around the force constant defect. IR phonon mode intensity is found to possess a universal dependence on the system parameters and the kink concentration. It is argued that the giant IR peak observed in a number of incommensurate charge density wave conductors can be explained in terms of dynamic charge transfer stimulated by kinks.
The exact solutions of the first order differential equation with delay are derived. The equation has been introduced as a model of traffic flow. The solution describes the traveling cluster of jam, which is characterized by Jacobi's elliptic function. We also obtain the family of solutions of such type of equations.
We integrate the one-dimensional nonlinear Schrödinger equation numerically for solitons moving in external potentials. In particular, we study the scattering off an interface separating two regions of constant potential modeled by a linear ramp. Transmission coefficients and inelasticities are computed as functions of the potential difference and the slope of the ramp. Our data show that the ramp's slope has a strong influence revealing unexpected windows of reflection in a transmission regime. The transmission coefficients for very small potential differences are compared with the theoretical predictions derived by perturbation theory. Also the time evolution of the solitary waves after the scattering is studied. We observed that they in general behave like solitons with an amplified amplitude. Due to this, they oscillate. The oscillation period is measured and compared with theoretical predictions. PACS number(s): 03.40.Kf, 0.3.65.Ge, 42.65.Tg, 42.81.Dp
We consider the damped hyperbolic equation (1) εu_{tt} + u_t = u_{xx} + F(u), x \in R, t \ge 0, where εis a positive, not necessarily small parameter. We assume that F(0) = F(1) = 0 and that F is concave on the interval [0,1]. Under these hypotheses, Eq.(1) has a family of monotone travelling wave solutions (or propagating fronts) connecting the equilibria u=0 and u=1. This family is indexed by a parameter c \ge c_* related to the speed of the front. In the critical case c=c_*, we prove that the travelling wave is asymptotically stable with respect to perturbations in a weighted Sobolev space. In addition, we show that the perturbations decay to zero like t^{-3/2} as t \to +\infty and approach a universal self-similar profile, which is independent of ε, F and of the initial data. In particular, our solutions behave for large times like those of the parabolic equation obtained by setting ε= 0 in Eq.(1). The proof of our results relies on careful energy estimates for the equation (1) rewritten in self-similar variables x/\sqrt{t}, \log t.
We study the stabilization of localized structures by discreteness in one-dimensional lattices of diffusively coupled nonlinear sites. We find that in an external driving field these structures may lose their stability by either relaxing to a homogeneous state or nucleating a pair of oppositely moving fronts. The corresponding bifurcation diagram demonstrates a cusp singularity. The obtained analytic results are in good quantitative agreement with numerical simulations.
Using the method of separation of variables, we developed a vector nonparaxial theory for the nonlinear wave equations in strong field approximation. We found that exact localized vortex soliton solution exists for the nonlinear 3D+1 vector wave equation and for 3D+1 vector nonlinear Schrödinger equation. This method is applicable for angular functions, which satisfy additional conditions. It is shown that exact vortex solitary wave exists only for solution with eigenvalue of momentum L=1
The concept of soliton complex in a nonlinear dispersive medium is proposed. It is shown that strongly interacting identical topological solitons in the medium can form bound soliton complexes which move without radiation. This phenomenon is considered to be universal and applicable to various physical systems. The soliton complex and its "excited" states are described analytically and numerically as solutions of nonlinear dispersive equations with the fourth and higher spatial or mixed derivatives. The dispersive sine-Gordon, double and triple sine-Gordon, and piecewise-linear models are studied in detail. Mechanisms and conditions of the formation of soliton complexes, and peculiarities of their stationary dynamics are investigated. A phenomenological approach to the description of the complexes and the classification of all the possible complex states are proposed. Some examples of physical systems, where the phenomenon can be experimentally observed, are briefly discussed.
Using the method of separation of variables, we developed a vector nonparaxial theory from the nonlinear wave equations in strong field approximation of a Kerr media. Investigating three waves on different frequencies, which satisfied the condition 2w3=w1+w2, we found that exact localized vortex parametric soliton solutions existed for the set of four nonlinear 3D+1vector wave equations. The solutions are obtained for a fixed angle between the main frequencies and the signal waves A=60 deg. This method is applicable for angular functions, which satisfy additional conditions. It is shown that exact parametric vortex solitary waves exist for solution with eigenrotation momentum L=1. The method is generalized for arbitrary number of signal waves appearing symmetrically in pairs in respect to w3. The existence of such type of vortex solitons, in the case of degenerate four-wave mixing is discussed. As all of the waves are localized and have momentum L=1, they are called optical mesons.