Rotating Charge Coupled to the Maxwell Field: Scattering Theory and Adiabatic Limit

We consider a spinning charge coupled to the Maxwell field. Through the appropriate symmetry in the initial conditions the charge remains at rest. We establish that any time-dependent finite energy solution converges to a sum of a soliton wave and an outgoing free wave. The convergence holds in global energy norm. Under a small constant external magnetic field the soliton manifold is stable in local energy seminorms and the evolution of the angular velocity is guided by an effective finite-dimensional dynamics. The proof uses a non-autonomous integral inequality method.Supported partly by the Wittgenstein 2000 Award of Peter Markowich, funded by the Austrian Science Foundation (FWF), research grants of DFG (436 RUS 113/615/0-1(R)) and RFBR (01-01-04002).On leave Department of Mechanics and Mathematics, Moscow State University, Moscow 119899, Russia. Supported partly by Max Planck Institute for the Mathematics in Sciences (Leipzig) and the Austrian Science Foundation (FWF) START Project (Y-137-TEC) of Norbert Mauser.     

Scattering of solitons for coupled wave-particle equations

We establish a long time soliton asymptotics for a nonlinear system of wave equation coupled to a charged particle. The coupled system has a six-dimensional manifold of soliton solutions. We show that in the large time approximation, any solution, with an initial state close to the solitary manifold, is a sum of a soliton and a dispersive wave which is a solution to the free wave equation. It is assumed that the charge density satisfies Wiener condition which is a version of Fermi Golden Rule, and that the momenta of the charge distribution vanish up to the fourth order. The proof is based on a development of the general strategy introduced by Buslaev and Perelman: symplectic projection in Hilbert space onto the solitary manifold, modulation equations for the parameters of the projection, and decay of the transversal component.     

Solitary waves in dusty plasmas with variable dust charge and two temperature ions

Propagation of nonlinear waves in dusty plasmas with variable dust charge and two temperature ions is analyzed. The Kadomtsev–Petviashivili (KP) equation is derived by using the reductive perturbation theory. A Sagdeev potential for this system has been proposed. This potential is used to study the stability conditions and existence of solitonic solutions. Also, it is shown that a rarefactive soliton can be propagates in most of the cases. The soliton energy has been calculated and a linear dispersion relation has been obtained using the standard normal-modes analysis. The effects of variable dust charge on the amplitude, width and energy of the soliton and its effects on the angular frequency of linear wave are discussed too. It is shown that the amplitude of solitary waves of KP equation diverges at critical values of plasma parameters. Solitonic solutions of modified KP equation with finite amplitude in this situation are derived.     

Soliton model of elementary electric charge

Conclusions Thus, we have shown that in the electrodynamics of the Klein-Gordon field there exist two spectra of three-dimensional electrostatic soliton solutions. One of them is dynamically stable, the other topologically stable. For each of the spectra, the value of the electrostatic potential at the center of the soliton is quantized, while the value of the electric field vanishes. At the periphery of the solitons, the electrostatic potential corresponds to the Coulomb law.The topological charge of the soliton is related to quantization of the value of the electrostatic potential at infinity: ,p=0, 1, 2, .... The topological soliton can obviously be regarded as a model of an elementary electric charge, an attractive feature of which is the absence of divergences of the integrals of the motion.For the dynamical solitons, we have _p =0,p=0, 1, 2, ....A rotating electrostatic soliton can be regarded as a soliton model of an elementary electric charge possessing an intrinsic magnetic moment. The magnetic field at the center of the rotating soliton is quantized, while at its periphery the field has a magneticdipole nature.Institute of Theoretical Physics, Ukrainian SSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 82, No. 3, pp. 349–359, March, 1990.     

Solitons via Lie-Bäcklund transformation for 5D low-energy string theory

We apply a non-linear matrix transformation of Lie-Bäcklund type on a seed soliton configuration in order to obtain a new solitonic solution in the framework of the 5D low-energy effective field theory of the bosonic string. The seed solution represents a stationary axisymmetric two-soliton configuration previously constructed through the inverse scattering method and consists of a massless gravitational field coupled to a non-trivial chargeless dilaton and to an axion field endowed with charge. We apply a fully parameterized non-linear matrix transformation of Ehlers type on this massless solution and get a massive rotating axisymmetric gravitational soliton coupled to charged axion and dilaton fields. We discuss on some physical properties of both the initial and the generated solitons and fully clarify the physical effect of the non-linear normalized Ehlers transformation on the seed solution.     

Spatiotemporal Chaotic Dynamics of Solitons with Internal Structure in the Presence of Finite-Width Inhomogeneities

We present an analytical and numerical study of the Klein–Gordon kink-soliton dynamics in inhomogeneous media. In particular, we study an external field that is almost constant for the whole system but that changes its sign at the center of coordinates and a localized impurity with finite-width. The soliton solution of the Klein–Gordon-like equations is usually treated as a structureless point-like particle. A richer dynamics is unveiled when the extended character of the soliton is taken into account. We show that interesting spatiotemporal phenomena appear when the structure of the soliton interacts with finite-width inhomogeneities. We solve an inverse problem in order to have external perturbations which are generic and topologically equivalent to well-known bifurcation models and such that the stability problem can be solved exactly. We also show the different quasiperiodic and chaotic motions the soliton undergoes as a time-dependent force pumps energy into the traslational mode of the kink and relate these dynamics with the excitation of the shape modes of the soliton.     

Weakly damped KdV soliton dynamics with the random force

The soliton dynamics in the random field is studied in the framework of the Korteweg–de Vries–Burgers equation. Asymptotic solution of this equation with weak dissipation is found and the average wave field is analyzed. All formulas can be given explicitly for the uniform (table-top) distribution function of the random field. Weakly damped KdV soliton on large times transforms to the “thick” soliton or KdV-like soliton depending from the statistical properties of the force. New scenario of KdV soliton transformation into the thick soliton and then again in KdV-like soliton is predicted for certain conditions.     

Reflection and Refraction of Solitons by the KdV–Burgers Equation in Nonhomogeneous Dissipative Media

We study the behavior of the soliton that encounters a barrier with dissipation while moving in a nondissipative medium. We use the Korteweg–de Vries–Burgers equation to model this situation. The modeling includes the case of a finite dissipative layer similar to a wave passing through air–glass–air and also a wave passing from a nondissipative layer into a dissipative layer (similar to light passing from air to water). The dissipation predictably reduces the soliton amplitude/velocity. Other effects also occur in the case of a finite barrier in the soliton path: after the wave leaves the dissipative barrier, it retains the soliton form, but a reflection wave arises as small and quasiharmonic oscillations (a breather). The breather propagates faster than the soliton passing through the barrier.     

Painlevé property, auto-Bäcklund transformation and analytic solutions of a variable-coefficient modified Korteweg-de Vries model in a hot magnetized dusty plasma with charge fluctuations

Under investigation in this paper is a variable-coefficient modified Korteweg-de Vries (vc-mKdV) model in a hot magnetized dusty plasma with charge fluctuations. With symbolic computation and bilinear method, Painlevé property is studied, auto-Bäcklund transformation is constructed, while soliton and other analytic solutions are obtained. Furthermore, influence of the coefficients on the dust-ion-acoustic (DIA) solitary wave propagation is investigated based on the soliton solution, which can be concluded as follows: (i) Amplitude of the DIA solitary wave is proportional to the square of the ratio of the coefficients of the dispersive to nonlinear terms; (ii) Velocity of the DIA solitary wave is controlled by the coefficients of the dispersive and dissipative terms; (iii) Propagation trajectory of the DIA solitary wave depends on the function forms of the coefficients of the dispersive, nonlinear and dissipative terms.     

Soliton decay in a Toda chain caused by dissipation

Consider a Toda chain with uniform friction. Starting with an initial condition that represents a soliton, we investigate its decay. The main result is that the solitary character is almost completely preserved.During the decay the wave activates other nonlinear modes. The corresponding actions appear to be bounded uniformly in time, proportional to the square of the friction coefficient.We focus upon the interaction with the same soliton but opposite direction of propagation. Comparing the numerical observations with an analytical model we conclude that the activated wave is well described by a linear equation, inhomogeneously driven by the main wave. The main wave itself decays as a nonlinear damped oscillator with one degree of freedom.     

The effects of strong magnetic fields and rotation on soliton stars at finite temperature

We study the effects of strong magnetic fields and uniform rotation on the properties of soliton stars in Lee-Wick model when a temperature dependence is introduced into this model. We first recall the properties of the Lee-Wick model and study the properties of soliton solutions, in particular, the stability condition, in terms of the parameters of the model and in terms of the number of fermions N inside the soliton (for very large N) in the presence of strong magnetic fields and uniform rotation. We also calculate the effects of gravity on the stability properties of the soliton stars in the simple approximation of coupling the Newtonian gravitational field to the energy density inside the soliton, treating this as constant throughout. Following Cottingham and Vinh Mau, we also make an analysis at finite temperature and show the possibility of a phase transition which leads to a model with parameters similar to those considered by Lee and his colleagues but in the presence of magnetic fields and rotation. More specifically, the effects of magnetic fields and rotation on the soliton mass and transition temperature are computed explicitly. We finally study the evolution on these magnetized and rotating soliton stars with the temperature from the early universe to the present time.     

Peakon,pseudo‐peakon,cuspon and smooth solitons for a nonlocal Kerr‐like media

This paper presents all possible exact explicit peakon, pseudo‐peakon, cuspon and smooth solitary wave solutions for a nonlocal Kerr‐like media. We apply the method of dynamical systems to analyze the dynamical behavior of the traveling wave solutions and their bifurcations depending on the parameters of the system. We present peakon, pseudo‐peakon, cuspon soliton solution in an explicit form. We also have obtained smooth soliton. Mathematical analysis and numeric graphs are provided for those soliton solutions of the nonlocal Kerr‐like media. Copyright © 2016 John Wiley & Sons, Ltd.     

A Method for Solving the Hopf Equation and Generalization of Sobolev–Schwartz Distributions

We introduce distributions that are functionals with values in a nonarchimedian field of Laurent series. These objects naturally generalize Sobolev–Schwartz distributions, and we use them to find generalized solutions of the Hopf equation in the form of infinitely narrow solitons and shock waves. We propose a method for computation of the profile of an infinitely narrow soliton and of a shock wave described by the Hopf equation.     

引力规范理论中的一类引力波方程

该文给出了Vierbein表述的局域Lorentz群引力规范理论中的一类引力波方程。证明了Bondi平面波方程和引力孤立波方程均被该类方程所包含,这些方程的解均为该类方程在一定条件下的特解。因而这些解是与量子场论协调一致的。     

Higher-Order Solitons in the N-Wave System

The soliton dressing matrices for the higher-order zeros of the Riemann–Hilbert problem for the N -wave system are considered. For the elementary higher-order zero, that is, whose algebraic multiplicity is arbitrary but the geometric multiplicity is 1, the general soliton dressing matrix is derived. The theory is applied to the study of higher-order soliton solutions in the three-wave interaction model. The simplest higher-order soliton solution is presented. In the generic case, this solution describes the breakup of a higher-order pumping wave into two higher-order elementary waves, and the reverse process. In non-generic cases, this solution could describe (i) the merger of a pumping sech wave and an elementary sech wave into two elementary waves (one sech and the other one higher order); (ii) the breakup of a higher-order pumping wave into two elementary sech waves and one pumping sech wave; and the reverse processes. This solution could also reproduce fundamental soliton solutions as a special case.     

Lumps and their interaction solutions of a (2+1)-dimensional generalized potential Kadomtsev-Petviashvili equation

A (2+1)-dimensional generalized potential Kadomtsev-Petviashvili (gpKP) equation which possesses a Hirota bilinear form is constructed. The lump waves are derived by using a positive quadratic function solution. By combining an exponential function with a quadratic function, an interaction solution between a lump and a one-kink soliton is obtained. Furthermore, an interaction solution between a lump and a two-kink soliton is presented by mixing two exponential functions with a quadratic function. This type of lump wave just appears to a line $k_2x+k_3y+k_4t+k_5 \sim 0$. We call this kind of lump wave is a special rogue wave. Some visual figures are depicted to explain the propagation phenomena of these interaction solutions.     

Finite Energy Scattering for the Lorentz–Maxwell Equation

In the case where the charge of the particle is small compared to its mass, we describe the asymptotics of the Lorentz–Maxwell equation (Abraham model) for any finite-energy data. As time goes to infinity, we prove that the speed of the particle converges to a certain limit, whereas the electromagnetic field can be decomposed into a soliton plus a free solution of the Maxwell equation. It is the first instance of a scattering result for general finite energy data in a field-particle equation. Submitted: October 31, 2007. Accepted: April 21, 2008.     

Nonlinear transparency regimes for three-component acoustic pulses in a system of electron and nuclear spins

We study acoustic solitons consisting of one longitudinal and two transverse components and propagating in the direction perpendicular to an external magnetic field in a crystal containing paramagnetic impurities of electron and nuclear spins. The coupling of the electron spin subsystem to the longitudinal sound allows making the velocity of the latter close to that of the transverse acoustic waves, which provides an effective interaction between all components of the elastic field by means of the nuclear spin subsystem. We derive a three-component system of material and reduced wave equations describing this process and construct its soliton solutions in the form of stationary and breather pulses. Based on them, we study the peculiarities of the dynamics of the elastic field components and reveal the differences from the two-component model. The existence of two families of breathers is an important distinctive feature of the considered case.