Modified sine-Gordon Equation Research Articles

Excitation of nonlinear solitary flexural waves in a moving domain wall

Numerical methods have been used to study the dynamics of a 180 °C domain wall described by a modified sine-Gordon equation, in an infinite ferromagnet with a weak ferromagnetism. The dynamics of solitary flexural waves, which appear upon the intersection of a domain wall with a local region of material with magnetic-anisotropy parameters differing from those of the basic volume of the ferromagnet, has been investigated. The case where the width of the domain wall and the size that characterizes the inhomogeneity of the magnetic-anisotropy parameter \( \tilde K \) are of the same order is considered. It is shown that a solitary flexural wave can be considered as one of the “kink-on-kink” solutions of the sine-Gordon equation.

Existence and stability analysis of finite0−π−0Josephson junctions

We investigate analytically and numerically a Josephson junction on finite domain with two $\pi$-discontinuity points characterized by a jump of $\pi$ in the phase difference of the junction, i.e. a 0-$\pi$-0 Josephson junction. The system is described by a modified sine-Gordon equation. We show that there is an instability region in which semifluxons will be spontaneously generated. Using a Hamiltonian energy characterization, it is shown how the existence of static semifluxons depends on the length of the junction, the facet length, and the applied bias current. The critical eigenvalue of the semifluxons is discussed as well. Numerical simulations are presented, accompanying our analytical results.

Fluxons interactions in a Josephson junction with a phase-shift

We consider a long Josephson junction with a discontinuity point characterized by a gauge phase-shift. The system is described by a modified sine-Gordon equation. We study, in particular, the interactions between a fluxon and a fractional fluxon. A perturbation theory is developed in the small phase-shift limit to understand the characteristics of the interaction. Finally, numerical computations of the threshold bias current and the threshold velocity for a fluxon running over a fractional fluxon are presented.

Flux-flow dynamics in a long Josephson junction with nonuniform bias current

The flux-flow dynamics in a long Josephson junction is investigated for a spatiallynonuniform bias current density. Starting from the linearization about a rotatingbackground, a simple approximate solution of the modified sine–Gordon equation ispresented, making possible to derive the current–voltage characteristic of the junction. Asan example of nonuniform driving, an overlap structure with an unbiased tail is discussed.In particular, it is shown how a nonuniform distribution of the bias current density mayaffect the dynamical resistance of a flux-flow oscillator. Analytical results are comparedwith direct numerical simulations and good agreement is found for a wide range of junctiondimensions.

Dynamics of ultrashort optical solitons in a system of nonsymmetric quantum objects

This paper theoretically discusses the propagation of an ultrashort laser pulse consisting of ordinary and extraordinary components in a system of anisotropic quantum centers. The treatment does not use the assumption that the density of quantum centers is small and makes no limitations on the transition dipole moments. It is shown that, if the spectral pulse width is much greater than the frequency of the quantum transitions, the dynamics of its ordinary component obeys a modified sine-Gordon equation that belongs to the class of integrable models. Soliton solutions of the given equation are constructed, and they are classified. Depending on the duration of the soliton, solutions are possible in the form of a 2π pulses (kinks and antikinks) or 0π steady-state traveling pulses (neutral kinks), which have no counterpart in the isotropic case. The behavior of the medium is studied when various types of solitons are transmitted.

Integrable models of the dynamics of longitudinal-transverse acoustic pulses in a paramagnetic crystal

We study the interaction between longitudinal-transverse acoustic pulses and a system of paramagnetic impurities with the effective spin S = 1 in a statically deformed crystal. We show that the dynamics of a pulse propagating at an arbitrary angle to the static-deformation direction and of the effective spins satisfy the modified reduced Maxwell-Bloch equations and, if the spectrum of the acoustic pulse overlaps the quantum transitions between spin sublevels, the modified sine-Gordon equation. These equations generalize the well-known models in the theory of the inverse scattering method and in the theory of self-induced transparency and also belong to the class of integrable equations. Analyzing soliton solutions shows that the pulse-medium interaction reveals some qualitatively new features in these models compared with the cases of purely transverse or purely longitudinal acoustic fields.

Soliton regimes of ultrashort pulse propagation through an array of asymmetric quantum objects

Nonlinear regimes of two-component ultrashort pulse propagation through an array of anisotropic quantum systems are analyzed theoretically under conditions when the slowly varying envelope approximation is inapplicable. It is shown that if the spectral width of the pulse is much larger than the quantum transition frequency, then the dynamics of its ordinary component can be described by a modified sine-Gordon equation that belongs to the class of integrable models but has never been used in physical applications. It is shown that the extraordinary pulse component responsible for the dynamic Stark shift of the transition frequency is coupled to its ordinary component by an algebraic relation. The soliton solutions to the derived equation are classified. Depending on the pulse duration, the solutions have the form of 2π pulses (kinks and antikinks) or steady traveling 0π pulses having no counterparts in isotropic models (neutral kinks). Interactions between the medium and solitons of different types and soliton collisions are analyzed.

New class of extremely short electromagnetic solitons

A physical realization is presented for the integrable modified sine-Gordon equation. In this case, this equation describes the propagation of an extremely short vector electromagnetic pulse in the system of asymmetric quantum objects that have permanent dipole moments in the self-energy states. Using soliton solutions for ordinary and extraordinary components, new regimes of the dynamics of the pulse and medium that are specific only to asymmetric media have been found.

Fluxon dynamics in an exponentially shaped Josephson junction

Dynamical properties of an exponentially shaped Josephson junction are investigated theoretically using a simple one-dimensional model described by a modified sine-Gordon equation. An approximate analytical solution is based on the linearization about a rotating background and consists of a flux-flow term accompanied by two oppositely directed plasma waves. Analytical results are compared with direct numerical simulations both for the magnetic field distribution within the junction and for the current-voltage characteristics.

Nonlinear sigma model for a condensate composed of fermionic atoms

A nonlinear sigma model is derived for the time development of a Bose–Einstein condensate composed of fermionic atoms. Spontaneous symmetry breaking of a Sp ( 2 ) symmetry in a coherent state path integral with anticommuting fields yields Goldstone bosons in a Sp ( 2 ) ⧹ U ( 2 ) coset space. After a Hubbard–Stratonovich transformation from the anticommuting fields to a local self-energy matrix with anomalous terms, the assumed short-ranged attractive interaction reduces this symmetry to a SO ( 4 ) ⧹ U ( 2 ) coset space with only one complex Goldstone field for the singlet pairs of fermions. This bosonic field for the anomalous term of fermions is separated in a gradient expansion from the density terms. The U ( 2 ) invariant density terms are considered as a background field or unchanged interacting Fermi sea in the spontaneous symmetry breaking of the SO ( 4 ) invariant action and appear as coefficients of correlation functions in the nonlinear sigma model for the Goldstone boson. The time development of the condensate composed of fermionic atoms results in a modified Sine–Gordon equation.

Unstable excited and stable oscillating gap 2π pulses

Dynamics of the gap 2π pulse dynamics in one-dimensional resonantly absorbing Bragg gratings are studied. A new family of stable oscillating and excited unstable gap 2π pulses is analytically and numerically described by a transition from the two-wave Maxwell–Bloch equation to the modified sine-Gordon equation and by direct integration of the two-wave Maxwell–Bloch equation.

The parametrically modified sine-Gordon equation: spectral properties, collisions, resonances and their connection to integrability

In this work, we revisit kinks and their interactions in the parametrically modified sine-Gordon (pmsG) equation. The central motivating question of our work is the following: what is the mechanism which in the absence of internal modes generates phonon radiation and causes the inelasticity of collisions in a near-integrable system? We do not give a complete answer to this question but rather we study a case example where we examine the penetration of an internal mode into the phonon band, and identify its consequences in the spectrum, the shape of the eigenfunctions and resonances of the spectrum with an external ac drive. In particular, by studying the linear spectrum around the kink solution we show that in the absence of the internal mode, some odd phonon modes can act as a localized wavepacket of modes depending on the parameter r of the potential. We also show by using the phenomenological theory developed in [Physica D 9 (1983) 1] that these modes are related to the quasi-resonance process and bound states suggested in [Physica D 9 (1983) 33], hence during the collision process the energy of the translational modes goes directly to the “localized” phonons. Finally, we investigate another problem for which these “localized” phonon modes also act as internal modes do. In particular, we study the resonance phenomena (related with either the internal mode or with the phonon modes) that appear in this system when it is subject to external ac drive. We attempt to connect such properties of the phonons (as well as properties of the internal modes) to the non-integrability of the model for r≠0.

Soliton shape stabilization in a superlattice with next-to-nearest neighbor spectrum in a field of a nonlinear wave

The effect of an electric field of a nonlinear (cnoidal) electromagnetic wave (pumping field) upon the shape of the solitary electromagnetic wave (soliton) in the quantum semiconductor superlattice with two harmonics of the electron spectrum is studied. Propagation of electromagnetic waves is shown in this case to be described by the modified double sine-Gordon equation. The possibility of the amplification of the pulse and its transformation into the dissipative soliton is noticed. The speed and width of a soliton depend on the presence of the second harmonic in the superlattice electron energy spectrum. The dependence of the dissipative soliton parameters on temperature and pumping field amplitude is also noticed. The possibility of propagation of electromagnetic waves described by solutions of the modified sine-Gordon equation in the superlattice with the spectrum under study is found.

Effect of the field of a nonlinear electromagnetic wave on the shape of a soliton in a semiconductor superlattice

The effect of the electric component of the field of a high-frequency (HF) nonlinear electromagnetic (EM) wave on the propagation of a solitary EM wave (soliton) in a quantum semiconductor superlattice is studied. It is noted that in the collisionless approximation, the solution of the modified sine-Gordon equation corresponding to the amplification of an EM pulse that, with allowance made for interminiband electron transitions, transforms into a dissipative soliton is possible. The effect of electron collisions with irregularities of the crystal lattice on the soliton dynamics under the action of the field of a HF nonlinear wave is considered. The condition for an increase in the traveling time of the solitary wave is found.

Topological defects with long-range interactions

We investigate a modified sine-Gordon equation which possesses soliton solutions with long-range interaction. We introduce a generalized version of the Ginzburg-Landau equation which supports long-range topological defects in D = 1 and D > 1. The interaction force between the defects decays so slowly that it is possible to enter the non-extensivity regime. These results can be applied to non-equilibrium systems, pattern formation and growth models.

Effect of Core Impurity on Kink Soliton Motion in Dislocation Line

Modified Sine-Gordon equation is established to investigate the kink soliton motion affected by core impurities. The dragging force that the core impurities exert on the moving kink soliton is determined by computer simulation of diffusing of solute atoms in dislocation core .The soliton motion can be dissolves into two kinds of movement : the changing of soliton position and soliton shape. These two kinds of movements correspond to the longitude core diffusion(LCD) and transverse core diffusion(TCD) of solute atoms respectively . We solve the equation of these motions numerically, temperature internal friction peak and stress amplitude internal friction peak are discussed. The results are compared with the anomalous internal friction peaks in Al-Mg,Al-Cu alloy.

Size effect of inner radius on circularly symmetric soliton states in annular Josephson junctions

The effect of the inner radius of an annular Josephson junetion on the excitation of circularly symmetric soliton states is explored by direct numerical simulation. In a circularly symmetric annular Josephson junction (CSAJJ hereafter), the phase difference in the order parameters Φ across the junction barrier obeys the modified sine-Gordon equation (SGE) in which there is a perturbation term Φp/ρ inversely proportional to the radial coordinate ρ, and this term does not appear in the case of a 1D in-line junction. The effect of the Φp/ρ term becomes prominent when ρ is small. It turns out that for junctions of given dissipation and annular width, there is a lower limit of the inner radii for the junctions to support stable soliton motion, while the lower limit depends strongly on the dissipation coefficient α. For example, for a CSAJJ with inner radius 5λJ, outer radius 10λJ, no stable soliton states would exist when α=0.01 or α=0.02. However, for 0.06 ≥ α ≥ 0.03, there are ring-shaped solitary wave solutions, presumably because the dissipation suppresses the effect of the Φp/ρ term. Finally, the behaviour of the soliton states and the relevant I-V characteristics are also disscussed.

Perturbation analysis of a parametrically changed sine-Gordon equation.

A long Josephson junction with a spatially varying inductance is a physical manifestation of a modified sine-Gordon equation with parametric perturbation. Soliton propagation in such Josephson junctions is discussed. First, for an adiabatic model where the inductance changes smoothly compared with soliton size, transmission or reflection of the soliton is described using a simple energy analysis. Next, the soliton propagation is solved on the basis of a perturbation theory constructed by McLaughlin and Scott. Radiation as well as soliton trajectories are presented numerically. Agreement between such solutions and the results of direct numerical integration by means of a finite-difference method is excellent.

Quasiperiodicity in Long RF-Biased Josephson Junctions

We present and discuss numerical simulations, using a modified sine-Gordon equation, of long Josephson junctions in a magnetic field and in the presence of rf radiation. Quasiperiodicity arises from the coexistence of the frequency of the driving force and the natural frequency of an oscillating spatial structure of soliton-like character. The quasiperiodic regime presents subharmonic response at either of those frequencies.

Effect of surface losses on soliton propagation in Josephson junctions

We have explored numerically the effects on soliton propagation of a third order damping term in the modified sine-Gordon equation. In Josephson tunnel junctions such a term corresponds physically to quasiparticle losses within the metal electrodes of the junction. We find that this loss term plays the dominant role in determining the shape and stability of the soliton at high velocity.