Approximate Soliton Research Articles

Coexistence of a self-induced transparency soliton and a Bragg soliton.

We theoretically show that a self-induced transparency (SIT) soliton and a Bragg soliton can coexist in a nonlinear photonic band gap (PBG) medium doped uniformly with inhomogeneous-broadening two-level atoms. The Maxwell-Bloch equations for the pulse propagating through such a uniformly doped PBG structure are derived first and further reduced to an effective nonlinear Schrödinger equation. This model describes an equivalent physical mechanism for a Bragg-soliton propagation resulting from the effective quadratic dispersion balancing with the effective third-order nonlinearity. Because the resonant atoms are taken into account, the original band gap can be shifted both by the dopants and the instantaneous nonlinearity response originating from an intense optical pulse. As a result, even if a SIT soliton with its central frequency deep inside the original forbidden band, it still can propagate through the resonant PBG medium as long as this SIT soliton satisfies the effective Bragg-soliton propagation. An approximate soliton solution describing such coexistence is found. We also show that the pulse width and group velocity of this soliton solution can be uniquely determined for given material parameters, atomic transition frequency, and input central frequency of the soliton. The numerical examples of the SIT soliton in a one-dimensional As2S3-based PBG structure doped uniformly with Lorentzian line-shape resonant atoms are shown. It is found that a SIT soliton with approximately 100-ps width in such a resonant PBG structure can travel with the velocity being two orders of magnitude slower than the light speed in an unprocessed host medium.

Approximate Stationary Solitons of the Fifth Order Singularly Perturbed KdV-Type Equation

Stationary solitons of the fifth order singularly perturbed KdV-type equation are investigated. The second order approximate solution to fifth order singularly perturbed KdV-type equation and third...

On The Pereira-Stenflo Solitons

A variational method is used to derive stationary solutions of the generalized nonlinear Schrödinger equation with complex coefficients that describes growth and damping which has been introduced and solved in the 1 + 1 dimensional case by Pereira and Stenflo (1977 Phys. Fluids 20 1733). In the 1+1 dimensional case, the exact classical Pereira-Stenflo soliton solution is reproduced. Application to the 1 + 2 dimensional case with cylindrical symmetry, provides an explicit approximate single soliton solution which is named the cylindrical Pereira-Stenflo soliton.

Evolution of solitons of nonlinear Schrödinger equation with variable parameters

The nonlinear Schrodinger equation with variable parameters is solved by means of variational technique. A set of evolution equations for the solitary-wave solution is derived. The propagation properties of the solitons in an adiabatic amplification system and in a dispersion-decreasing fiber are analyzed. An explicit analytical approximate soliton solution in the exponentially dispersion-decreasing fiber is obtained using the derived dynamical equations.

Kink chains from instantons on a torus

We describe how the procedure of calculating approximate solitons from instanton holonomies may be extended to the case of soliton crystals. It is shown how sine-Gordon kink chains may be obtained from CP1 instantons on T2. These kink chains turn out to be remarkably accurate approximations to the true solutions. Some remarks on the relevance of this work to Skyrme crystals are also made.

Excitation of Josephson junction by an electromagnetic field in the nonlinear process formalism

The numerical analysis of the excitation of the Josephson junction (Jj) by an external electromagnetic field has been reported. The fully nonlinear approach requires using the quasiperiodic solutions of the sine-Gordon equation (sGe) which are expressed by the Riemann theta functions. Their local soliton approximation of the nondissipative Jj is presented. Using the self-consistent phenomenological model of a long Jj, the influence of the external electromagnetic field on the volt-ampere characteristics is considered.

Soliton model of electron transfer in mixed-valence dimers and trimers

The soliton approximation is examined for mixed-valence dimers and trimers with one migrating electron. The calculations use Hamilton's canonical equations and a time-dependent wave function of the soliton type. In the two limiting cases (strong and weak vibronic coupling), the soliton approximation correlates well with the exact analytical solution. As in the adiabatic approximation at g2/v=1, the delocalized limit (quasifree electron) is transformed to the localized one (electron locked on the center). For molecular trimers, the soliton approximation allows us to reveal the cases where the transfer of an electron from the first to the third center is accompanied by electron density concentration on the intermediate center.

Nonlinear approach to the excitation of a Josephson junction by an electromagnetic field

The excitation of a Josephson junction ( J j) by an external electromagnetic field has been reported. The proposed numerical model has been obtained by considering the class of quasi-periodic solutions of the unperturbed sine-Gordon equation (sGe) and their soliton approximation. The five-phase sGe solution contains bouncing fluxons, bias and the breather-type phase which modulate the electromagnetic field on the edge of the junction. Using a self-consistent phenomenological model of the long, nondissipative J j, the influence of the external electromagnetic field to volt-ampere ( I–V) characteristics is discussed.

Solitons on lattices

We examine a variety of numerical and approximate analytical methods to study families of solitary waves on lattices. Such waves, when they exist, travel through the lattice without loss of energy, and have approximate soliton properties on collision. Corresponding quantum problems are also briefly described.

PHENOMENOLOGICAL IMPLICATIONS OF HIGH-Tc SUPERCONDUCTIVITY

We develop a phenomenological model for high-T c superconductors. Some main features are emerging in copper oxides: characteristic quasi two-dimensional Cu-O planes, strong correlation of antiferromagnetism, existence of a vortex lattice structure, and observation of a small coherence length and a large penetration depth. These features indicate that the superconductive pair is reasonably constrained to a small volume in real space and may be conceived of as a string-carrying vortex, and therefore can be well simulated by the dual phenomenological local boson fields [Formula: see text] and Φ. The various mean-field ground states of the system are discussed. The field equations of motion are originally solved to get approximate analytical soliton solutions. The effective Hamiltonian is formulated by a variational method for finite temperatures. The model parameter behaviour described by the relationship of the variational parameters is investigated. We discuss the critical temperature T c , the specific heat cV and its jump Δc at T c , and the critical magnetic fields H c1 and H c2 . These results are in agreement with experimental observations, especially the critical behaviours and the zero temperature values. The model also allows interpretation of the variation of T c with oxygen vacancy x and that with doping fraction δ in Cu-O planes, as well as the dependence of γ (defined as the specific heat coefficient of the T-linear term) on δ.

Femtosecond soliton propagation in fibers with slowly decreasing dispersion

We obtain the adiabatic compression of femtosecond solitons in fibers with slowly decreasing dispersion (FSDD’s) and discover the effect of soliton pulse-width stabilization in FSDD’s. A theoretical model for the description of femtosecond-pulse propagation in FSDD’s is presented. The model takes into account higher-order nonlinear and dispersive effects in fibers. We present numerical simulations of a generalized nonlinear Schrodinger equation and the adiabatic soliton approximation theory as well. Good quantitative agreement between the theory and the experiments is seen. The effect of soliton pulse-width stabilization in FSDD’s is described as a result of the combined action of the third-order dispersion and Raman self-scattering effects. We also show the possibility of obtaining high-quality pulses with less than a 15-fs pulse width in the 1.6-μm spectral region by soliton compression in FSDD’s.

Approximate solution of optical soliton in lossless fibres with third-order dispersion

A near zero-dispersion regime, fundamental soliton with third-order dispersion is considered. There are three different waves radiated from the soliton. We find a threshold condition for the radiated waves. Below the threshold condition a stable soliton exists; an approximate soliton solution is obtained.

Momentum projection of the nucleon in the colour-dielectric model

We use the method of Lübeck et al. to project the mean-field solutions of the colour-dielectric model onto momentum eigenstates, so removing the spurious c.m. motion. A coherent state is used to provide a quantum description of the colour-dielectric field. In addition the nucleon mass is varied after projection. A range of parameter sets is considered, but we find that this technique is only appropriate to those with a simple quadratic potential. These results show a significant improvement over the mean-field approximation, the product of the proton mass and r.m.s. radius being within 30% of its observed value. Projection of solutions for a double-minimum potential results in a mass-radius project which is an order of magnitude too small, due to the high mean-square momentum in the soliton approximation. We conclude that the MFA is not a good starting point for parameter sets of this kind.

Phenomenological electrodynamics of the Josephson junction

Starting from the self-consistent phenomenological model of the “long” Josephson junction, the I–V curve for the first zero field steps can be determined also in the case of a junction without dissipation. This is achieved considering the solutions of the nonperturbed sine-Gordon equation (sGe) which contain the static and dynamic phases simultaneously. Such approach gives rise to the unified description of dc and/or ac processes as following from the same solution of sGe and thus reflecting the real physical situation.The considered solutions are expressed by the Riemann theta functions, which in this problem cannot be reduced to the elliptic functions, but for simplicity, the results presented here were obtained by means of their local soliton approximation. The different aspects of the proposed model, importance of the periodicity in time for the correct determination of observable quantities and the type of the boundary conditions, other than those adopted in the literature, are discussed also. At last follows an application of the open system theory which is sketched in the appendix.

Approximate soliton solutions around an exact soliton solution of the toda lattice equation

It is shown both analytically and numerically that the Toda lattice equation exp [-(u n-u n-1)]- exp[-(u n+1-u n)] - u ̈ n = 0 exhibits approximate one- or multi-soliton solutions exp [-( u n - u n-1 )] -1 = p(d 2/d t 2)ln( f n ) with p = arbitrary constant #62; 0 corresponding to the exact one with p = 1 for soliton wave number at least up to 1.0. Such a flexibility of soliton amplitude for given soliton frequency and wave number implies that the Toda lattice shares some of its properties with the linear lattice, though it is highly nonlinear.

Continuum Version for the PPP Model of Polyacetylene and Other Half-Filled Peierls Systems

A continuum version is developed for the Pariser-Parr-Pople (PPP) model of polyacetylene which includes both the on-site and different site Coulomb interactions. A continuum SCF equation for π electronic and lattice structures is obtained in the approximation to take into account the terms up to O(a2) in the expansion by the lattice constant a. The SCF equation contains four gap parameters, spin dependent SCF potential and Fermi velocity which are slowly varying functions of position and depend on the Coulomb interaction. The weak coupling limit approximation which includes only the terms up to O(a) is insufficient both qualitatively and quantitatively. When the lowest correction to the weak coupling limit is taken into account, the average bond order, the effective Fermi velocity and the effective coupling constants are renormalized to have reasonable values. In this approximation, however, only the SCF solutions with one non-vanishing gap parameter are possible. Inclusion of further correction term gives the SCF solutions with coexistence of two non-vanishing gap parameters. Solitons in the PPP model are not pure amplitude solitons but amplitude-phase solitons. A soliton in the PPP model can be approximated as a phase soliton when the long range part of the Coulomb interaction is screened. The analytical expressions for neutral and charged solitons are obtained in the phase soliton approximation. The solitons connecting the SCF states with two gap parameters can have a fractional charge or a fractional spin.

Reflection dissipation of an ion-acoustic soliton

The dynamic interaction between an ion-acoustic soliton and adiabatically reflected particles is investigated. A perturbed Korteweg–de Vries (KdV) equation for the wave electric potential is derived from the Vlasov–Poisson equations and solved in the adiabatic soliton approximation. The soliton attenuation/amplification rate is found to be proportional to χi‖Φm‖1/2, where Φm is the initial soliton amplitude, and χi is the imaginary part of the linear susceptibility of ions/electrons when the wave is compressive/rarefactive mode. The amplitude variation is caused by momentum exchange between the soliton and reflected particles, as viewed from the soliton reference frame.

Higgs-type Lagrangians and vortex-line and soliton solutions

Approximate vortex-line and soliton solutions of the Higgs-type Lagrangian are obtained analytically, and the magnetic field, magnetic energy and its root mean square radius, Higgs energy, and interaction energy between the vector field A/sub ..mu../ and the Higgs scalar phi are calculated. The soliton solution for another case is commented on. (auth)