Continuous Soliton Research Articles

REACTION-DIFFUSION FISHER’S EQUATIONS VIA DECOMPOSITION METHOD

The effect of the source, initial or boundary conditions in the use of Adomian decomposition method (ADM) on nonlinear partial differential equation or nonlinear equation in general is enormous. Sometimes the equation in question result to continuous exact solution in series form, other times it result to discrete approximate analytical solutions. In this paper, we show that continuous exact solitons can be obtained on application of ADM to the Fisher's equation with the deployment Taylor theorem to the terms(s) in question. And, the resulting series is split into the integral equations during the solution process. Resulting to multivariate Taylor's series of the exact solitons with the help of Adomian polynomials of the nonlinear reaction term correctly calculated. More physical results are further depicted in 2D, 3D and contour plots.

Matter-wave solitons in an array of spin-orbit-coupled Bose-Einstein condensates.

We investigate matter-wave solitons in a binary Bose-Einstein condensate (BEC) with spin-orbit (SO) coupling, loaded in a one-dimensional (1D) deep optical lattice and a three-dimensional anisotropic magnetic trap, which creates an array of elongated sub-BECs with transverse tunneling. We show that the system supports 1D continuous and discrete solitons localized in the longitudinal (along the array) and the transverse (across the array) directions, respectively. In addition, such solitons are always unpolarized in the zero-momentum state but polarized in finite-momentum states. We also show that the system supports stable two-dimensional semidiscrete solitons, including single- and multiple-peaked ones, localized in both the longitudinal and transverse directions. Stability diagrams for single-peaked semidiscrete solitons in different parameter spaces are identified. The results reported here are beneficial not only for understanding the physical property of SO-coupled BECs but also for generating new types of matter-wave solitons.

Spontaneous symmetry breaking and ghost states in two-dimensional fractional nonlinear media with non-Hermitian potential

The interaction of fractional diffraction and parity-time ({{{{{mathcal{PT}}}}}}) symmetric would bring some unique properties to certain physical system. Here we report a spontaneous symmetry breaking (SSB) phenomenon and ghost states of solitons supported by the two-dimensional (2D) fractional nonlinear Schrödinger (FNLS) equation with focusing and defocusing Kerr media under 2D non-Hermitian {{{{{mathcal{PT}}}}}}-symmetric potential. The continuous asymmetric soliton bifurcates out in a SSB way. For the focusing regime, as the power exceeds a critical value, the asymmetric soliton bifurcates from fundamental symmetric solitons, and when it turns to the defocusing regime, the symmetry breaking phenomenon is found in the branch of the first excited state. The symmetric solitons (including fundamental one and first excited state) are destabilized at this point. Intriguingly, the branches of asymmetry solitons are existing with complex conjugate propagation constants (alias ghost states), which is solely in fractional media. Moreover, we investigate the dependence of fractional Lévy index (α) on the symmetry breaking of solitons in detail. And the stabilities of fundamental symmetric soliton, asymmetric solitons, as well as the first excited states are explored. Moreover, The management of optical field propagation is achieved by modulation of the external potential. Meanwhile, we find the stable excitations of solitons via adiabatic excitations of system parameters. These results will provide some theoretical basis for the study of SSB phenomena and related physical experiments in the 2D fractional media with {{{{{mathcal{PT}}}}}}-symmetric potentials.

Controlling superfluid flows using dissipative impurities

We propose and analyze a protocol to create and control the superfluid flow in a one dimensional, weakly interacting Bose gas by noisy point contacts. Considering first a single contact in a static or moving condensate, we identify three different dynamical regimes: I. a linear response regime, where the noise induces a coherent flow in proportion to the strength of the noise, II. a Zeno regime with suppressed currents, and III. a regime of continuous soliton emission. Generalizing to two point contacts in a condensate at rest we show that noise tuning can be employed to control or stabilize the superfluid transport of particles along the segment which connects them.

Spontaneous symmetry breaking and ghost states supported by the fractional PT-symmetric saturable nonlinear Schrödinger equation.

We report a novel spontaneous symmetry breaking phenomenon and ghost states existed in the framework of the fractional nonlinear Schrödinger equation with focusing saturable nonlinearity and PT-symmetric potential. The continuous asymmetric soliton branch bifurcates from the fundamental symmetric one as the power exceeds some critical value. Intriguingly, the symmetry of fundamental solitons is broken into two branches of asymmetry solitons (alias ghost states) with complex conjugate propagation constants, which is solely in fractional media. Besides, the dipole and tripole solitons (i.e., first and second excited states) are also studied numerically. Moreover, we analyze the influences of fractional Lévy index ( α) and saturable nonlinear parameters (S) on the symmetry breaking of solitons in detail. The stability of fundamental symmetric soliton, asymmetric, dipole, and tripole solitons is explored via the linear stability analysis and direct propagations. Moreover, we explore the elastic/semi-elastic collision phenomena between symmetric and asymmetric solitons. Meanwhile, we find the stable excitations from the fractional diffraction with saturation nonlinearity to integer-order diffraction with Kerr nonlinearity via the adiabatic excitations of parameters. These results will provide some theoretical basis for the study of spontaneous symmetry breaking phenomena and related physical experiments in the fractional media with PT-symmetric potentials.

Nonlinear Schrödinger equationswith amplitude-dependent Wadati potentials.

Complex Wadati-type potentials of the form V(x)=-w^{2}(x)+iw_{x}(x), where w(x) is a real-valued function, are known to possess a number of intriguing features, unusual for generic non-Hermitian potentials. In the present paper, we introduce a class of nonlinear Schrödinger-type problems which generalize the Wadati potentials by assuming that the base function w(x) depends not only on the transverse spatial coordinate, but also on the amplitude of the field. Several examples of prospective physical relevance are discussed, including models with the nonlinear dispersion or with the derivative nonlinearity. The numerical study indicates that the generalized model inherits the remarkable features of standard Wadati potentials, such as the existence of continuous soliton families, the possibility of symmetry-breaking bifurcations when the model obeys the parity-time symmetry, the existence of constant-amplitude waves, and the eigenvalue quartets in the linear-instability spectra. Our results deepen the current understanding of the interplay between nonlinearity and non-Hermiticity and expand the class of systems which enjoy the exceptional combination of properties unusual for generic dissipative nonlinear models.

Continuous time soliton resolution for two-bubble equivariant wave maps

Continuous time soliton resolution for two-bubble equivariant wave maps

Solitons in Kerr media with two-dimensional non-parity-time-symmetric complex potentials

Our results indicate that continuous soliton families can exist and be stable in Kerr media with two-dimensional (2D) non-parity-time (PT)-symmetric complex potentials. There are several discrete eigenvalues in the linear spectra of these complex potentials. Fundamental solitons bifurcate from the largest discrete eigenvalue while the dipole solitons bifurcate from the second- or third- largest discrete eigenvalue. We further find that eigenvalues of the soliton linear-stability spectra emerge as complex conjugate pairs. The effect of different parameters of the complex potentials on soliton stability is discussed in detail. Moreover, we study the transverse energy flow vector of the solitons in these complex potentials.

Generation of soliton rain in a passive mode-locked Tm-doped fiber laser at 2 µm

In this paper, the experimental investigation and theoretical analysis on the soliton rain generated in the Tm-doped fiber laser have been carried out. On the basis of nonlinear polarization rotation technology, the fundamental and harmonic soliton rain pulses were obtained, the influence of net negative dispersion cavity length on the output performances of soliton rain pulse were studied, and the spectral sideband characteristics of soliton rain were numerically analyzed. By comparing the spectra of continuous wave mode-locking and soliton rain states, it can be found that the quasi-periodic modulation exists for the spectral sideband of soliton rain resulted from the complex interaction between the dispersive wave and the soliton wave. The interaction is one of the major factors for the formation of soliton rain. The research results are helpful for the better understanding of the soliton rain dynamics and mechanism in passive mode-locked fiber lasers.

Wavelength conversion through soliton self-frequency shift in tellurite microstructured fiber with picosecond pump pulse

Wavelength conversion to the wavelength range that is not covered by commercially available lasers could be accomplished through the soliton self-frequency shift (SSFS) effect. In this study, the phenomenon of SSFS pumped by a picosecond-order pulse in a tellurite microstructured fiber is investigated both theoretically and experimentally. The balance between the dispersion and the nonlinearity achieved by a 1958 nm pump laser induces a distinct SSFS effect. Attributed to the large spectral distance between the pump pulse and the fiber zero-dispersion wavelength, the SSFS is not cancelled due to energy shedding from the soliton to the dispersive wave. Details about the physical mechanisms behind this phenomenon and the variations of the wavelength shift, the conversion efficiency are revealed based on numerical simulations. Owing to the large soliton number N, the pulse width of the first split fundamental soliton is approximately 40 fs, producing a pulse compression factor of ∼38, much higher than that pumped by a femtosecond pulse. Experiments were also conducted to confirm the validity of the simulation results. By varying the pump power, a continuous soliton shift from 1990 nm to 2264 nm was generated. The generation of SSFS in tellurite microstructured fibers with picosecond pump pulse can provide a new approach for wavelength conversion in the mid-infrared range and could be useful in medical and some other areas.

Lossless polariton solitons

Photons and excitons in a semiconductor microcavity interact to form exciton-polariton condensates. These are governed by a nonlinear quantum-mechanical system involving exciton and photon wavefunctions. We calculate all non-traveling harmonic soliton solutions for the one-dimensional lossless system. There are two frequency bands of bright solitons when the inter-exciton interactions produce an attractive nonlinearity and two frequency bands of dark solitons when the nonlinearity is repulsive. In addition, there are two frequency bands for which the exciton wavefunction is discontinuous at its symmetry point, where it undergoes a phase jump of π. A band of continuous dark solitons merges with a band of discontinuous dark solitons, forming a larger band over which the soliton far-field amplitude varies from 0 to ∞; the discontinuity is initiated when the operating frequency exceeds the free exciton frequency. The far fields of the solitons in the lowest and highest frequency bands (one discontinuous and one continuous dark) are linearly unstable, whereas the other four bands have linearly stable far fields, including the merged band of dark solitons.

Continuous Solitons in a Lattice Nonlinearity.

We study theoretically and experimentally the propagation of optical solitons in a lattice nonlinearity, a periodic pattern that both affects and is strongly affected by the wave. Observations are carried out using spatial photorefractive solitons in a volume microstructured crystal with a built-in oscillating low-frequency dielectric constant. The pattern causes an oscillating electro-optic response that induces a periodic optical nonlinearity. On-axis results in potassium-lithium-tantalate-niobate indicate the appearance of effective continuous saturated-Kerr solitons, where all spatial traces of the lattice vanish, independently of the ratio between beam width and lattice constant. Decoupling the lattice nonlinearity allows the detection of discrete delocalized and localized light distributions, demonstrating that the continuous solitons form out of the combined compensation of diffraction and of the underlying periodic volume pattern.

Continuous and discontinuous dark solitons in polariton condensates

Bose-Einstein condensates of exciton-polaritons are described by a Schr\"odinger system of two equations. Nonlinearity due to exciton interactions gives rise to a frequency band of dark soliton solutions, which are found analytically for the lossless zero-velocity case. The soliton's far-field value varies from zero to infinity as the operating frequency varies across the band. For positive detuning (photon frequency higher than exciton frequency), the exciton wavefunction becomes discontinuous when the operating frequency exceeds the exciton frequency. This phenomenon lies outside the parameter regime of validity of the Gross-Pitaevskii (GP) model. Within its regime of validity, we give a derivation of a single-mode GP model from the initial Schr\"odinger system and compare the continuous polariton solitons and GP solitons using the healing length notion.

Optical Amplification of Tweezers and Bright Soliton Using an Interferometer Ring Resonator System

System of optical soliton amplification which consists of an add/drop multiplexer system is presented. Optical dark and bright soliton pulses can be amplified significantly. Dark soliton and Gaussian laser beam are input into the configured system and travel within the waveguide. Continuous soliton pulse will be sliced into smaller pulses by the nonlinear effect known as chaos, thus large bandwidth of optical signals can be obtained. Amplification occurs when the soliton propagates along the add/drop multiplexer system. Amplified optical tweezers signals with power of 487 W and FWHM of 6.8 nm are simulated. The amplified bright soliton could be obtained with maximum power of 360 W and FWHM of 1.5 nm. Therefore amplification of soliton pulses can be used to perform the long distance optical communication.

Simulation Of Soliton Amplification In Micro Ring Resonator For Optical Communication

A system consisting of a series of micro ring resonator (MRR) is proposed. Optical dark and bright soliton pulses propagating through the nonlinear waveguides are amplified. This system can be used in long distance communication system. The dark and bright soliton is input into the designed system. The nonlinear effect contributes to segregation of continuous soliton pulse into smaller pulses. In this way large bandwidth of optical signals can be obtained. The power amplification occurs when the soliton propagates along the MRRs systems. In this research the concern is the generation of amplified pulse of optical dark and bright soliton while propagating in the MRR device. Simulated results show the amplification of bright soliton in which the input power increases from 0.6 W to 10.9331 W and 7.684 W at the trapped wavelength of 1520.428 nm and 1519.912 nm respectively. Key words: Optical soliton; dark soliton; bright soliton; soliton amplification

Soliton interactions in some semidiscrete integrable systems

In this paper, soliton interactions in three semidiscrete integrable systems are studied. It will be shown that in some cases the solitary waves of the semidiscrete integrable systems can interact elastically, i.e., the shapes, the amplitudes, and the velocities of waves preserve after undergoing collision. However, soliton interactions in the semidiscrete integrable systems possess different properties from the continuous solitons, for example, the shorter waves can travel faster than the taller ones. We also find that two stable solitary waves can be destroyed after interaction, and the elastic properties of collisions between two solitary waves are sensitively dependent on the choices of some parameters in solution. We therefore conclude that soliton interactions in semidiscrete integrable systems are richer than the ones in continuous soliton systems.

A real nonlinear integrable couplings of continuous soliton hierarchy and its Hamiltonian structure

Some integrable coupling systems of existing papers are linear integrable couplings. In the Letter, beginning with Lax pairs from special non-semisimple matrix Lie algebras, we establish a scheme for constructing real nonlinear integrable couplings of continuous soliton hierarchy. A direct application to the AKNS spectral problem leads to a novel nonlinear integrable couplings, then we consider the Hamiltonian structures of nonlinear integrable couplings of AKNS hierarchy with the component-trace identity.

Transient Raman response and soliton self-frequency shift in tellurite microstructured fiber

We report on the Raman response function and the soliton self-frequency shift in a tellurite microstructured fiber pumped at 1557 nm. The Raman response function in a tellurite fiber quantifies a contribution of the Raman fraction to the nonlinear optical response. Continuous soliton wavelength shift from 1582 to 1851 nm is observed in a tellurite microstructured optical fiber with the length of 6.5 cm. The experimental results strongly correlated with the numerical simulations based on a generalized nonlinear Schrödinger equation.

Transition from discrete to continuous Townes solitons in periodic media

We present a comprehensive analysis of how the properties of two-dimensional lattice (``discrete'') solitons in a Kerr medium are influenced by their peak intensity and width. We are able to quantitatively relate the Townes solution for solitons in a two-dimensional, homogeneous media to two distinct regimes of the lattice solitons: to narrow, high-intensity, highly nonlinear solitons and to broad, low-intensity, weakly nonlinear solitons, which experience the periodic potential as an effective homogeneous medium. Both regimes, although they support a different power flow and are affected by completely different diffraction dynamics, are thus traced back to the same physical phenomenon. They are separated by a range of unstable and stable solutions, directly caused by the periodicity of the lattice.

An investigation of dark soliton behaviors within the nonlinear micro and nano ring resonators

We firstly propose the interesting results of a dark soliton pulse propagating within the nonlinear micro and nano waveguides. The system consists of nonlinear micro and nano ring resonators, whereas the dark soliton is input into the system and traveling within the waveguide. A continuous dark soliton pulse is chopped to be the smaller pulses by the nonlinear effects known as chaos. The nonlinear behaviors such as chaos, bistability and bifurcation are analyzed and discussed. The power amplification is the property that can be used to perform the long distance link, where the security is the dominant reason.