Nonlinear Schr Dinger Equation Research Articles

THE NUMERICAL APPROXIMATION OF STATIONARY WAVE SOLUTIONS FOR TWO-COMPONENT SYSTEM OF NONLINEAR SCHRÖDINGER EQUATIONS BY USING GENERALIZATION PETVIASHVILI METHOD

The Petviashvili method is a numerical method for obtaining fundamental solitary wave solutions of stationary scalar nonlinear wave equations with power-law nonlinearity: , where is a positive definite and self-adjoint operator and is constant. Due to the case being a system of solitary nonlinear wave equations, we generalize the Petviashvili method. We apply this generalized method for a two-component system of Nonlinear Schr dinger Equations (NLSE) for 2-D.

Quasi discreteness analysis of a 2-dimensional Heisenberg ferromagnetic spin system with biquadratic interactions

The dynamical theory of soliton excitations in two-dimensional ferromagnets is studied by introducing a Hamiltonian that includes biquadratic interaction along with uniaxial anisotropy. To obtain a dynamical equation of motion, we use the Dyson-Maleev transformation and the coherent state ansatz. We obtain a Nonlinear Schrdinger equation by applying the multiple-scale and quasi-discreteness methods. For the more general nonintegrable case, we perform a multiple scale perturbation analysis to determine the discreteness effect on the soliton excitations. Finally, we examine modulational instability (MI) under the influence of small perturbations.

Global well-posedness and critical norm concentration for inhomogeneous biharmonic NLS

We consider the inhomogeneous biharmonic nonlinear Schr dinger (IBNLS) equation in $${\mathbb {R}}^N$$ , $$\begin{aligned} i \partial _t u +\Delta ^2 u -|x|^{-b} |u|^{2\sigma }u = 0, \end{aligned}$$ where $$\sigma > 0$$ and $$b > 0$$ . We first study the local well-posedness in $${\dot{H}}^{s_c}\cap \dot{H}^2 $$ , for $$N\ge 5$$ and $$0<s_c<2$$ , where $$s_c=\frac{N}{2}-\frac{4-b}{2\sigma }$$ . Next, we established a Gagliardo-Nirenberg type inequality in order to obtain sufficient conditions for global existence of solutions in $$\dot{H}^{s_c}\cap \dot{H}^2$$ with $$0\le s_c<2$$ . Finally, we study the phenomenon of $$L^{\sigma _c}$$ -norm concentration for finite time blow up solutions with bounded $$\dot{H}^{s_c}$$ -norm, where $$\sigma _c=\frac{2N\sigma }{4-b}$$ . Our main tool is the compact embedding of $$\dot{L}^p\cap \dot{H}^2$$ into a weighted $$L^{2\sigma +2}$$ space, which may be seen of independent interest.

Influence of Higher-order Effects on N-soliton Solution of the Coupled Nonlinear Schrdinger Equation

Influence of Higher-order Effects on N-soliton Solution of the Coupled Nonlinear Schrdinger Equation

On the local fractional variable‐coefficient Ablowitz–Kaup–Newell–Segur hierarchy: Hamiltonian structure, localization of nonlocal symmetries, and exact solution of reduced equations

Starting from a local fractional variable‐coefficient Ablowitz–Kaup–Newell –Segur (AKNS) spectral problem involving an arbitrary function , we derive a hierarchy of fractional variable‐coefficient nonlinear evolution equations (NLEEs), which possesses Hamiltonian structure. It is interesting that the hierarchy is explicitly related to many important equations such as the fractional variable‐coefficient nonlinear Schr dinger equation, the fractional variable‐coefficient Burgers equation, and the fractional variable‐coefficient mKdV equation involving time‐dependent damping and dispersion. Then, the truncated Painlevé method is generalized to find the multiple nonlocal symmetries of the fractional variable‐coefficient Burgers equation. Then, starting from the initial value problem of the localized symmetries for the expanded system, we derive the auto‐B cklund transformation of the fractional variable‐coefficient Burgers equation. Furthermore, exact solutions of the fractional variable‐coefficient mKdV equation is obtained by the improved solitary ansatze methods.

Stability Analysis of Magnetohydrodynamics Waves in Compressible Turbulent Plasma

Here, the solitary waves propagation in the cold plasma and their stability conditions are studied. The governing equations are expanded by using the reductive perturbation method with taken under the influence of the magnetic field under consideration. A new nonlinear wave equation is obtained that reconciles the derivative nonlinear Schrdinger equation with a modified form. By considering the magnetic field is constant along the x-direction, the complete set of equations is obtained, and the stable solitary waves are observed. A compariso between the soliton solutions of the modified nonlinear Schrdinger evolution equation (MNLS) and the solutions of the compressible magnetohydrodynamic (MHD) equations has been performed. It is shown that stable solitons can be created in such nonrelativistic fluids in the presence of the magnetic field. The modulation instability for a one-dimensional MNLS equation is carried as well. The regions of stability and instability fo the present system are well determined.

Four-dark Soliton Solution of the Coupled Nonlinear Schrdinger Equation

Four-dark Soliton Solution of the Coupled Nonlinear Schrdinger Equation

N-soliton Solution of the Generalized Coupled Nonlinear Schrdinger Equation

N-soliton Solution of the Generalized Coupled Nonlinear Schrdinger Equation

Cross-phase modulation instability under delayed response and walk-off effects in the anomalous regime of dispersion

The dynamics of the modulation instability induced by cross phase modulation is studied by considering the influence of the walk-off and noninstantaneous response effects for two copropagating optical fields travelling in the anomalous regime of dispersion. To do so, we make use of extensions of the nonlinear Schrdinger equation jointly with the Debye model for polarization, which is shown to be effective and simplified when compared to the implementation of other methods for the noninstantaneous nonlinear response. In analyzing the sideband formation, two bands are observed with different behaviors with respect to the way the maximum gain and the respective frequency vary with increasing phase mismatch. Further, we also show the manner in which the maximum gain as well as its corresponding frequency scale with the delay parameter τ is substantially different from the cases of both fields experiencing the normal group-velocity dispersion regime and the case of mixed regimes. These facts may give rise to many new possibilities for the evolution of the modulated wave when compared to the nonanomalous cases studied in the literature.

变系数耦合非线性薛定谔方程的二孤子解及其相互作用

变系数耦合非线性薛定谔方程的二孤子解及其相互作用

群速度与相速度对层级结构NSLE一孤子解的影响

群速度与相速度对层级结构NSLE一孤子解的影响

Pt-Symmetric Nonlinear Directional Fiber Couplers with Gain and Loss for Ultrashort Optical Pulses

In this paper, we study the higher-order nonlinear Schrdinger (HNLS) equations for a directional fiber optic coupler with the third-order dispersion (TOD), self-steeping (SS) and stimulated Raman scattering (SRS) effects and gain in one waveguide and loss in the other. We have reduced the system equations to an uncoupled HNLS equation which has bright and dark soliton solutions. It is shown that the Hamiltonian for this equation has a PT-symmetric form. Furthermore the equilibrium points are found and the discussion about their stability is presented.

All-fiber spectral compression of femtosecond pulse for coherent anti-Stokes Raman scattering excitation source

Coherent anti-Stokes Raman scattering (CARS) imaging of femtosecond pulses has been a research hotspot in recent years, but the wide spectrum of the femtosecond pulse limits the spectral resolution of CARS imaging. Spectral compression is considered as an effective method to solve this problem. In this work, an all-fiber chirp spectral compression method of graded-index multi-mode fiber/single-mode fiber (GI-MMF/SMF) structure based on fiber pre-chirp and self-phase modulation is presented. It can be used as a CARS excitation source to increase the spectral resolution of CARS imaging. In the section of numerical simulation, the mean group velocity dispersion value of GI-MMF is used as a numerical parameter of the chirp analysis, which is estimated by analyzing modes of GI-MMF. On one hand, the mode field distributions in GI-MMF are simulated numerically by the finite-difference time-domain method, and these different modes are divided into eight mode groups. On the other hand, the energy proportion of each mode group is regarded as a weight value. Then we can obtain a mean group velocity dispersion value of 50/125 m GI-MMF, which is -2.28710-5 fs2/nm, by calculating the sum of group velocity dispersion weight values of mode groups. The results of spectral compression with different length ratios of 50/125 m GI-MMF to 780HP SMF are also analyzed based on the generalized nonlinear Schrdinger equation and split-step Fourier algorithm. The spectral width of 2.486 nm and the compression ratio of 5.230 are calculated, when the length ratio of 50/125 m GI-MMF to 780HP SMF is 1.2. In the section of experiment, three kinds of GI-MMFs with different core diameters are used in the experiment, the influences of the core diameter and the length ratio of GI-MMF to 780HP SMF on the spectral compression are investigated. The results show that the spectral width of 2.243 nm, corresponding to the compression ratio of 5.796 is obtained, when the length ratio of 50/125 m GI-MMF to 780HP SMF is 1.2, which is consistent with the simulation result. Under the condition of the same length ratio, the use of 105/125 m GI-MMF can raise the compression ratio to 152.941, and the spectral width of output pulse is 0.085 nm. When the pulse is applied to CARS spectrum detection, the theoretical spectral resolution can be 1.386 cm-1. The experimental results show that the spectral compression way to improve spectral resolution of CARS imaging is effective. This spectral compression system is characterized by simple structure, and high and controllable compression ratio, which provides theoretical and experimental basis for the all-fiber high spectral resolution CARS excitation source research.

Application of fixed point theorem for stability analysis of a nonlinear Schrodinger with Caputo-Liouville derivative

Using the new Caputo-Liouville derivative with fractional order, we have modified the nonlinear Schrdinger equation. We have shown some useful in connection of the new derivative with fractional order. We used an iterative approach to derive an approximate solution of the modified equation. We have established the stability of the iteration scheme using the fixed point theorem. We have in addition presented in detail the uniqueness of the special solution.

Nonlinear optical vibrations of single-walled carbon nanotubes.

We demonstrate a new specific phenomenon of the long-time resonant energy exchange in carbon nanotubes (CNTs), which is realized by two types of optical vibrations, the Circumferential Flexure Mode (CFM) and the Radial Breathing Mode (RBM). We show that the modified nonlinear Schrdinger equation, obtained in the framework of the nonlinear theory of elastic thin shells, allows us to describe the nonlinear dynamics of CNTs for specified frequency bands. Comparative analysis of the oscillations of the CFM and RBM branches shows the qualitative difference of nonlinear effects for these branches. While the nonlinear resonant interaction of the low-frequency modes in the CFM branch leads to energy capture in some domains of the CNT, the same interaction in the RBM branch does not demonstrate any tendency for energy localization. The reason lies in the distinction in the nonlinear terms in the equations of motion. While CFMs are characterized by soft polynomial nonlinearity, RBM dynamics is characterized by hard gradient nonlinearity. Moreover, in contrast to the CFM, the importance of nonlinearity in the case of RBM oscillations decreases as the length to radius ratio increases. Numerical integration of the equations of thin shell theory confirms the results of the analytical study.

Collapse and evolution of wave field based on a generalized nonlinear Schrdinger equation

A generalized nonlinear Schrdinger equation is numerically studied using the split-step Fourier method. For a fixed external potential field and an initial pulse disturbance, the effects of the complex coefficients p and q in the nonlinear Schrdinger equation on the evolution of the wave field are investigated. From a large number of simulations, it is found that the evolution of the wave field remains similar for different signs of the real parts of p and q, and different values of the real part of p. The initial pulse consisting of the longest wavelength modes (in the smallest-|k| corner of the phase space) of the spectrum first suffers modulational instability. Collapse begins at t～0.1, followed by inverse cascade of the shortest wavelength modes to longer wavelength ones, so that the whole k space becomes turbulent. For p = 1+0.04i, and q = 1+0.6i, it is found that first modulational instability occurs in the longer wavelength regime and the wave energy is transferred to the larger |k| modes. Then the wave collapse appears with increasing wave energy. Next, the large-|k| modes condense into a smaller-|k| mode by inverse cascade before spreading to the center of the phase space, until a turbulent state occurs there. Finally, most of the wave energy is condensed to the neighborhoods of three modes.

Theoretical study on generating mid-infrared ultrashort pulse in mode-locked Er3+: ZBLAN fiber laser

Fiber lasers show several advantages over other types of lasers. They are efficient, compact, and rugged since they require few bulk components and are virtually unaffected by the surrounding environment. Mode-locked mid-infrared (mid-IR) lasers are essential for a wide variety of applications. The promising applications of mode-locked fiber lasers at wavelengths near 3 m include combs generation (metrology), spectroscopic sensors, infrared countermeasures, laser surgery, high-efficient pump sources for longer-wavelength oscillators and mid-IR supercontinuum source pumping. Based on the nonlinear Schrdinger equation (NLSE), a theoretical model of passively mode-locked Er3+-doped fluoride fiber laser using a saturable absorber is set up. Some mechanisms for generating mid-IR ultrashort pulse in fiber lasers are investigated. When the dispersion of the cavity is managed properly, the numerical simulation mainly focuses on the evolution process of mid-IR ultrashort pulse in fluoride fiber oscillators. Influences of the intracavity net dispersion and the small-signal gain on the generation of mode-locked pulses are analyzed in detail. And the reasonable parameter windows are given. Just as the simulated results showed, for a case of 4 m Er3+-doped fluoride fiber, small-signal gain g0= 0.6 m-1 and unsaturated loss l0 = 0.7, the stable mode-locked pulses are achieved by tuning the net intracavity dispersion within a certain range from 0.72 ps2 to 0.83 ps2. As the net intracavity dispersion increases, the output pulse duration increases gradually, while the spectrum width (FWHM) and peak power decrease accordingly. In addition, for the case of 4 m Er3+-doped fluoride fiber, unsaturated loss l0 = 0.7 and net intracavity dispersion of 0.8 ps2, the stable mode-locked pulses can also be obtained by tuning the small-signal gain within a certain range from 0.55 to 0.70 m-1. As the small-signal gain increases, the output pulse duration, spectral width, and peak power increase gradually. This work may be beneficial to the design of experiments for achieving more narrow pulse duration, wide spectral width, and high peak power mid-infrared ultrashort pulse.

Perturbed solution and analyses for single photon transmission equation in optical fiber

As is well known, quantum optics has developed significantly in recent years and advanced several hot research topics, such as quantum communications, quantum sensing, quantum calculations, etc. Among these researches, it is important to understand the quantum information transmitting in optical fiber. For realizing longer transmission distance and better transmission quality, great effort has devoted to the researches of encoding and decoding at the transmitter and the receiver end. However, less attention was paid to the fading of signal in the transmission channel. In this work, we mainly focus on the transmission model of optical quantum transmission and the influences of loss, dispersion and nonlinear effect on fiber transmission of optical quantum information are also discussed.Quantum information transmission can be influenced by loss, dispersion and nonlinear effect in optical fiber, leading to transmission state evolution and energy transfer. Based on the transmission equation of single mode fiber and quantum theory of electromagnetic field, the fundamental mode field of single mode fiber is quantized. A quantum transmission equation is deduced from the classical optical transmission equation through quantizing the amplitude of electromagnetic field. Compared with classic wave theory, the photon transmission equation quantizing the slowly-varying amplitude in the coupled nonlinear Schrdinger equation is obtained. In the classic wave equation, light is interpreted as energy which propagates as waves. The photon transmission equation is obtained by quantizing the slowly-varying amplitude of light, that is, the particle nature of light. The energy propagates through alternative interaction between creation and annihilation operator on photons. The transmission equations show that photons will interact with the transmission medium during propagation and be influenced by dispersion, nonlinear effect, loss, etc. By giving a trail solution and introducing a perturbation term, the transmission equation is solved for the complicated case where the dispersion, loss and nonlinear effect are all involved. A dispersion equation that should be satisfied for nontrivial solution is then obtained. From this dispersion equation, the relation between photon power and perturbation frequency is calculated and analyzed. The change of photon power in generalized field with perturbation frequency is discussed, and the influences of fiber dispersion and nonlinearity on the solution are analyzed.Some conclusions are obtained by perturbed solution and analyses of single photon transmission equation in optical fiber. It is found that photon power decreases with the increase of perturbation frequency and reaches its maximum value for zero perturbation frequency. At the same time, the optical power is affected by the dispersion of the optical fiber. Photon power decreases with the GVD coefficient far from the zero dispersion point. It is also found that photon power decreases with the increase of nonlinear coefficient. This work may contribute to the research of the properties of quantum fiber transmission system.

Nonlinear Faraday rotation in electromagnetically induced transparency medium of semiconductor three quantum dots

In the past few years, many interesting optical phenomena, such as electromagnetically induced transparency, coherent optical control of a biexciton, slow light and optical solitons, have been investigated in single quantum dot (QD). However, in an actual semiconductor device there exist many quantum dots (QDs). Recently, QD molecule, which is comprised of double semiconductor QDs coupled by tunneling coupling, has been proposed. In this new semiconductor structure, many complex but interesting phenomena have been discovered. In fact, three QD molecules may also be composed of three QDs, which can be coupled by interdot tunneling coupling. For the three semiconductor QDs molecules, the influence of the interdot tunneling coupling strength must be considered. So, in this paper, with considering that a weak, -linear-polarized probe field can form left- and right-polarized components under the control of the parallel magnetic field, and when they are combined with the tunneling coupling among the QDs, an electromagnetically induced transparency medium of a five-level M configuration semiconductor three QDs is proposed. Subsequently, the nonlinear Faraday rotation in the semiconductor three QDs is analytically studied. For the linear case, the linear dispersion relation is driven by a method of multiple scales. Then, by studying the linear optical properties, it is found that the system exhibits a single tunneling induced transparency window due to the quantum destructive interference effect driven by the interdot tunneling coupling under appropriate conditions, and the width of the tunneling induced transparency window can be effectively controlled by the strength of the interdot tunneling coupling. Meanwhile, the switch regulatory effect, which changes from the anomalous dispersion regime to the normal dispersion regime, is likely to be achieved by changing the strength of the interdot tunneling coupling. For the nonlinear case, two coupled nonlinear Schrdinger equations, which govern the evolutions of left- and right-polarized components of the weak, -linear-polarized probe field under the applied longitudinal magnetic field, are derived. By studying the nonlinear properties, it is shown that a large nonlinear Faraday rotation angle can be obtained due to the quantum interference effect which is induced by the interdot tunneling coupling with a very low absorption of the weak, -linear-polarized probe field. In addition, it is also found that the nonlinear Faraday rotation direction is opposite to line Faraday rotation for the same magnetic field. What is more, the nonlinear Faraday rotation angle grows bigger than the linear Faraday rotation. These results mean that the Faraday rotation of the three semiconductor QDs with the electromagnetically induced transparency can be more effectively controlled by the nonlinear effect.

Influences of Raman gain and self-steepening on the propagation characteristic of Airy pulse

By using nonlinear Schrdinger equation including Raman gain and self-steepening but ignoring fiber loss situation, the propagation characteristics of Airy pulse are simulated and analyzed in the single-mode fiber. Simulations show that Airy pulse can be converted into soliton and its propagation direction is skewed due to the effects of Raman gain and self-steepening under a certain condition. In time domain, the number of small peaks of Airy pulse reduces rapidly. Airy pulse becomes a peak structure containing a main peak and sub-peaks where the energies can be ignored by changing the coefficient a reasonablely, which is approximated as the soliton structure. Therefore, Airy pulse is regarded as transforming into soliton. Meanwhile, in the case of small values b, there exists a significant difference in shape between Airy pulse and soliton. With the value of parameter b increasing slowly, the shape of Airy pulse is very close to soliton's, therefore Airy pulse can transform into soliton by changing value b reasonablely. Compared with by changing b value, Airy pulse convered into the soliton is stable by changing the a value reasonablely. Simultaneously, with the increases of values of coefficient a and amplitude b, the time-shift of Airy pulse increases. However, the time-shift of Airy pulse would decrease when Raman gain and Self-steepening become strong, no matter what the values of a and b are. Further, the acceleration properties of Airy pulse are investigated. It is found that Airy pulse autoacceleration is not a stable value at the beginning but it gradually stabilizes with the increase of transmission distance. The reason is that the energies of secondary peaks exert a tremendous influence on the main lobe of Airy pulse at the beginning, however, secondary peaks diffuse fast with the increase of transmission and then the influence can be ignored to a certain extent. So, the main peak gradually stabilizes with the increase of transmission distance.