Coupled Nonlinear Schrodinger Equations Research Articles

Bifurcation analysis of stationary solutions of two-dimensional coupled Gross–Pitaevskii equations using deflated continuation

Recently, a novel bifurcation technique known as deflated continuation was applied to the single-component nonlinear Schrödinger (NLS) equation with a parabolic trap in two spatial dimensions. This bifurcation analysis revealed previously unknown solutions, shedding light on this fundamental problem in the physics of ultracold atoms. In the present work, we take this a step further by applying deflated continuation to two coupled NLS equations, which – feature a considerably more complex landscape of solutions. Upon identifying branches of solutions, we construct the relevant bifurcation diagrams and perform spectral stability analysis to identify parametric regimes of stability and instability and to understand the mechanisms by which these branches emerge. The method reveals a remarkable wealth of solutions. These include both well-known states arising from the Cartesian and polar small amplitude limits of the underlying linear problem, but also a significant number of more complex states that arise through (typically pitchfork) bifurcations.

Diverse composite waves in coherently coupled inhomogeneous fiber systems with external potentials

In this paper, we focus on the coherently coupled nonlinear Schrodinger (NLS) equations with variable coefficient including four-wave mixing, external potential and gain/loss terms, and derive three types of similarity transformations under different constraint conditions. Based on these transformations, one can transform the variable coefficient coupled NLS equations into constant coefficient coupled NLS equations that can be decoupled by using of a linear transformation. Thus, various composite waves superposed by different nonlinear waves can be investigated in homogeneous and inhomogeneous systems with the aid of the solutions of NLS equation and these transformations. The diversity of various composite waves and the energy exchange between two coherently coupled components in homogeneous fiber system are demonstrated. Furthermore, based on the obtained three types of similarity transformations, the characteristics of the composite waves are investigated in tunneling system and periodic perturbation system, respectively. These results could be helpful to explore the diverse dynamics of the composite waves in birefringent fiber.

Dynamics of the smooth positons of the coupled nonlinear Schrödinger equations

In this paper, we formulate the compact determinant representation formula of n-soliton solution for the coupled nonlinear Schrödinger equations (CNLS) equations, and obtain the expression of the n-positon solutions by using the Taylor expansion to degenerate eigenvalues λ3j−2→λ1(j=1,2,…,n). Especially, the expressions of two-positon and three-positon solutions are obtained by the corresponding formulas. Furthermore, the dynamical properties of the smooth positons of the CNLS equations are analyzed by using the method of decomposition of the modulus square, which can be used to approximately describe the trajectories and time-dependent ‘phase shifts’ of positons after the collision. In addition, the collision between soliton and positon, and mutual collision of two two-positons are graphically discussed, and these collision properties are remarkably different from before.

Breather waves, high-order rogue waves and their dynamics in the coupled nonlinear Schrödinger equations with alternate signs of nonlinearities

We consider the analytic vector breather and high-order rogue wave solutions for the coupled nonlinear Schrödinger (NLS) equations with alternate signs of nonlinearities via Darboux dressing transformation. By adjusting spectral parameters, we indicate that the breather wave solutions contain temporally periodic nonlinear waves and spatially periodic nonlinear waves, respectively. The vector rogue wave solutions include the bright one-peak-two-valleys rogue wave and the bright rogue wave without valleys. Additionally, we successfully show different types of the distributions for the second-order vector rogue waves. The existence condition for the rogue waves is discussed. For the coupled NLS equations with alternate signs of nonlinearities, the rogue wave solutions exist if the baseband modulation instability (MI) is present.

An efficient split-step compact finite difference method for the coupled Gross–Pitaevskii equations

ABSTRACTIn this study, we propose an efficient split-step compact finite difference (SSCFD) method for computing the coupled Gross–Pitaevskii (CGP) equations. The coupled equations are divided into two parts, nonlinear subproblems and linear ones. Commonly, the nonlinear subproblems could be integrated directly and accurately, but it fails when the time-dependent potential cannot be integrated exactly. In this case, the midpoint and trapezoidal rules are applied approximately. At the same time, the split order is not reduced. For the linear ones, compact finite difference cannot be designed directly. To circumvent this problem, a linear transformation is introduced to decouple the system, which can make the split-step method be used again. Additionally, the proposed SSCFD method also holds for the coupled nonlinear Schrödinger (CNLS) system with time-dependent potential. Finally, numerical experiments for CGP equations and CNLS equations are well simulated, conservative properties and convergence rates are demonstrated as well. It is shown from the numerical tests that the present method is efficient and reliable.

Nonlinear interaction of co-directional shear horizontal waves in a two-layered elastic medium

The nonlinear interaction of co-directional shear horizontal (SH) waves propagating in a two-layered plate of uniform thickness is considered. It is assumed that the constituent materials of both layers are nonlinear, homogeneous, isotropic and incompressible elastic and having different mechanical properties. It is also assumed that between the linear shear velocities of the top layer, $$c_1$$ , and the bottom layer, $$c_2$$ , the inequality $$c_1<c_2$$ is valid. For the existence of a SH wave, the phase velocity of the wave, c, must satisfy either the inequality $$c_1<c_2 \le c$$ or the one $$c_1< c\le c_2 $$ and here the problem is examined under the second condition. By employing an asymptotic perturbation method and balancing the weak nonlinearity and dispersion in the analysis, it is shown that the first order slowly varying amplitudes of interacting waves are governed asymptotically by a coupled nonlinear Schrodinger (CNLS) equations. Then the effects of nonlinearity of the materials and the ratio of layers’ thickness on the linear instabilities of solutions of the CNLS equations and the existence of solitary wave solutions of the CNLS are examined.

Polarization-Maintaining Single-Frequency Fiber Laser With Quadruple Wavelengths at the C-Band

In this paper, a polarization-maintaining quad-wavelength (QW) single-frequency fiber laser (SFFL) at the C-band with the novel and compact structure is demonstrated. Applying the rate equations and multiple coupled nonlinear Schrodinger equations, the theoretical model of QW fiber lasers is established, and then the optical spectra, temporal evolution, and stability of the QW fiber laser are analyzed numerically. Based on the theoretical analysis, a QW-SFFL having an ultrashort linear distributed Bragg reflector cavity is proposed. Utilizing a wideband fiber Bragg grating coupled with a dual-channel polarization maintaining fiber Bragg grating as the wavelength selection and a 15-mm-long Er 3+ /Yb 3+ co-doped phosphate fiber as the gain medium has realized a robust QW laser with the wavelength spacing of 0.4 nm. All the wavelengths maintain the single-frequency operation independently with the linewidths of 20 kHz. This type of compact SFFLs with quadruple wavelengths is widely adapted to lidar systems and fiber sensing.

One-pump fiber optical parametric amplifiers: from the pulsed to the continuous wave operation

The characteristics of one-pump fiber optical parametric amplifiers (1-P FOPAs) are theoretically investigated as the temporal width of the input pump increases from the pulsed to the continuous wave (CW) regime. Coupled nonlinear Schrodinger equations are numerically solved, and the pulse width of the pump is gradually increased to simulate the evolution of the gain and output spectra of the 1-P FOPA from the pulsed to the CW operation. The results show that the quasi-CW operation is achieved when the pulse width is about 500 fs. Also, when the temporal width of the pump increases and lies in the picosecond regime, the characteristics of FOPA recover those of the FOPA operated in the CW regime.

Three-Dimensional Coupled NLS Equations for Envelope Gravity Solitary Waves in Baroclinic Atmosphere and Modulational Instability

Envelope gravity solitary waves are an important research hot spot in the field of solitary wave. And the weakly nonlinear model equations system is a part of the research of envelope gravity solitary waves. Because of the lack of technology and theory, previous studies tried hard to reduce the variable numbers and constructed the two-dimensional model in barotropic atmosphere and could only describe the propagation feature in a direction. But for the propagation of envelope gravity solitary waves in real ocean ridges and atmospheric mountains, the three-dimensional model is more appropriate. Meanwhile, the baroclinic problem of atmosphere is also an inevitable topic. In the paper, the three-dimensional coupled nonlinear Schrödinger (CNLS) equations are presented to describe the evolution of envelope gravity solitary waves in baroclinic atmosphere, which are derived from the basic dynamic equations by employing perturbation and multiscale methods. The model overcomes two disadvantages: (1) baroclinic problem and (2) propagation path problem. Then, based on trial function method, we deduce the solution of the CNLS equations. Finally, modulational instability of wave trains is also discussed.

Solitons in two attractive semiconductor nanowires

In this paper, by using two semiconductor nanowires attracted to each other by means of Lorentz force, we construct through similarity transformations, explicit solutions to the coupled nonlinear Schrodinger equations (CNSE) with potentials as a function of time and spatial coordinates. We find explicit solutions of electrons and holes such as periodic, bright and dark solitons. We also study the instability of the modulation (MI) of (CNSE) and note that the velocity of the electrons influences the gain MI spectrum.

A Riemann–Hilbert approach for a new type coupled nonlinear Schrödinger equations

We develop a Riemann–Hilbert approach to the Cauchy problem on the line for a new type of coupled nonlinear Schrödinger (CNLS) equationsiq1,t+q1,xx+2(|q1|2−2|q2|2)q1−2q22q1⁎=0,iq2,t+q2,xx+2(2|q1|2−|q2|2)q2+2q12q2⁎=0. This approach allows us to give a representation of the solution to the Cauchy problem of the CNLS equations in terms of the solution of a 4×4 Riemann–Hilbert problem formulated in the complex k-plane. Due to the energy conservation law of above system is ∫−∞+∞(|q1|2−|q2|2)dx, it is difficult to obtain a solution for this system by using the energy estimate method of PDE's. Therefore, this approach efficiently provides a new way in studying the nonlinear problems that PDE's theory can't solve. Furthermore, this representation is then used for retrieving the soliton solutions.

Bright–dark soliton solutions for the (2+1)-dimensional variable-coefficient coupled nonlinear Schrödinger system in a graded-index waveguide

Under investigation in this paper is the (2[Formula: see text]+[Formula: see text]1)-dimensional coupled nonlinear Schrödinger (NLS) system with variable coefficients, which describes the propagation of an optical beam inside the two-dimensional graded-index waveguide amplifier with the polarization effects. Through a similarity transformation, we convert that system into a set of the integrable defocusing (1[Formula: see text]+[Formula: see text]1)-dimensional coupled NLS equations, and subsequently construct the bright–dark soliton solutions for the original system which are converted from the ones of the latter set. With the graphic analysis, we discuss the soliton propagation and collision with r(t), which is related to the nonlinear, profile and gain/loss coefficients. When r(t) is a constant, one soliton propagates with the amplitude, width and velocity unvaried, while velocity and width of the one soliton can be affected, and two solitons possess the elastic collision; When r(t) is a linear function, velocity and width of the one soliton varies with t increasing, and collision of the two solitons is altered. Besides, bound-state solitons are seen.

Copropagation of coupled laser pulses in magnetized plasmas: Modulational instability and coupled solitons

The nonlinear propagation of two circularly polarized strong laser pulses in a magnetized plasma is considered. In a weakly relativistic regime, it is shown that the evolution of amplitudes of the laser pulses is governed by two coupled cubic nonlinear Schrödinger (NLS) equations. The modulational instability of coupled plane wave solutions is investigated based on coupled NLS equations. The dependence of the growth rate on relevant parameters of the system is addressed. The possibility of formation of various kinds of coupled solitary wave solutions in magnetized and unmagnetized cases is considered, and the variation of the profile of these solutions with different parameters is explored.

On Preconditioners Based on HSS for the Space Fractional CNLS Equations

AbstractThe space fractional coupled nonlinear Schrödinger (CNLS) equations are discretized by an implicit conservative difference scheme with the fractional centered difference formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix which can be written as the imaginary unit times the identity matrix and a symmetric Toeplitz-plusdiagonal matrix. In this paper, we present new preconditioners based on Hermitian and skew-Hermitian splitting (HSS) for such Toeplitz-like matrix. Theoretically, we show that all the eigenvalues of the resulting preconditioned matrices lie in the interior of the disk of radius 1 centered at the point (1,0). Thus Krylov subspace methods with the proposed preconditioners converge very fast. Numerical examples are given to illustrate the effectiveness of the proposed preconditioners.

New Solutions for Nonlinear Schrodinger Equations using Adomian Decomposition and Homotopy Perturbation Methods

Coupled non-linear Schrodinger equation (CNLS) with two mode propagations in an optical fibre was studied. Using Painleve analysis, we are successfully solving the focusing Manakov system, bright and dark solitons using Adomian Decomposition Method (ADM) and Homotopy Perturbation Method (HPM). Both methods are fast convergent, do not need linearization and eliminates massive computation work. Its hows that HPM does not involve the Adomian polynomials when dealing with non-linear problems.

On partially inexact HSS iteration methods for the complex symmetric linear systems in space fractional CNLS equations

The space fractional coupled nonlinear Schrödinger (CNLS) equations are discretized by an implicit conservative difference scheme with the fractional centered difference formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix and a symmetric positive definite diagonal-plus-Toeplitz matrix. The Hermitian and skew-Hermitian splitting (HSS) method and the partially inexact HSS (PIHSS) method are employed to solve the discretized linear system. In the inner iteration processes of the HSS method, we only need to solve the linear sub-systems associated with the Hermitian part inexactly by the conjugate gradient (CG) method, resulting in PIHSS iteration method. Theoretical analyses show that both HSS and PIHSS methods are unconditionally convergent. Numerical examples are given to demonstrate the effectiveness of the HSS iteration and the PIHSS iteration.

Vector combined and crossing Kuznetsov–Ma solitons in $$\mathcal {PT}$$ PT -symmetric coupled waveguides

A variable-coefficient coupled nonlinear Schrodinger equation with balanced gain and loss is studied, and two kinds of analytical Kuznetsov–Ma soliton solutions including combined Kuznetsov–Ma soliton solution and crossing two-Kuznetsov–Ma soliton solution are obtained. In a diffraction decreasing system, the control for the excitation of two kinds of Kuznetsov–Ma soliton including rear excitation, peak excitation and initial excitation is investigated. The different peak locations of Kuznetsov–Ma soliton along the propagation direction appear repeatedly, which makes rear excitation, peak excitation and initial excitation happen again and again.

A coupled "AB" system: Rogue waves and modulation instabilities.

Rogue waves are unexpectedly large and localized displacements from an equilibrium position or an otherwise calm background. For the nonlinear Schrödinger (NLS) model widely used in fluid mechanics and optics, these waves can occur only when dispersion and nonlinearity are of the same sign, a regime of modulation instability. For coupled NLS equations, rogue waves will arise even if dispersion and nonlinearity are of opposite signs in each component as new regimes of modulation instability will appear in the coupled system. The same phenomenon will be demonstrated here for a coupled "AB" system, a wave-current interaction model describing baroclinic instability processes in geophysical flows. Indeed, the onset of modulation instability correlates precisely with the existence criterion for rogue waves for this system. Transitions from "elevation" rogue waves to "depression" rogue waves are elucidated analytically. The dispersion relation as a polynomial of the fourth order may possess double pairs of complex roots, leading to multiple configurations of rogue waves for a given set of input parameters. For special parameter regimes, the dispersion relation reduces to a cubic polynomial, allowing the existence criterion for rogue waves to be computed explicitly. Numerical tests correlating modulation instability and evolution of rogue waves were conducted.

Quaternion Approach to Solve Coupled Nonlinear Schrödinger Equation and Crosstalk of Quarter-Phase-Shift-Key Signals in Polarization Multiplexing Systems**Supported by the National Natural Science Foundation of China under Grant No 61275075, and the Beijing Natural Science Foundation under Grant Nos 4132035 and 4144080.

The quaternion approach to solve the coupled nonlinear Schrödinger equations (CNSEs) in fibers is proposed, converting the CNSEs to a single variable equation by using a conception of eigen-quaternion of coupled quaternion. The crosstalk of quarter-phase-shift-key signals caused by fiber nonlinearity in polarization multiplexing systems with 100 Gbps bit-rate is investigated and simulated. The results demonstrate that the crosstalk is like a rotated ghosting of input constellation. For the 50 km conventional fiber link, when the total power is less than 4 mW, the crosstalk effect can be neglected; when the power is larger than 20 mW, the crosstalk is very obvious. In addition, the crosstalk can not be detected according to the output eye diagram and state of polarization in Poincaré sphere in the trunk fiber, making it difficult for the monitoring of optical trunk link.

Maximum-norm error analysis of a difference scheme for the space fractional CNLS

The difference method for the space fractional coupled nonlinear Schrödinger equations (CNLS) is studied. The fractional centered difference is used to approximate the space fractional Laplacian. This scheme conserves the discrete mass and energy. Due to the nonlocal nature of fractional Laplacian, in the classic Sobolev space, it is hard to obtain the error estimation in l∞. To overcome this difficulty, the fractional Sobolev space Hα/2 and a fractional norm equivalence in Hα/2 are introduced. Then the convergence of order O(h2+τ2) in l∞ is proved by fractional Sobolev inequality, where h is the mesh size and τ is the time step. Numerical examples are given to illustrate the theoretical results at last.