Nonautonomous Solitons Research Articles

Controllable behaviors of nonautonomous solitons on background of continuous wave and cnoidal wave in $$\mathcal {PT}$$ PT -symmetric dimer with inhomogeneous effect

We obtain exact \(\mathcal {PT}\)-symmetric and \(\mathcal {PT}\)-antisymmetric nonautonomous soliton solutions on background waves. These solutions indicate that dispersion and nonlinear coefficients influence form factors of nonautonomous solitons such as amplitude, width and center; however, linear coupling coefficient and gain/loss parameter only influence phase of solitons. Based on these solutions, the controllable behaviors such as postpone, sustainment and restraint on continuous wave background in an exponential decreasing dispersion system are discussed. Moreover, the propagation behaviors of solitons on the cnoidal wave background in different dispersion systems are also studied.

Characteristics of nonautonomous W-shaped soliton and Peregrine comb in a variable-coefficient higher-order nonlinear Schrödinger equation

In this paper, we study the nonautonomous characteristics of the W-shaped solitons on constant backgrounds for a variable-coefficient higher-order nonlinear Schrödinger (vc-HNLS) equation. Several types of solitons are obtained in exponentially, linearly and periodically fluctuating decreasing dispersion profiles, respectively. We also show the compression effect of the third-order dispersion (TOD) coefficient on the W-shaped soliton. We further analyze the formation and spatiotemporal characteristics of the Peregrine combs (PCs) when TOD coefficient is of the periodic form. Moreover, the PC can be converted into the Peregrine wall if the modulation amplitude is equal to 1.

Nonautonomous solitons, breathers and rogue waves for the cubic-quintic nonlinear Schrödinger equation in an inhomogeneous optical fibre

Under investigation in this paper is a cubic-quintic nonlinear Schrödinger equation which can describe the propagation of ultrashort pulses in an inhomogeneous optical fibre. Lax pair and conservation laws are constructed from which the integrability of the equation can be verified. Through a gauge transformation, spectral problem for the equation is converted to an Ablowitz–Kaup–Newell–Segur spectral problem. Nonautonomous solitons and breathers are derived based on the Darboux transformation (DT), while nonautonomous rogue waves are obtained via the generalized DT. Influence of the group-velocity dispersion, gain-or-loss coefficient and group-velocity coefficient on the propagation and interaction of the nonautonomous solitons, breathers and rogue waves is also discussed. The gain-or-loss coefficient influences the amplitude of nonautonomous soliton. The characteristic line and velocity for the nonautonomous soliton are dependent on the group-velocity dispersion and group-velocity coefficient, but independent of the gain-or-loss coefficient. For the second-order nonautonomous soliton, if the two spectral parameters have the same real part, the bound-state structure can be formed, and the intensity becomes amplified in the attractive procedure. There exist two types of the nonautonomous breathers. Quasi-periods for the nonautonomous breathers are given, and effects of those coefficients on the quasi-periods are also discussed: The group-velocity dispersion can influence the trajectories of the nonautonomous breathers and rogue waves, while the gain-or-loss coefficient affects the backgrounds for the nonautonomous breathers and rogue waves

Interaction of the nonautonomous soliton in the optical fiber

Investigated in this paper is a generalized nonautonomous nonlinear Schrödinger equation with both external potential and phase modulation, which can be used to described the nonautonomous soliton in the optical fiber. With our results as follows: Lax pair, Daboux transformation, condition of the modulation instability and N−soliton solutions have been obtained; Propagation and interaction of the solitons have been analysed: (1) the group velocity and linear potential affect the direction and trajectory of one/two solitons′ propagation, respectively; (2) two solitons have the same average velocity at some fixed time or a period.

Darboux transformation and nonautonomous solitons for a generalized inhomogeneous Hirota equation

In this paper, a generalized inhomogeneous Hirota equation with spatial inhomogeneity and nonlocal nonlinearity is investigated in detail. Firstly, the Darboux transformation is constructed based on corresponding nonisospectral linear eigenvalue problem. This transformation has an essential difference from the isospectral case. Furthermore, the nonautonomous soliton solutions are obtained via the Darboux transformation. Finally, properties of these solutions in the inhomogeneous media are discussed graphically to illustrate the influences of the variable coefficients. It is found that the velocity and amplitude of the solitons can be controlled by the inhomogeneous parameters. Especially, a special two-soliton solution which are localized both in space and time exhibits the feature of the so-called rogue waves but with a zero background.

On the Darboux transformation of a generalized inhomogeneous higher-order nonlinear Schrödinger equation

Recently, a paper about the Nth-order rogue waves for an inhomogeneous higher-order nonlinear Schrodinger equation using the generalized Darboux transformation is published. Song et al. (Nonlinear Dyn 82(1):489–500. doi: 10.1007/s11071-015-2170-6 , 2015). However, the inhomogeneous equation which admits a nonisospectral linear eigenvalue problem is mistaken for having a constant spectral parameter by the authors. This basic error causes the results to be wrong, especially regarding the Darboux transformation (DT) in Sect. 2 when the inhomogeneous terms are dependent of spatial variable x. In fact, the DT for inhomogeneous equation has an essential difference from the isospectral case, and their results are correct only in the absence of inhomogeneity which was already discussed in detail before. Consequently, we firstly modify the DT based on corresponding nonisospectral linear eigenvalue problem. Then, the nonautonomous solitons are obtained from zero seed solutions. Properties of these solutions in the inhomogeneous media are discussed graphically to illustrate the influences of the variable coefficients. Finally, the failure of finding breather and rogue wave solutions from this modified DT is also discussed.

Nonautonomous solitons and Wronskian solutions for the (3+1)-dimensional variable-coefficient forced Kadomtsev–Petviashvili equation in the fluid or plasma

Under investigation in this paper is a ( 3 + 1 )-dimensional variable-coefficient forced Kadomtsev–Petviashvili equation which can describe the nonautonomous solitons in such areas as fluids and plasmas. The first- and second-order nonautonomous solitons are constructed via the Hirota bilinear method. Propagation and interaction of the nonautonomous solitons are analyzed. Perturbation coefficient affects the amplitude of the nonautonomous soliton. The background where the nonautonomous soliton exists can be influenced by the external force coefficient. Breathers and resonant interaction, which are the special interaction structures for the second-order nonautonomous solitons, are also presented. Nonuniformity coefficient influences the period of the breather. For the resonant interaction, the two nonautonomous solitons merge into a single solitary wave and form three branches, the amplitudes of which are influenced by the perturbation coefficient. Solutions in terms of the Wronskian determinants are constructed and verified via the direct substitution into the bilinear form.

Nonautonomous solitons in terms of the double Wronskian determinant for a variable-coefficient Gross–Pitaevskii equation in the Bose–Einstein condensate

Under investigation in this paper is a variable-coefficient Gross–Pitaevskii equation which describes the Bose–Einstein condensate. Lax pair, bilinear forms and bilinear Bäcklund transformation for the equation under some integrable conditions are derived. Based on the Lax pair and bilinear forms, double Wronskian solutions are constructed and verified. The [Formula: see text]th-order nonautonomous solitons in terms of the double Wronskian determinant are given. Propagation and interaction for the first- and second-order nonautonomous solitons are discussed from three cases. Amplitudes of the first- and second-order nonautonomous solitons are affected by a real parameter related to the variable coefficients, but independent of the gain-or-loss coefficient [Formula: see text] and linear external potential coefficient [Formula: see text]. For Case 1 [Formula: see text], [Formula: see text] leads to the accelerated propagation of nonautonomous solitons. Parabolic-, cubic-, exponential- and cosine-type nonautonomous solitons are exhibited due to the different choices of [Formula: see text]. For Case 2 [Formula: see text], if the real part of the spectral parameter equals 0, stationary soliton can be formed. If we take the harmonic external potential coefficient [Formula: see text] as a positive constant and let the real parts of the two spectral parameters be the same, bound-state-like structures can be formed, but there are only one attractive and two repulsive procedures. For Case 3 [[Formula: see text] and [Formula: see text] are taken as nonzero constants], head-on interaction, overtaking interaction and bound-state structure can be formed based on the signs of the two spectral parameters.

Nonautonomous solitons and interactions for a variable-coefficient resonant nonlinear Schrödinger equation

A variable-coefficient resonant nonlinear Schrödinger (vc-RNLS) equation is considered in this paper. Binary Bell polynomials are employed to obtain the bilinear form and multi-soliton solutions under the integrable conditions. Four types of nonautonomous solitons are derived including the parabolic soliton, compressed soliton, phase-shifted soliton and periodic soliton. Propagation dynamics for each type is analyzed in detail. Nonautonomous resonant and intermediate-state soliton interactions are found to be existent under certain conditions. Specially, periodic soliton interactions are discussed, which shows that the periodic dispersion has no effect on the generation of resonance and intermediate-state solitons. Those analysis might have the applications in optical communication systems with the black hole physics flavor.

Nonautonomous soliton, controllable interaction and numerical simulation for generalized coupled cubic–quintic nonlinear Schrödinger equations

The study of soliton interactions is a significance for improving pulse qualities in nonlinear optics. In this paper, a generalized coupled cubic–quintic nonlinear Schrodinger (GCCQNLS) equation with the group-velocity dispersion, fiber gain-or-loss and nonlinearity coefficient functions is studied, which describes the evolution of a slowly varying wave packet envelope in the inhomogeneous optical fiber. In particular, based on the similarity transformation, we report several families of nonautonomous wave solutions of the GCCQNLS equation. It is reported that there are possibilities to manipulate the interactions of nonautonomous wave solution through manipulating nonlinear and gain/loss functions. Interactions between the different-type bright two solitons have been asymptotically analyzed and presented. And, the two parabolic-type bright solitons propagating with the opposite directions both change their directions after the interaction. Interactions between the linear-, parabolic- and periodic-type bright two solitons are elastic. At last, the numerical simulations on the evolution and collision of two soliton solutions are performed to verify the prediction of the analytical formulations. We present the general approach can provide many possibilities to manipulate soliton waves experimentally and consider the potential applications for the optical self-routing, non-Kerr media and Bose–Einstein condensates (BEC).

Conservation laws, bilinear Bäcklund transformations and solitons for a nonautonomous nonlinear Schrödinger equation with external potentials

Under investigation in this paper is a nonautonomous nonlinear Schrödinger equation with external potentials, which can govern the dynamics of nonautonomous solitons in the nonlinear optical medium non-uniformly distributed in both the transverse and longitudinal directions. Based on the Lax pair, we present an infinite sequence of the conservation laws. Bilinear forms, bilinear Bäcklund transformations, one-, two- and N-soliton solutions under a known variable-coefficient constraint are generated via the Hirota method. With G(t)=0 and R(t)B(t) being a constant, amplitude of the soliton remains unvarying during the propagation, where t is the scaled time, G(t) is the gain/loss coefficient, B(t), the group velocity dispersion coefficient, and R(t), the nonlinearity coefficient. If we set G(t) ≠ 0 or R(t)B(t) as a variable, the amplitude becomes varying. Due to the different choices of the linear oscillator potential coefficient α(t), periodic-, parabolic-, S- and V-type solitons are observed. Meanwhile, we find that α(t) has no influence on the soliton amplitude. Interaction between the two amplitude-unvarying solitons and that between the two amplitude-varying ones are displayed, respectively. The velocity of a moving soliton always keeps varying.

Nonautonomous solitons, breathers and rogue waves for the Gross–Pitaevskii equation in the Bose–Einstein condensate

Under investigation in this paper is the Gross–Pitaevskii equation which describes the dynamics of the Bose–Einstein condensate. Lax pair, conservation laws and Darboux transformation (DT) are constructed. Nonautonomous solitons and breathers are derived based on the DT obtained. A kind of modulation instability process is generated. Nonautonomous rogue waves are obtained via the generalized DT. Influence of the nonlinearity, linear external potential, harmonic external potential, and spectral parameter on the propagation and interaction of the nonautonomous solitons, breathers and rogue waves is also discussed.Amplitude of the first-order nonautonomous soliton is proportional to the imaginary part of the spectral parameter and inversely proportional to the nonlinearity parameter. Linear external potential parameter affects the location of the first-order nonautonomous soliton. Head-on interaction, overtaking interaction and bound-state-like nonautonomous solitons can be formed based on the signs of the real parts of the spectral parameters. Quasi-periodic behaviors are exhibited for the nonautonomous breathers. If the harmonic external potential parameter is negative, quasi-period decreases along the positive time axis, with an increase in the amplitude and a compression in the width. Quasi-period decreases with the increase of the nonlinearity parameter. The second-order nonautonomous rogue wave can split into three first-order ones. Nonlinearity parameter has an effect on the amplitude of the rogue wave. Linear external potential parameter influences the location of the rogue wave, while harmonic external potential parameter affects the curved direction of the background.

Nonautonomous Solitons for the Coupled Variable-Coefficient Cubic-Quintic Nonlinear Schrödinger Equations with External Potentials in the Non-Kerr Fibre

AbstractVariable-coefficient nonlinear Schrödinger (NLS)-type models are used to describe certain phenomena in plasma physics, nonlinear optics, arterial mechanics, and Bose–Einstein condensation. In this article, the coupled variable-coefficient cubic-quintic NLS equations with external potentials in the non-Kerr fibre are investigated. Via symbolic computation, similarity transformations and relevant constraints on the coefficient functions are obtained. Based on those transformations, such equations are transformed into the coupled cubic-quintic NLS equations with constant coefficients. Nonautonomous soliton solutions are derived, and propagation and interaction of the nonautonomous solitons in the non-Kerr fibre are investigated analytically and graphically. Those soliton solutions are related to the group velocity dispersionr(x) and external potentialsh1(x) andh2(x,t). With the different choices ofr(x), parabolic, cubic, and periodically oscillating solitons are obtained; with the different choices ofh2(x,t), we can see the dromion-like structures and nonautonomous solitons with smooth step-like oscillator frequency profiles, to name a few.

Influence of generalized external potentials on nonlinear tunneling of nonautonomous solitons: Soliton management

Soliton control and management using generalized external potentials in an inhomogeneous fiber to the design of high speed optical devices and ultrahigh capacity transmission systems are investigated based on solving the variable-coefficient generalized nonautonomous nonlinear Schrödinger equation with the help of symbolic computation. We construct Lax pair for GNLS equation by means of AKNS method and two soliton solutions are obtained by virtue of the Darboux transformation. With symbolic computation, we manipulate the control parameters and external potentials to investigate the propagation behaviors of nonautonomous solitons. Moreover, the main evolution features of obtained two soliton solutions are exposed by some interesting figures through computer simulation. Especially, we analyze the influence of external potentials such as periodic, exponential and parabolic potential on soliton propagation. Finally, soliton propagation under the absence (vanishing) of external potential is also discussed. Obtained results confirmed that external potentials has strong influence on the soliton dynamics. Our results might provide a new method to achieve the soliton pulse compression while they passing through the potential barrier or well under the influence of external potentials.

Nonautonomous Matter-Wave Solitons in a Bose–Einstein Condensate with an External Potential

Nonautonomous matter-wave solitons in a Bose–Einstein condensate with an external potential are reported. Via the non-isospectral Ablowitz–Kaup–Newell–Segur system, the Gross–Pitaevskii equation is found to have the double Wronskian solutions. The verification of such solutions is finished through some double Wronskian identities. With the zero-potential Lax pair, the double Wronskian solutions give the nonautonomous N-soliton solutions that contain 2N parameters. For characterizing the asymptotic behavior of the bright two-soliton solutions, the explicit expressions of asymptotic solitons are given. Effects of the linear and harmonic potentials on the bound states between two matter-wave solitons are discussed.

Nonautonomous discrete bright soliton solutions and interaction management for the Ablowitz-Ladik equation.

We present the nonautonomous discrete bright soliton solutions and their interactions in the discrete Ablowitz-Ladik (DAL) equation with variable coefficients, which possesses complicated wave propagation in time and differs from the usual bright soliton waves. The differential-difference similarity transformation allows us to relate the discrete bright soliton solutions of the inhomogeneous DAL equation to the solutions of the homogeneous DAL equation. Propagation and interaction behaviors of the nonautonomous discrete solitons are analyzed through the one- and two-soliton solutions. We study the discrete snaking behaviors, parabolic behaviors, and interaction behaviors of the discrete solitons. In addition, the interaction management with free functions and dynamic behaviors of these solutions is investigated analytically, which have certain applications in electrical and optical systems.

Analytical non-autonomous wave solitons for the dispersive cubic-quintic Gross–Pitaevskii equation and the interactions

A high-order dispersive cubic-quintic Gross–Pitaevskii (HDCQGP) equation (a generalized variable coefficients nonlinear Schrödinger equation with the third and fourth-order and the cubic-quintic nonlinear terms) is considered, and is transformed into a standard cubic-quintic nonlinear Schrödinger equation (NLSE). By using the generalized tanh-function method, we study exact solutions of the HDCQGP equation with time-modulated potential and nonlinearity. In particular, based on the similarity transformation, we report several families of non-autonomous wave solutions of the HDCQGP equation with snaking behaviors and different amplitude surfaces. At last, we consider the numerical simulation of two solitons collision for the NLSE with different parameters. These results may raise the possibility of relative experiments and potential applications.

Investigation on nonautonomous soliton management in generalized external potentials via dispersion and nonlinearity

We investigate the controllable behavior of nonautonomous soliton in external potentials with variable dispersion and nonlinearity management functions, which describes the propagation of optical pulses in an inhomogeneous fiber system. We derive the Lax pair with a variable spectral parameter and the exact multi-soliton solution is generated via Darboux transformation. Based on these solutions, several novel optical solitons are constructed by selecting appropriate functions and the main evolution features of these waves are shown by some interesting figures with computer simulation. As few examples, breathers in periodic potential, soliton compression in an exponentially dispersion decreasing fiber and interaction of boomerang solitons are discussed. The presented results have applications in the study of nonautonomous soliton birefringence-managed switching architecture. These results are potentially useful in the management of nonautonomous soliton with external potentials in the optical soliton communications and long-haul telecommunication networks.

Nonautonomous solitons in modified inhomogeneous Hirota equation: soliton control and soliton interaction

We have investigated the femtosecond soliton propagation in inhomogeneous fiber, which is described by the modified inhomogeneous Hirota equation with variable coefficient (MIH-vc). With the aid of AKNS method, corresponding Lax pair is constructed. By virtue of the Darboux transformation method and symbolic computation, the analytic one- and two-soliton solutions are explicitly obtained. Using obtained solutions, we graphically discuss the features of femtosecond solitons in modified inhomogeneous Hirota system by changing the profile of variable coefficients. We analyze various form of group velocity dispersion, third order dispersion and nonlinearity parameter for periodic amplification system, exponentially distributed system, parabolic solitons, periodic exponentially modulated system, which will be observable in the future experiments. These results are potentially useful in future experiments and soliton control for long-distance optical communication. Finally, the soliton solutions of the MIH-vc equation in double Wronskian form is constructed and further verified using the Wronskian technique by substitute in bilinear equations.

Nonautonomous spatiotemporal solitons in inhomogeneous nonlinear media with linear and nonlinear gain/loss and different distributed diffraction/dispersion

We analytically derive nonautonomous spatiotemporal soliton solutions of the (3 + 1)-dimensional inhomogeneous nonlinear Schrödinger equation with linear and nonlinear gain/loss and different distributed diffraction/dispersion from two kinds of variable transformation connected with a solvable differential equation. Diffraction/dispersion and nonlinearity play important roles in the evolutional characteristics such as amplitude, width, time shift, linear and nonlinear chirp phase, phase offset and phase shift. Diffraction and dispersion management of spatiotemporal solitons with and without nonlinear chirp factor is discussed in a decreasing diffraction/dispersion fiber with exponential, hyperbolic, logarithmic, Gaussian and linear profiles and a periodic modulated system. In a decreasing diffraction/dispersion fiber with exponential profile, the nonlinear chirp factor affects the trace of propagation and the amplitude of the soliton, while the nonlinear chirp factor only influences the amplitude of the soliton in other media. Moreover, we discuss the impact of nonlinear chirp phase on the total phase of spatiotemporal bright and dark solitons.