Variable-coefficient Nonlinear Schrodinger Research Articles

Effects of dispersion terms on optical soliton propagation in a lossy fiber system

In this paper, a variable-coefficient nonlinear Schrodinger equation that describes the optical soliton propagation in dispersion management fiber systems is studied. Two- and three-soliton solutions are obtained by using the Hirota bilinear method. Based on those solutions, the effects of related parameters on optical soliton propagation are discussed. By choosing different values of the third-order dispersion, the amplification of optical solitons can be realized. In addition, the interactions among the solitons can be reduced by setting a proper value of the group velocity dispersion. The results of this paper may be helpful to design optical amplifiers or to improve the quality of optical communications.

Interactions among solitons for a fifth-order variable coefficient nonlinear Schrödinger equation

A fifth-order variable coefficient nonlinear Schrodinger equation based on original inhomogeneous model is proposed and studied. Bright multi-soliton analytic solutions of the equation are calculated through the Hirota method and auxiliary function. The effect of different constraints between fifth-order dispersion with third-order dispersion on soliton transmission is researched. Besides, their propagation and interaction dynamics are analyzed. Moreover, based on the obtained three-soliton solutions, we change the value of $$\beta (x)$$ to discuss the soliton propagation. Three different propagation and interaction structures are derived via adjusting the nonlinear term. The results can enrich the inhomogeneous model and apply in nonlinear optical fiber.

Phase-shift controlling of three solitons in dispersion-decreasing fibers

Phase-shift controlling can attenuate the interactions between solitons and gives practical advantage in optical communication systems. For the variable-coefficient nonlinear Schrodinger equation, which can be imitated the transmission of solitons in the dispersion-decreasing fiber, analytic three solitons solutions are derived via the Hirota method. Based on the obtained solutions, influences of the second-order dispersion parameters and other related parameters in different function types on the soliton transmission are discussed. Results declare that phase-shift controlling of solitons in dispersion-decreasing fiber can be achieved when the dispersion function is Gaussian one. In addition, by adjusting the constraint value, propagation distance of solitons can be further extended. This may be useful in the optical logic devices and ultra-short pulse lasers.

Periodic attenuating oscillation between soliton interactions for higher-order variable coefficient nonlinear Schrödinger equation

According to the change in the amplitude of the oscillation, it can be divided into equal-amplitude oscillation, amplitude-reduced oscillation (attenuating oscillation) and amplitude-increasing oscillation (divergence oscillation). In this paper, the periodic attenuating oscillation of solitons for a higher-order variable coefficient nonlinear Schrodinger equation is investigated. Analytic one- and two-soliton solutions of this equation are obtained by the Hirota bilinear method. By analyzing the soliton propagation properties, we study how to choose the corresponding parameters to control the soliton propagation and periodic attenuation oscillation phenomena. Results might be of significance for the study of optical communications including soliton control, amplification, compression and interactions.

Semirational rogue waves for the three coupled variable-coefficient nonlinear Schrödinger equations in an inhomogeneous multicomponent optical fibre

In this paper, we investigate the three coupled variable-coefficient nonlinear Schrodinger equations, which describe the amplification or attenuation of the picosecond pulse propagation in the inhomogeneous multicomponent optical fibre with different frequencies or polarisations. Based on the Darboux dressing transformation, semirational rogue wave solutions are derived. Semirational rogue waves, Peregrine combs and Peregrine walls are obtained and demonstrated. Splitting behaviour of the semirational Peregrine combs and attenuating phenomenon of the semirational Peregrine wall are exhibited. Effects of the group velocity dispersion, nonlinearity and fibre gain / loss are discussed according to the different fibres. We find that the maximum amplitude of the hump of the semirational rogue wave is less than nine times the background height due to the interaction between the soliton part and rogue wave part. Further, there is a bent in the soliton part of the semirational rogue.

Control of interaction between femtosecond dark solitons in inhomogeneous optical fibers

In this paper, we present exact femtosecond one- and two-dark soliton solutions for a variable-coefficient higher-order nonlinear Schrodinger equation via modified Hirota method. The propagation and interaction of femtosecond dark solitons are investigated in inhomogeneous fiber systems. Elastic collision, bound oscillation and parallel propagation can be achieved in both Gaussian distributed parameter system and exponentially periodic distributed parameter system by choosing the appropriate distributed parameters and soliton parameters. The results may be beneficial to the realization of interaction control of femtosecond dark solitons in communication systems.

Solitons and breather-to-soliton transitions for an integrable higher-order variable-coefficient nonlinear Schrödinger equation in an optical fiber

Under investigation in this paper is an integrable eighth-order variable-coefficient nonlinear Schrodinger equation in an optical fiber. One-, two-, three-soliton and the first-, second-order breather solutions are obtained via the Darboux transformation. Properties of the solitons are discussed graphically. Breather-to-soliton transitions are studied under certain constraints. Discussions indicate that the soliton amplitude is not related to the variable coefficients, but related to some spectral parameters, while the soliton velocity is related to both the variable coefficients and spectral parameters. We find that there are two types of the breather-to-soliton transitions, M-shaped and W-shaped, which are determined through the spectral parameters.

Nonautonomous multi-peak solitons and modulation instability for a variable-coefficient nonlinear Schrödinger equation with higher-order effects

In this paper, we study a variable-coefficient nonlinear Schrodinger (vc-NLS) equation with fourth-order effects describing an inhomogeneous one-dimensional continuum anisotropic Heisenberg ferromagnetic spin chain or alpha helical protein. The first-order nonautonomous breather solution of the fourth-order vc-NLS equation is derived. The state transition between nonautonomous breather and nonautonomous multi-peak soliton can be realized when group velocity dispersion (GVD) coefficient is proportional to the fourth-order dispersion (FOD) coefficient. We also display how the higher-order effects influence the nonautonomous multi-peak solitons. Our results show that the velocity and localization of the nonautonomous multi-peak soliton are affected by the FOD coefficient, and the peak number is controlled by the GVD coefficient. Further, we also show the compression effect and motion with variable velocity of nonautonomous multi-peak soliton in two kinds of dispersion management systems. Finally, we reveal the relation between the state transition and the modulation instability (MI) analysis.

Two-dimensional localized Peregrine solution and breather excited in a variable-coefficient nonlinear Schrödinger equation with partial nonlocality

Hierarchies of Peregrine solution and breather solution are derived in a (2+1)-dimensional variable-coefficient nonlinear Schrodinger equation with partial nonlocality. Based on these solutions, we study the control of the excitation of Peregrine solution and breather solution in different planes. In particular, the localized Peregrine solution and breather solution are firstly reported in two-dimensional space. It is expected that our analysis and results may give new insight into higher-dimensional localized rogue waves in nonlocal media.

Optical Solitons and Cnodial Waves in Two Kinds of Inhomogeneous Nonlinear Media with Constant and Variable Dispersions

Abstract Based on F -expansion method of variable-coefficient, we obtain nonautonomous optical solitons and cnodial waves constructed by Jacobian elliptic function solutions of variable-coefficient nonlinear Schrodinger equation. Further, we discuss optical solitons and cnodial waves in two kinds of inhomogeneous nonlinear media with constant and variable dispersions.

Solitons, breathers and rogue waves for a sixth-order variable-coefficient nonlinear Schrödinger equation in an ocean or optical fiber

Under investigation in this paper is a sixth-order variable-coefficient nonlinear Schrodinger equation in an ocean or optical fiber. Through the Darboux transformation (DT) and generalized DT, we obtain the multi-soliton solutions, breathers and rogue waves. Choosing different values of \(\alpha\)(x), \(\beta\)(x), \(\gamma\)(x) and \(\delta\)(x), which are the coefficients of the third-, fourth-, fifth- and sixth-order dispersions, respectively, we investigate their effects on those solutions, where x is the scaled propagation variable. When \(\alpha\)(x), \(\beta\)(x), \(\gamma\)(x) and \(\delta\)(x) are chosen as the linear, parabolic and periodic functions, we obtain the parabolic, cubic and quasi-periodic solitons, respectively. Head-on and overtaking interactions between the two solitons are presented, and the interactions are elastic. Besides, with certain values of the spectral parameter \(\lambda\), a shock region between the two solitons appears, and the interaction is inelastic. Interactions between two kinds of the breathers are also studied, and we find that the interaction regions are similar to those of the second-order rogue waves. Rogue waves are split into some first-order rogue waves when \(\alpha\)(x), \(\beta\)(x), \(\gamma\)(x) and \(\delta\)(x) are the periodic or odd-numbered functions.

Solitons and dromion-like structures in an inhomogeneous optical fiber

In this paper, a generalized higher-order variable-coefficient nonlinear Schrodinger equation is studied, which describes the propagation of subpicosecond or femtosecond pulses in an inhomogeneous optical fiber. We derive a set of the integrable constraints on the variable coefficients. Under those constraints, via the symbolic computation and modified Hirota method, bilinear equations, one-, two-,three-soliton solutions and dromion-like structures are obtained. Properties and interactions for the solitons are studied: (a) effects on the solitons resulting from the wave number k, third-order dispersion $$\delta _1(z)$$ , group velocity dispersion $$\alpha (z)$$ , gain/loss $$\varGamma _2(z)$$ and group-velocity-related $$\gamma (z)$$ are discussed analytically and graphically where z is the normalized propagation distance along the fiber; (b) bound state with different values of $$\alpha (z)$$ , $$\delta _1(z)$$ , $$\gamma (z)$$ and $$\varGamma _2(z)$$ are presented where some periodic or quasiperiodic formulae are derived. Interactions between the two solitons and between the bound states and a single soliton are, respectively, discussed; and (c) single, double and triple dromion-like structures with different values of $$\alpha (z)$$ , $$\delta _1(z)$$ , $$\gamma (z)$$ are also presented, distortions of which are found to be determined by those variable coefficients.

Soliton solution for the Landau-Lifshitz equation of a one-dimensional bicomponent magnonic crystal.

We investigate nonlinear localized magnetic excitations in a one-dimensional bicomponent magnonic crystal under a periodic magnetic field of spatially varying strength. The governing Landau-Lifshitz equation is transformed into a variable coefficient nonlinear Schrödinger (VCNLS) equation using stereographic projection. In general, the VCNLS equation is nonintegrable and by using Painlevé analysis, we obtain necessary conditions for the VCNLS equation to pass the Weiss-Tabor-Carnevale Painlevé test. A sufficient integrability condition is obtained by further exploring a transformation, which can map the VCNLS equation into the well-known standard nonlinear Schrödinger equation. The transformation builds a systematic connection between the solution of the standard nonlinear Schrödinger equation and VCNLS equation. The results show that the excitation of magnetization in the form of a soliton exists on the oscillatory background with a structure similar to the form of spin Bloch waves. Such a solution exists only when certain conditions on the coefficient of the VCNLS equation are satisfied. To corroborate the analytical results, we performed the numerical simulation by solving the governing VCNLS equation with integrability conditions using the split step Fourier method and the result agrees well with analytical results, and it suggests a way to control the dynamics of magnetization in the form of solitons by an appropriate spatial modulation of the nonlinearity coefficient in the governing VCNLS equation, which depends on the ferromagnetic materials which form the bicomponent magnonic crystal.

Soliton dynamics in the three-spine $$\alpha $$ α -helical protein with inhomogeneous effect

We study a three-coupled variable-coefficient nonlinear Schrodinger equation, which describes soliton dynamics in the three-spine \(\alpha \)-helical protein with inhomogeneous effect, and analytically obtain multi-soliton solutions, whose formation originates from the dynamical balance between the dispersion owing to the resonant interaction of intrapeptide dipole vibrations and the nonlinear interaction provided by those vibrations with the local displacements of the peptide groups. Using these analytical solutions as initial solutions, we discuss the dynamical behaviors of solitonic interactions and the influence of the protein inhomogeneity on shape-changing collisions of solitons by direct numerical simulations.

Dark soliton collisions for a fourth-order variable-coefficient nonlinear Schrödinger equation in an inhomogeneous Heisenberg ferromagnetic spin chain or alpha helical protein

Under investigation in this paper is a fourth-order variable-coefficient nonlinear Schrodinger equation, which can describe an inhomogeneous one-dimensional anisotropic Heisenberg ferromagnetic spin chain or alpha helical protein. With the aid of the Hirota method and symbolic computation, bilinear forms, dark one- and two-soliton solutions are obtained. Influences of the variable coefficients on the dark one and two solitons are graphically shown and discussed. Amplitude and shape of the dark one soliton keep invariant during the propagation when the variable coefficients are chosen as the constants. With the variable coefficients being the functions, amplitude of the dark soliton keeps unchanged during the propagation, but direction of the soliton curves. Head-on and overtaking collisions between the dark two solitons are displayed with the variable coefficients chosen as the constants, and it is shown that the shapes of the two solitons do not change during the collision. When we choose the variable coefficients as the functions, directions of the two solitons change and elastic collisions occur between the two solitons.

Transformations and Soliton Solutions for a Variable-coefficient Nonlinear Schrödinger Equation in the Dispersion Decreasing Fiber with Symbolic Computation

Describing the dispersion decreasing fiber, a variable-coe fficient nonlinear Schrodinger equation is hereby under investigation. Three transformations have been obtained from such a equation to the known standard and cylindrical nonlinear Schrodinger equations with the relevant constraints on the variable coefficients presented, which t urn out to be more general than those previously published in the literature. Meanwhile, several families of exact dark-soliton-like and bright-soliton-like solutions are constructed. Also, we o btain some similarity solutions, which can be illustrated in terms of the elliptic and the second Painle ve transcendent equations.

Localized structures of the (3+1)-dimensional nonlinear Schrödinger equation with different diffractions and power-law nonlinearities in PT-symmetric potentials

We study a (3+1)-dimensional variable-coefficient nonlinear Schrodinger equation with different diffractions and power-law nonlinearity in PT-symmetric potentials. Considering different PT-symmetric potentials, we obtain two kinds of analytical sech-type localized soliton solutions. From these solutions, we analytically discuss the powers and power-flow densities. Moreover, we study compression and expansion of localized structures in the periodic distributed amplification system.

Current-Induced Magnetization Dynamics in 1-D Bicomponent Magnonic Crystal

Nonlinear localized magnetic excitations in 1-D bicomponent magnonic crystal are investigated under periodic magnetic field of spatially varying strength with spin current. The governing modified Landau–Lifshitz equation with spin current is transformed into the variable coefficient nonlinear Schrodinger (VCNLS) equation using a stereographic projection. In general, the VCNLS equation is nonintegrable and by using similarity transformation technique, the VCNLS equation is reduced into the standard NLS equation with certain integrability conditions. A systematic connection between the solution of VCNLS and NLS equations is established. The solution of VCNLS equation is obtained using the solution of the NLS equation. The excitation of magnetization is in the form of solitons on the oscillatory background with structure similar to the form of spin Bloch waves, and in addition, the spin current controls the soliton velocity. The velocity of soliton is directly proportional to the strength of current density and inversely proportional to the average saturation magnetization value, which is the value of saturation magnetization of the composite material at the interface of the two ferromagnetic materials, which are used to form the bicomponent magnonic crystal.

Oscillating solitons in nonlinear optics

Oscillating solitons are obtained in nonlinear optics. Analytical study of the variable-coefficient nonlinear Schrodinger equation, which is used to describe the soliton propagation in those systems, is carried out using the Hirota’s bilinear method. The bilinear forms and analytic soliton solutions are derived, and the relevant properties and features of oscillating solitons are illustrated. Oscillating solitons are controlled by the reciprocal of the group velocity and Kerr nonlinearity. Results of this paper will be valuable to the study of dispersion-managed optical communication system and mode-locked fibre lasers.

Controllable combined Peregrine soliton and Kuznetsov–Ma soliton in $${\varvec{\mathcal {PT}}}$$ PT -symmetric nonlinear couplers with gain and loss

We investigate a (2+1)-dimensional-coupled variable coefficient nonlinear Schrodinger equation in parity time symmetric nonlinear couplers with gain and loss and analytically obtain a combined structure solution via the Darboux transformation method. When the imaginary part of the eigenvalue \(n\) is smaller or bigger than 1, we can obtain the combined Peregrine soliton and Akhmediev breather, or Kuznetsov–Ma soliton, respectively. Moreover, we study the controllable behaviors of this combined Peregrine soliton and Kuznetsov–Ma soliton structure in a diffraction decreasing system with exponential profile. In this system, the effective propagation distance \(Z\) exists a maximal value \(Z_m\) and the maximum amplitude of the KM soliton appears in the periodic locations \(Z_{i}\). By modulating the relation between values of \(Z_m\) and \(Z_i\), we realize the control for the excitation of the combined Peregrine soliton and Kuznetsov–Ma soliton, such as the restraint, maintenance, and postpone.