Modified Hirota Method Research Articles

Hirota [formula omitted]-operator forms, multiple soliton waves, and other nonlinear patterns of a 2D generalized Kadomtsev–Petviashvili equation

In the present paper, extensive research is conducted on a 2D generalized Kadomtsev–Petviashvili (2D-gKP) equation which models water waves with long wavelengths. The study starts by establishing Hirota D-operator forms of the governing equation using the Bell polynomial method (BPM). Through checking the three-soliton condition for the 2D-gKP equation, its integrability is then examined, and as a consequence, single-, double-, and (special) triple-soliton waves are derived from the modified Hirota method. In the end, after deriving lump and breather waves of the 2D-gKP equation through ansatz methods, a theoretical discussion along with its proof is presented. Attempts have been made to elucidate the physical features of nonlinear waves by displaying such waves in 2D and 3D postures. The paper’s findings contribute certainly to research on nonlinear waves of KP-type equations.

Propagation characteristics of bright and mixed solitons based on the variable coefficient (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation

In the study of telecommunication system, the variable coefficient (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation is used as the optical solitons transmission model, which not only explains the physical meaning of the existing model with quintic terms, but also has more nonlinear dynamics characteristics of the higher dimensional system than the lower dimensional system. In this paper, the analytical soliton solutions of the (3+1)-dimensional cubic-quintic CGL equations with variable coefficients are obtained by using the modified Hirota method. By selecting certain parameters of the nonlinear coefficients and spectral filtering terms, a special kind of mixed soliton solution is obtained, which has the characteristics of bright soliton, dark soliton and kinked soliton at the same time. Subsequently, the influence of changing the nonlinear, spectral filtering, linear loss parameters and other parameters on the transmission characteristics of solitons is discussed respectively, so as to realize the control of optical solitons, which can not only control the propagation of optical solitons in different forms, but also can realize the adjustment of the amplitude and pulse width of the pulse and control the propagation direction and energy of the pulse for the mixed solitons of a particular form. The research results of high dimensional CGL system in this paper can be applied to nonlinear optical system, ultra-fast optical digital logic system and other different experiments and application fields.

Soliton Rectangular Pulses and Bound States in a Dissipative System Modeled by the Variable-Coefficients Complex Cubic-Quintic Ginzburg–Landau EquationSupported by the National Natural Science Foundation of China (Grant Nos. 11875008 and 12075034), and the Beijing University of Posts and Telecommunications Excellent Ph.D. Students Foundation (Grant No. CX2021129).

The complex cubic-quintic Ginzburg–Landau equation (CQGLE) is a universal model for describing a dissipative system, especially fiber laser. The analytic one-soliton solution of the variable-coefficients CQGLE is calculated by a modified Hirota method. Then, phenomena of soliton pulses splitting and stable bound states of two solitons are investigated. Moreover, rectangular dissipative soliton pulses of the variable-coefficients CQGLE are realized and controlled effectively in the theoretical research for the first time, which breaks through energy limitation of soliton pulses and is expected to provide theoretical basis for preparation of high-energy soliton pulses in fiber lasers.

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.

Dynamics of coupled mode solitons in bursting neural networks.

Using an electrically coupled chain of Hindmarsh-Rose neural models, we analytically derived the nonlinearly coupled complex Ginzburg-Landau equations. This is realized by superimposing the lower and upper cutoff modes of wave propagation and by employing the multiple scale expansions in the semidiscrete approximation. We explore the modified Hirota method to analytically obtain the bright-bright pulse soliton solutions of our nonlinearly coupled equations. With these bright solitons as initial conditions of our numerical scheme, and knowing that electrical signals are the basis of information transfer in the nervous system, it is found that prior to collisions at the boundaries of the network, neural information is purely conveyed by bisolitons at lower cutoff mode. After collision, the bisolitons are completely annihilated and neural information is now relayed by the upper cutoff mode via the propagation of plane waves. It is also shown that the linear gain of the system is inextricably linked to the complex physiological mechanisms of ion mobility, since the speeds and spatial profiles of the coupled nerve impulses vary with the gain. A linear stability analysis performed on the coupled system mainly confirms the instability of plane waves in the neural network, with a glaring example of the transition of weak plane waves into a dark soliton and then static kinks. Numerical simulations have confirmed the annihilation phenomenon subsequent to collision in neural systems. They equally showed that the symmetry breaking of the pulse solution of the system leaves in the network static internal modes, sometime referred to as Goldstone modes.

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.

Dark soliton control in inhomogeneous optical fibers

Analytic two-dark soliton solutions for a variable–coefficient nonlinear Schrödinger equation are obtained via modified Hirota method. Parallel solitons are observed and soliton control such as the soliton compression is realized with different group velocity dispersion profiles. Besides, soliton interactions are investigated with the interaction distance being adjusted. In addition, soliton repulsive structures as well as attractive ones are obtained with exponential dispersion profile. Results in our research may be useful for the soliton control in inhomogeneous optical fibers, which will be a benefit to the realistic optical communication systems.

Analytic soliton solutions of cubic‐quintic Ginzburg‐Landau equation with variable nonlinearity and spectral filtering in fiber lasers

In fiber lasers, the study of the cubic‐quintic complex Ginzburg‐Landau equations (CGLE) has attracted much attention. In this paper, four families (kink solitons, gray solitons, Y‐type solitons and combined solitons) of exact soliton solutions for the variable‐coefficient cubic‐quintic CGLE are obtained via the modified Hirota method. Appropriate parameters are chosen to investigate the properties of solitons. The influences of nonlinearity and spectral filtering effect are discussed in these obtained exact soliton solutions, respectively. Methods to amplify the amplitude and compress the width of solitons are put forward. Numerical simulation with split‐step Fourier method and fourth‐order Runge‐Kutta algorithm are carried out to validate some of the analytic results. Transformation from the variable‐coefficient cubic‐quintic CGLE to the constant coefficients one is proposed. The results obtained may have certain applications in soliton control in fiber lasers, and may have guiding value in experiments in the future. image

Dromion-like structures and stability analysis in the variable coefficients complex Ginzburg–Landau equation

The study of the complex Ginzburg–Landau equation, which can describe the fiber laser system, is of significance for ultra-fast laser. In this paper, dromion-like structures for the complex Ginzburg–Landau equation are considered due to their abundant nonlinear dynamics. Via the modified Hirota method and simplified assumption, the analytic dromion-like solution is obtained. The partial asymmetry of structure is particularly discussed, which arises from asymmetry of nonlinear and dispersion terms. Furthermore, the stability of dromion-like structures is analyzed. Oscillation structure emerges to exhibit strong interference when the dispersion loss is perturbed. Through the appropriate modulation of modified exponent parameter, the oscillation structure is transformed into two dromion-like structures. It indicates that the dromion-like structure is unstable, and the coherence intensity is affected by the modified exponent parameter. Results in this paper may be useful in accounting for some nonlinear phenomena in fiber laser systems, and understanding the essential role of modified Hirota method.

Soliton amplification in gain medium governed by Ginzburg–Landau equation

With the modified Hirota method, analytic soliton solutions for the generalized cubic complex Ginzburg–Landau equation with variable coefficients are derived for the first time. Based on the analytic solutions, soliton amplification is realized by choosing corresponding parameters properly. Besides, physical effects affecting the soliton amplification are discussed. Furthermore, stability analysis is presented. Results in this paper may be of value in further understanding the soliton amplification in fiber laser, and helpful for the generation of supercontinuum.

Symbolic computation on pulse compression without pedestal in an inhomogeneous medium

The coupled nonlinear Schrodinger equations with the harmonic potential and variable coefficients are studied for the pulse propagation in an inhomogeneous medium. With the modified Hirota method and symbolic computation, the bilinear form and analytic one-soliton solutions are obtained. A type of pulse compression technique is proposed, which can have the optical pulses compressed without any external devices. Moreover, the compressed pulses are pedestal free. The influences of the inhomogeneity of the refractive index, Kerr nonlinearity and diffraction are analyzed as well. The proposed technique may provide a different method for the pulse compression.

Symbolic computation of solitons in the normal dispersion regime of inhomogeneous optical fibres

A nonlinear Schrödinger equation with varying dispersion, nonlinearity and gain (or absorption) is studied for ultrashort optical pulses propagating in inhomogeneous optical fibres in the case of normal dispersion. Using the modified Hirota method and symbolic computation, the bilinear form and analytic soliton solution are derived. Stable bright and dark solitons are observed in the normal dispersion regime. A periodically varying soliton and compressed soliton without any fluctuation are obtained. Combined and kink-shaped solitons are observed. Possibly applicable soliton control techniques, which are used to design dispersion-managed systems, are proposed. The proposed techniques may find applications in soliton management communication links, soliton compression and soliton control.

A new approach to the analytic soliton solutions for the variable-coefficient higher-order nonlinear Schrödinger model in inhomogeneous optical fibers

In this paper, for the propagation of the ultra-short optical pulses in the normal dispersion regime of inhomogeneous optical fibers, the variable-coefficient higher-order nonlinear Schrödinger equation is investigated. A bilinear form and analytic soliton solutions are obtained with the help of the modified Hirota method and symbolic computation. Through choosing the different dispersion profiles of the inhomogeneous optical fibers, relevant properties of the soliton solution are graphically discussed. Parabolic-type evolution of the soliton is observed. Additionally, periodic and s-shaped solitons are derived, respectively. Finally, a possibly applicable compression technique for the dark soliton is proposed. The results might be of potential application to soliton control, soliton compression, signal amplification and dispersion management.

Transmission of solitary pulse in dissipative nonlinear transmission lines

A class of dissipative complex Ginzburg–Landau (DCGL) equations that govern the wave propagation in dissipative nonlinear transmission lines is solved exactly by means of the Hirota bilinear method. Two-soliton solutions of the DCGL equations, from which the one-soliton solutions are deduced, are obtained in analytical form. The modified Hirota method imposes some restrictions on the coefficients equations. Namely, the second-order dispersion must be real. The physical requirement of the solutions imposes complementary conditions on the combination of the dispersion and nonlinear gain/loss terms of the equation, as well as on the coefficient of the Kerr nonlinearity. The analytical solutions for one-solitary pulses are tested in direct simulations.

Coupled modified Kadomtsev–Petviashvili equations in dispersive elastic media

The present work considers two-dimensional non-linear wave propagation in an infinite, homogeneous micropolar elastic medium. The reductive perturbation method is directly applied to a Lagrangian whose Euler–Lagrange equations give the field equations for a geometrically non-linear micropolar elastic medium. It is shown that the behavior of non-linear waves in the long-wave approximation is governed by the two-coupled modified Kadomtsev–Petviashvili equations. Travelling wave solutions of the non-linear evolution equations are considered by means of a modified Hirota method.

Nonlinear Transverse Waves in a Generalized Elastic Solidand the Complex Modified Korteweg–deVries Equation

By using an asymptotic expansion technique based on the Lagrangiandensity formulation, it is demonstrated that the nonlinear interactionbetween two transverse waves propagating in a generalized elasticsolid is described asymptotically by the complex modifiedKorteweg–deVries (CMKdV) equation. Also, the CMKdV equation isshown to be Hamiltonian and its solitary wave solutions areinvestigated by means of a modified Hirota method.

Four-wave interaction equations: Hamiltonian formulation, conservation laws and the Hirota method

Some of the analytical properties of the four-wave interaction equations derived previously by the present authors are studied. The equations are shown to be Hamiltonian and to have, at least, four conserved densities. Solitary wave solutions of the equations are also considered by means of a modified Hirota method.