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The Schrödinger equation depends on the physical circumstance, which describes the state function of a quantum‐mechanical system and gives a characterization of a system evolving with time. The essential focus of proposed research is to observe the solution for fractional generalized nonlinear Schrödinger (FGNS) equation using q‐homotopy analysis transform method (q‐HATM). The fractional order derivative is taken in the Atangana‐Baleanu (AB) sense. The physical behaviours of achieved solution for FGNS equation are discussed and sketch out graphically. The existence of the solution for the FGNS equation is presented through theorems 4.1 to 4.3. The proposed numerical simulations confirm the advantages of the AB derivative through q‐HATM. Few numerical experiments were carried out to validate the proposed method. Moreover, numerical simulations are carried out to verify efficiency and robustness of the derived results by considering two cases. [ABSTRACT FROM AUTHOR]

In this study, the fractional variational iteration method using He’s polynomials is employed for constructing semi-analytical solutions of the fractional-in-time generalized nonlinear Schrödinger equation involving Jumarie’s modified Riemann–Liouville derivative. As a result, we obtain the solutions of the original problems in the forms of convergent series that are calculated easily. Our calculation results indicate that the method is quite efficient and convenient for these problems.

The generalized nonlinear Schrödinger equation with anti-cubic and cubic-quintic-septic nonlinearities is considered. A series of direct transformations is performed for the traveling wave reduction of the original equation. The implicit function method is used for nonlinear ordinary differential equation with some constraints on the parameters. Some analytical solutions are found in the form of periodic and solitary waves. Three conservation laws, corresponding to the generalized nonlinear Schrödinger equation are obtained and conserved quantities corresponding to the solution in implicit form are found [ABSTRACT FROM AUTHOR]

In the current paper, a generalized nonlinear Schrödinger (gNLS) equation with logarithmic nonlinearity is studied as a model for the propagation of optical pulses. More precisely, after applying a specific hypothesis for the solution of the governing equation, its Gaussian solitary wave is retrieved using the ansatz method. Some numerical simulations in two- and three-dimensional postures are presented to investigate the impact of different physical parameters on Gaussian solitary wave' dynamics. Results confirm that the physical parameters of the gNLS equation have a key role in controlling the dynamics of the Gaussian solitary wave. [ABSTRACT FROM AUTHOR]

A local differential quadrature method based on Fourier series expansion numerically solves the generalized nonlinear Schrodinger equation. For time integration, a Runge-Kutta fourth-order method is used. Matrix stability analysis is used to examine the method's stability. Three test problems involving the motion of a single solitary wave, the interaction of two solitary waves, and a solution that blows up in finite time, respectively, demonstrate the accuracy and efficiency of the provided method. Finally, the numerical results obtained from the presented method are compared with the exact solution and those obtained in earlier works available in the literature.

In this paper, we investigate several important properties of the generalized nonlinear Schrödinger equation with P T -symmetric δ -sech optical potentials, such as the phase breaking of P T -symmetric δ -sech potentials, and existence, dynamics and excitations of new soliton solutions. Specifically, we identify the fully real spectral region of the non-Hermitian Hamiltonian and observe the phase-breaking phenomenon. Additionally, we discover a new soliton solution that represents a peakon solution, a smooth soliton solution, and a double-hump soliton solution under different potential parameters. We analyze the stability of these three types of solutions and determine their stability domains. Furthermore, we study the numerical peakon solution and its stability. In particular, we investigate the interaction of peakon solutions and observe semi-elastic interactions. Finally, we explore the stable adiabatic excitations of peakons. These results contribute to a deeper understanding of P T -symmetric optical fields and provide insights for related experimental works. [ABSTRACT FROM AUTHOR]

Complex systems are prevalent in various disciplines encompassing the natural and social sciences, such as physics, biology, economics, and sociology. Leveraging data science techniques, particularly those rooted in artificial intelligence and machine learning, offers a promising avenue for comprehending the intricacies of complex systems without necessitating detailed knowledge of underlying dynamics. In this paper, we demonstrate that multiscale entropy (MSE) is pivotal in describing the steady state of complex systems. Introducing the multiscale entropy dynamics (MED) methodology, we provide a framework for dissecting system dynamics and uncovering the driving forces behind their evolution. Our investigation reveals that the MED methodology facilitates the expression of complex system dynamics through a Generalized Nonlinear Schrödinger Equation (GNSE) that thus demonstrates its potential applicability across diverse complex systems. By elucidating the entropic underpinnings of complexity, our study paves the way for a deeper understanding of dynamic phenomena. It offers insights into the behavior of complex systems across various domains. [ABSTRACT FROM AUTHOR]

The higher-order nonlinear Schrödinger equation with combined nonlinearities is derived by an asymptotic reduction from the (2+4) Korteweg–de Vries model for weakly nonlinear wave packets for the context of interfacial waves in a three-layer symmetric media. Focusing properties and modulation instability effects are discussed for the considered physical context. Instability growth rate, maximum of the increment and the boundaries of the instability interval are derived in terms of three-layer density stratification, their structure on the parameter planes of relative layer depth, carrier wavenumber and envelope amplitude, are considered in detail. [ABSTRACT FROM AUTHOR]

In optical fibers, the generalized nonlinear Schrödinger equations with self-steepening (SS), self-frequency shift (SFS), intermodal dispersion (IMD), and third-order dispersion (TOD) play an important role. Our investigation covers a variety of physical parameters based on how optical solitons change their structure as they move through an optical medium. Our study shows that modifying the coefficients for SS, SFS, IMD, and TOD can affect optical solitons' profiles either by altering their nature or without doing so. We used the extended rational sinh–cosh method, which works with various types of soliton profiles. These profiles include dark, kink-dark, kink, and anti-kink solitons. By selecting appropriate physical parameter values, the behavior of various optical solitons is graphically depicted. As a result, we utilize the eigenvalue spectrum to investigate linear stability analysis. [ABSTRACT FROM AUTHOR]

Today, nonlinear Schrödinger-type equations are the focus of explorers and scientists. Hereby, we look into a generalized nonlinear Schrödinger equation by constructing the modified generalized Darboux transformation and analysing the type-I, type-II and type-III degenerate solitons for generalized nonlinear Schrödinger equation via some semirational solutions. Type-I degenerate solitons refer to the degenerate solitons, type-II degenerate solitons mean the interactions between the solitons and the degenerate solitons, and type-III degenerate solitons denote the bound states among a series of the degenerate solitons. We hope that the mathematical research method used in this paper could provide some theoretical assistance for future research on the nonlinear Schrödinger-type equations. [ABSTRACT FROM AUTHOR]

Complex systems are prevalent in various disciplines encompassing the natural and social sciences, such as physics, biology, economics, and sociology. Leveraging data science techniques, particularly those rooted in artificial intelligence and machine learning, offers a promising avenue for comprehending the intricacies of complex systems without necessitating detailed knowledge of underlying dynamics. In this paper, we demonstrate that multiscale entropy (MSE) is pivotal in describing the steady state of complex systems. Introducing the multiscale entropy dynamics (MED) methodology, we provide a framework for dissecting system dynamics and uncovering the driving forces behind their evolution. Our investigation reveals that the MED methodology facilitates the expression of complex system dynamics through a Generalized Nonlinear Schrödinger Equation (GNSE) that thus demonstrates its potential applicability across diverse complex systems. By elucidating the entropic underpinnings of complexity, our study paves the way for a deeper understanding of dynamic phenomena. It offers insights into the behavior of complex systems across various domains.

The higher-order nonlinear Schrödinger equation with combined nonlinearities is derived by an asymptotic reduction from the (2+4) Korteweg–de Vries model for weakly nonlinear wave packets for the context of interfacial waves in a three-layer symmetric media. Focusing properties and modulation instability effects are discussed for the considered physical context. Instability growth rate, maximum of the increment and the boundaries of the instability interval are derived in terms of three-layer density stratification, their structure on the parameter planes of relative layer depth, carrier wavenumber and envelope amplitude, are considered in detail.

In this paper, we study the dynamics of spatiotemporal soliton solutions to a generalized nonlinear Schrödinger equation with inhomogeneous coefficients in analytical and numerical forms. The characteristics and physical applications of the soliton solutions are investigated under different modulations of the coefficients depicting diffraction, potential, nonlinearity, source, and gain. We find that structures of these soliton solutions can be effectively manipulated by appropriately selecting the source term. Besides, the impact of gate effect on soliton propagation is investigated by employing the hyper-Gaussian and rectangle functions as diffraction coefficients. The results show that (i) the hyper-Gaussian modulation can be replaced by the rectangle modulation for large exponent and (ii) the solitons undergo a broadening in width and a sudden decrease in amplitude when passing through the gate functions. Our findings may have potential applications for the investigation of soliton control in the fields of nonlinear optics and Bose–Einstein condensations.

We consider a generalized (3+1)-dimensional nonlinear Schrödinger with cubic-quintic nonlinear and self-frequency shift and self-steepening terms. We disrupt the plane wave to study the stability of the wave in this media. We examine how various factors—such as the amplitude of the plane wave, cubic and quintic nonlinear terms, self-steepening, and self-frequency shift—affect the modulational instability (MI). Our findings reveal that the amplitude of the plane wave and the quintic nonlinear term can expand the MI bands and increase the amplitude of the MI growth rate. Conversely, the self-steepening and self-frequency shift terms exert opposite effects, narrowing the MI bands and reducing the amplitude of the MI growth rate. We investigate the existence of modulated chirped rational and polynomial Jacobi elliptic function solutions and chirped optical solitons. [ABSTRACT FROM AUTHOR]

We provide Python tools enabling numerical simulation and analysis of the propagation dynamics of ultrashort laser pulses in nonlinear waveguides. The modeling approach is based on the widely used generalized nonlinear Schrödinger equation for the pulse envelope. The presented software implements the effects of linear dispersion, pulse self-steepening, and the Raman effect. The focus lies on the implementation of input pulse shot noise, i.e. classical background fields that mimic quantum noise, which are often not thoroughly presented in the scientific literature. We discuss and implement commonly adopted quantum noise models based on pure spectral phase noise, as well as Gaussian noise. Coherence properties of the resulting spectra can be calculated. We demonstrate the functionality of the software by reproducing results for a supercontinuum generation process in a photonic crystal fiber, documented in the scientific literature. The presented Python tools are open-source and released under the MIT license in a publicly available software repository.

The family of the generalized Schrödinger equations with Kerr nonlinearity of unrestricted order is considered. The solutions of equations are looked for using traveling wave reductions. The Painlevé test is applied for finding arbitrary constants in the expansion of the general solution into the Laurent series. It is shown that the equation does not pass the Painlevé test but has two arbitrary constants in local expansion. This fact allows us to look for solitary wave solutions for equations of unrestricted order. The main result of this paper is the theorem of existence of optical solitons for equations of unrestricted order that is proved by direct calculation. The optical solitons for partial differential equations of the twelfth order are given in detail. [ABSTRACT FROM AUTHOR]

In this paper, an efficient nonconforming mixed finite element method (MFEM) is studied with E Q 1 rot element and zero-order Raviart–Thomas element for a generalized nonlinear Schrödinger equation. On the one hand, by introducing p → = ∇ u as an intermediate variable, we split the equation into two low-order equations and present a semi-discrete scheme for nonconforming MFEM and prove its existence and uniqueness. And the superconvergence results for original variable u in broken H 1 -norm and flux p → = ∇ u in L 2 norm are derived by using the proved properties of the above nonconforming MFEs. On the other hand, a linearized Crank–Nicolson fully discrete scheme is constructed and the superclose and superconvergence results with order O (h 2 + τ 2) for above variables are also derived. The keys to our analysis are the following two skills: one is an important novel property for the above MFEs (see Lemma 2.3) and another is a splitting technique for nonlinear terms, while previous literature always only obtain the convergent estimates in a routine way. Finally, two numerical examples are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and τ is the time step. [ABSTRACT FROM AUTHOR]

Research on the shear horizontal (SH) waves propagating in a nonlinear elastic half-space coated with a nonlinear elastic layer having slow variation in the boundary surfaces is presented. It is assumed that both the free surface and interface change as a function of the distance in the direction of propagation of the waves. By employing a perturbation method, nonlinear self-modulation of the surface SH waves for a general geometry is characterized by a generalized nonlinear Schrödinger (GNLS) equation with variable coefficients depending on functions representing the irregularities of boundary surfaces in addition to linear and nonlinear material parameters of the medium. The solitary wavelike solutions of this GNLS equation are obtained for certain special cases of the irregular boundary surfaces. It has been demonstrated by graphs that the propagation characteristics of SH waves are significantly affected by slowly varying layer and nonlinear properties of the media. [ABSTRACT FROM AUTHOR]

Internal solitary waves are very common physical phenomena in the ocean, which play an important role in the transport of marine matter, momentum and energy. Because the generalized nonlinear Schrödinger equation can well explain the effects of nonlinearity and dispersion in the ocean, it is more suitable for describing the deep-sea internal wave propagation and evolution than other mathematical models. At first, by designing skillfully the trial-Lagrange functional, different kinds of variational principles are successfully established for a generalized nonlinear Schrödinger equation by the semi-inverse method. Then, the constructed variational principles are proved correct by minimizing the functionals with the calculus of variations. Furthermore, some kinds of internal solitary wave solutions are obtained and demonstrated by semi-inverse variational principle for the generalized nonlinear Schrödinger equation.

We investigate the generalized nonlinear Schrödinger equation (NLSE) of third order analytically, that accept the single-parameter family of one hump embedded soliton. This equation has been utilized to model ultra-short pulses in optical fibers. The physical phenomenon of this dynamical equation is revealed through the exact solutions for example solitons, elliptic function solutions, etc. These types of novel explicit solutions are constructed of generalized third-order NLSE via the modified extended direct algebraic method (MEDAM), that have key applications in engineering and applied sciences. Movements of some achieved solutions are described graphically, that assists to know the physical interpretation of this equation. Furthermore, the configuration conditions of dark-bright solitons are also presented. The influence of this scheme is shown in computational work and achieved results. [ABSTRACT FROM AUTHOR]

This article deals with new soliton solutions of the generalized nonlinear Schrodinger equation with variable coefficients by the direct algebraic method. Once the variables of this technique are considered as special values. we could achieve the solitary waves that are unique from those attained by the other methods. It can be inferred that the solution methodology in conjunction with symbolic computing eligible to solve nonlinear partial differential equations precisely. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is the generalized nonlinear Schrödinger (g-NLS) equation which has extensive applications in various physical fields. Firstly, we prove Liouville integrability of this equation by deriving its bi-Hamiltonian structures applying the variational identity. Nextly, we calculate the modulational instability for the possible reason of the formation of the rogue wave. Moreover, based on the generalized (2 , N - 2) -fold Darboux transformation (DT), we can derive several mixed localized wave solutions such as breathers, rogue waves and semi-rational solitons for this equation, and accurately analyze a lot of important physical quantities. Finally, we present these solutions graphically by choosing appropriate parameters and discuss their dynamic behavior. It is worth noting that all of these solutions can change from a strong interaction to a weak interaction by choosing the parameters. This may also be one of the reasons why relevant wave structures presenting diversity, and useful to explain some physical phenomena in nonlinear optics. [ABSTRACT FROM AUTHOR]

We propose a numerical model to simulate amplification in a pulsed synchronously pumped chirped pulse amplifier (CPA) operating at a low repetition rate. The generalized nonlinear Schrödinger equation and dynamic rate equations are employed to facilitate an accurate treatment of time-space-dependent-gain, wavelength-dependent saturation energy, nonlinearity, and dispersion. Because of an accurate instantaneous gain term, the model can adequately explain pulse distortion and spectral redshift during amplification. The bidirectional amplified spontaneous emission, final output energy, and evolution of temporal and spectral intensities are also investigated through simulations. In addition, a pulsed synchronously pumped CPA fiber system is estbalished to validate the model. The experimental results for the wave shape, spectrum and energy show excellent agreement with those of simulations.

Slot waveguides comprised of Ge23Sb7S70 and Si with and without graphene are considered for super‐continuum generation in the mid‐infrared range. The dispersion characteristics are optimized for the proposed waveguides comprised of Si and Ge23Sb7S70 chalcogenide as the core and cladding, respectively with and without graphene sheets embedded in the proposed structure. The proposed waveguide without graphene possesses all‐normal dispersion and the super‐continuum spectra with bandwidth of 1.75 µm is obtained for a pump with wavelength of 2 µm, peak power of 100 W, and time duration of 50 fs, at 20 dB level. Also, the total length of the waveguide is 8 mm. The optimized structure with three graphene sheets presents flat anomalous dispersion and the super‐continuum spectra with bandwidth of 3 µm is obtained for a pump with wavelength, peak power, and time duration of 3 µm, 7 mW, and 50 fs, respectively at 10 dB level. It is demonstrated that the proper dispersion region for super‐continuum generation depends on the sign of Kerr‐index of the nonlinear material. The proposed waveguides are highly compatible with fabrication technologies for mid infrared applications. [ABSTRACT FROM AUTHOR]

We study the modulation instability (MI) for optical wave propagation in the presence of slowly varying saturable nonlinearity, group velocity dispersion and a PT-symmetric external potential over the length of tapered graded-index waveguide. First we study the MI with space-dependent dispersion and saturated nonlinearity and then with constant dispersion and saturated nonlinearity. We analytically compute the growth rate and analyze the possibility of existence of MI for focussing and defocussing nonlinearities for different choices of group velocity dispersion terms and saturated nonlinearities. We observe that MI is independent of PT symmetric potential term and typically depends upon the choice of group velocity dispersion term, nonlinearity, saturated nonlinearity and wave amplitude. MI gain decreases with increase in cw amplitude however it increases with increase in saturated nonlinearity both for focussing and defocussing nonlinearities. [ABSTRACT FROM AUTHOR]

In this paper, we find analytic solutions of generalized nonlinear Schrödinger equation (GNLSE), the underlying equation for studying pulse propagation through optical waveguides, by implementing the extended Sinh-Gordon equation expansion method (ShGEEM). Here, the terms corresponding to second and third order dispersion, waveguide’s loss, Kerr Effect, self-steepening and Raman Effect are included in GNLSE. The obtained analytic solutions represent bright, dark and combined bright-dark solitons. The constraint conditions for the existence of valid solutions are given. Also, the new solutions are graphically shown for pulse propagation along a photonic crystal fiber.

The family of the generalized Schrödinger equations with Kerr nonlinearity of unrestricted order is considered. The solutions of equations are looked for using traveling wave reductions. The Painlevé test is applied for finding arbitrary constants in the expansion of the general solution into the Laurent series. It is shown that the equation does not pass the Painlevé test but has two arbitrary constants in local expansion. This fact allows us to look for solitary wave solutions for equations of unrestricted order. The main result of this paper is the theorem of existence of optical solitons for equations of unrestricted order that is proved by direct calculation. The optical solitons for partial differential equations of the twelfth order are given in detail.

Internal solitary waves are very common physical phenomena in the ocean, which play an important role in the transport of marine matter, momentum and energy. Because the generalized nonlinear Schrödinger equation can well explain the effects of nonlinearity and dispersion in the ocean, it is more suitable for describing the deep-sea internal wave propagation and evolution than other mathematical models. At first, by designing skillfully the trial-Lagrange functional, different kinds of variational principles are successfully established for a generalized nonlinear Schrödinger equation by the semi-inverse method. Then, the constructed variational principles are proved correct by minimizing the functionals with the calculus of variations. Furthermore, some kinds of internal solitary wave solutions are obtained and demonstrated by semi-inverse variational principle for the generalized nonlinear Schrödinger equation. [ABSTRACT FROM AUTHOR]

We propose a numerical model to simulate amplification in a pulsed synchronously pumped chirped pulse amplifier (CPA) operating at a low repetition rate. The generalized nonlinear Schrödinger equation and dynamic rate equations are employed to facilitate an accurate treatment of time-space-dependent-gain, wavelength-dependent saturation energy, nonlinearity, and dispersion. Because of an accurate instantaneous gain term, the model can adequately explain pulse distortion and spectral redshift during amplification. The bidirectional amplified spontaneous emission, final output energy, and evolution of temporal and spectral intensities are also investigated through simulations. In addition, a pulsed synchronously pumped CPA fiber system is estbalished to validate the model. The experimental results for the wave shape, spectrum and energy show excellent agreement with those of simulations. [ABSTRACT FROM AUTHOR]

A generalized nonlinear Schrödinger equation with higher-order dispersive and nonlinear effects is presented to govern the pulse propagation in negative index materials. Via the mapping method, Jacobian elliptic function solutions and their corresponding soliton solutions are found. Parameter modulation of higher-order dispersive effect and nonlinearities for periodic waves and solitons is studied. With the addition of the third dispersion parameter, or the quintic nonlinear parameter, the change of real and imaginary parts of periodic waves and solitons is same; namely, their amplitudes increase, and yet their widths decrease. Moreover, phase transitions of solutions with parameter modulation are found. [ABSTRACT FROM AUTHOR]

This paper introduces the fractal form of the generalized nonlinear Schrödinger equation, newly named as the Biswas–Milovic model (BM). The BM equation theoretically explains the transmission of solitons for transatlantic and transcontinental distances utilizing optical fibers. The BM equation relating to Kerr law, parabolic law and nonlinearity quadratic law was studied using a variational approach for optical soliton solutions. Essential novel conditions are presented that guarantee the existence of the appropriate solitons. Besides, the physical action of the solution obtained was recorded in terms of 3D and contour plots for distinct parameters for the three different nonlinearities. This study shows the relevance and huge potential of the variational approach to the generalized nonlinear Schrödinger equation. [ABSTRACT FROM AUTHOR]

We prove theoretically, that the evolution of optical solitons can be dramatically influenced in the course of nonlinear interaction with much smaller group velocity matched pulses. Even weak pump pulses can be used to control the solitons, e.g., to compensate their degradation caused by Raman-scattering. [ABSTRACT FROM AUTHOR]

Modulational collapse of a wave pulse in a complex nonlinear plasma described by a complex nonlinear Schrödinger equation (CNSE) including the effects of viscous heating and nonlinear damping, and the adjusting by an external potential are investigated. Theoretical and numerical results reveal that the original pulse first suffers modulational instability, the growth rate of the modulational instability increases with the increase of viscous heating, but it is not sensitive to the nonlinear damping. The instability induces the self-similar collapsing, then the fields rapidly transform into a turbulent state with short-wavelength modes. The results are explained in terms of the evolution of the total energy, which can become invariant during the evolution even though the system is nonconservative. The external potential can delay the self-defocusing process of the pulse. Those results can apply to the phenomena described by the CNSE. [ABSTRACT FROM AUTHOR]

In this paper, the dispersion coefficients and the nonlinear parameter of a wagon wheel fiber is obtained by finite element method. Efficient broadband near infrared supercontinuum generation is predicted in a 15 cm of the single-mode small core silicon wagon wheel fiber by solving the generalized nonlinear Schrödinger equation when the fiber is pumped by femtosecond pulses in the abnormal dispersion regime. Here, it is shown that the flattened supercontinuum spectrum over 850 nm wide is achieved. [ABSTRACT FROM AUTHOR]

• Numerical study of solitonic nonlinear directional coupler in a dual-core fibre. • High-contrast nonlinear glass dual-core fibre optimization from aspect of geometry. • Advantageous switching performance by self-trapping of femtosecond solitons. • Near IR excitation wavelength and fs pulse width effect on the switching performance. • Predicted 46 dB time confined switching contrast at pulse parameters 1500 nm/75 fs. A systematic numerical study of ultrafast nonlinear directional coupler performance based on soliton self-trapping in a novel type of dual-core optical fibre is presented. The considered highly nonlinear fibre structure is composed of a real, intentionally developed soft glass-pair with high refractive index contrast at the level of 0.4 in the near infrared. Nonlinear propagation of picojoule level femtosecond pulses was studied numerically with the aim to identify the best switching performance in input parameter space of 1400–1800 nm in terms of excitation wavelengths, and of 75–150 fs in terms of pulse width, respectively. For every combination of excitation wavelength and pulse width, the switching energies together with the optimal fibre length were determined and their relation to the input and switching parameters is discussed. The highest switching contrast of 46 dB in the time window of the ultrashort soliton was predicted at combination of 1500 nm excitation wavelength and 75 fs pulse width considering 43 mm fibre length. These results represent significant improvement both from point of view of switching contrast and switching energies, which are only at level of 20 pJ, in comparison to the previously published case of air-glass dual-core photonic crystal fibre. Moreover, the simpler fibre design without cladding microstructure together with the all-solid approach holds promise of improved dual-core symmetry and therefore offers high probability of the successful realization of a low power, compact and simple switching device. [ABSTRACT FROM AUTHOR]

The present paper investigates the features of specific waves in a weakly nonlocal nonlinear medium. More precisely, a generalized nonlinear Schrödinger equation (gNLSE) involving a high-order dispersion term, a high-order nonlinear term, and the effect of a term that accounts for the weak nonlocality is considered. By applying the Jacobi elliptic function (JEF) method, a wide range of Jacobi elliptic function solutions to the governing model is first derived. The characteristics of modulation instability (MI) and modulated wave (MW) topological and non-topological solitons are then analyzed by giving a series of two- and three-dimensional graphs. A major accomplishment of the study is the discovery of stable and unstable zones, as well as the control of the amplitude of topological and non-topological solitons. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of spatiotemporal soliton solutions to a generalized nonlinear Schrödinger equation with inhomogeneous coefficients in analytical and numerical forms. The characteristics and physical applications of the soliton solutions are investigated under different modulations of the coefficients depicting diffraction, potential, nonlinearity, source, and gain. We find that structures of these soliton solutions can be effectively manipulated by appropriately selecting the source term. Besides, the impact of gate effect on soliton propagation is investigated by employing the hyper-Gaussian and rectangle functions as diffraction coefficients. The results show that (i) the hyper-Gaussian modulation can be replaced by the rectangle modulation for large exponent and (ii) the solitons undergo a broadening in width and a sudden decrease in amplitude when passing through the gate functions. Our findings may have potential applications for the investigation of soliton control in the fields of nonlinear optics and Bose–Einstein condensations. • Analytical and numerical solutions for spatiotemporal solitons are obtained. • Structures of solutions can be manipulated by appropriately selecting the source term. • The impact of gate effect on soliton propagation is investigated. [ABSTRACT FROM AUTHOR]

Conservation laws of the generalized nonlinear Schrödinger equation with arbitrary refractive index of the fourth order are found. Conservative quantities corresponding to the bright optical soliton are calculated. Direct calculations of three conservation laws for the considered equation are used. This approach allows us to obtain the conservative densities and fluxes of the equation. It is shown that there exist three conservation laws corresponding to the studied equation. The bright optical solitons of the equation are obtained. Using these solutions, conservative quantities of the soliton for the equation are calculated by means of integration. Some properties of conservation laws are discussed. [ABSTRACT FROM AUTHOR]

The complete discrimination system method is employed to find exact solutions for a dispersive cubic–quintic nonlinear Schrödinger equation with third order and fourth order time derivatives. As a result, we derive a range of solutions which include triangular function solutions, kink solitary wave solutions, dark solitary wave solutions, Jacobian elliptic function solutions, rational function solutions and implicit analytical solutions. Numerical simulations are presented to visualize the mechanism of Eq. (1) by selecting appropriate parameters of the solutions. The comparison between our results and other's works are also given. [ABSTRACT FROM AUTHOR]

The photonic crystal fiber allows the control of dispersion and nonlinear effects. The propagation of pulses is described by the generalized nonlinear Schrödinger equation. High order dispersion and nonlinear effects like SPM, self steepening and Raman scattering render the analytical solution of the GNLSE inaccessible. The fourth-order Range-Kutta interaction picture (RK4IP) is an efficient and stable method to determine the numerical solution. In this paper, we present our numerical simulating results relating to the influence of the high order dispersion and nonlinear effects on the soliton self-frequency shift (SSFS). Using the RK4IP algorithm we compute the numerical solutions of an initial hyperbolic secant pulse for some lengths z of a PCF. We calculate the corresponding SSFS, where only the GVD and Raman or GVD, TOD and Raman effects act simultaneously. We show that our numerical results agree well with the theoretical results found in the literature. [ABSTRACT FROM AUTHOR]

In this paper, a new generalized nonlinear Schrödinger (GNLS) equation is investigated by Darboux matrix method. Firstly, the n-fold Darboux transformation (DT) of the GNLS equation is constructed. Then, the soliton solutions, breather solutions, and rogue wave solutions of the GNLS equation are studied based on the DT by choosing different seed solutions. Furthermore, the dynamic features of these solutions are explicitly delineated through some figures with the help of Maple software. [ABSTRACT FROM AUTHOR]