Nonlinear Schrodinger Equation Research Articles

Integrable reductions of the multi-component Kaup–Newell equations

We investigate the local and nonlocal integrable reductions of the multi-component Kaup–Newell equations and derive a range of novel integrable derivative nonlinear Schrödinger (NLS) equations. Firstly, we demonstrate that the multi-component Kaup–Newell equations can undergo one local reduction and three nonlocal reductions, specifically space-reversal, time-reversal, and space–time-reversal reductions. These reductions result in four multi-component derivative NLS equations. Secondly, focusing on the four-component Kaup–Newell equations, we introduce six joint reductions, which are combinations of the aforementioned local and nonlocal reductions. Through these joint reductions between the local reduction and nonlocal reductions, we discover three scalar nonlocal derivative NLS equations, including a T-reversal derivative NLS equation. Furthermore, through the joint reductions among the nonlocal reductions, we identify three scalar nonlocal derivative NLS equations with four variables, namely (x,t), (-x,t), (x,−t), and (−x,−t). All these reduced systems are integrable systems, meaning they possess infinitely many conserved integrals.

Application of the Dbar-method to a nonlocal coupled modified NLS equation and nonlocal reduction

A nonlocal modified NLS (mNLS) equation is studied by using Dbar method and nonlocal reduction. The nonlocal coupled modified NLS (cmNLS) equation and its Lax representation are derived by introducing bi-Dbar problem and parity condition. The special spectral transform matrices are defined to get the solutions of nonlocal cmNLS equation. The general reduction conditions from the nonlocal cmNLS equation to the nonlocal mNLS equation are presented. As applications, some soliton solutions, breather solutions, periodic solutions and mixed solutions of the nonlocal mNLS equation are given.

Unique Approach to Exactly Solve Optical Pulses in Nonlinear Meta-materials

Ultra-short pulse propagation in nonlinear NRM has been investigated where a wide class of solutions for bright and dark solitons has been analyzed for distinct parameter ranges. Here the pulse propagation in nonlinear meta-material has been analytically by solving the nonlinear Schrödinger's equation (NLSE) in composite media expressing frequency dispersion in the dielectric permittivity (ε) and the magnetic permeability (µ). The solutions are exactly shown to be of trigonometric & localized types. The analytical and simulation based result has been utilized to study the intensity variation in case of a nonlinear meta-material which typically behaves as a negative refractive medium (NRM), for which both ε and µ exhibit frequency dispersion and are negative in nature. The peak of the pulse-intensity curve slowly decreases as the frequency increases towards the magnetic plasma frequency. The stability of the solitonic solutions has also been established.

Spatiotemporal mode-locking of an all-fiber laser based on InP quantum dot saturable absorber

Passive mode-locking based on saturable absorbers (SAs) is an effective way to generate ultrafast pulses, while the spatiotemporal mode-locking (STML) based on SAs are rarely studied. We construct an all-fiber laser with InP quantum dots (QDs) SA, and realize multi-mode Q-switching (MMQS) and STML. In addition, by adjusting the polarization controller (PC) in the laser cavity, we obtain two different single-pulse STML states. The narrowest pulse widths of the two states are 57 ps and 32 ps, respectively, and the pulse width can be tuned continuously with increasing the pump power. By decoupling the effects of the PC and the SA in our experiment, we find that the polarization plays a key role in the selection of transverse modes. The effect of independent polarization on spatiotemporal mode-locked pulses has not been investigated before. To further analyze and understand the experimental results, we numerically solve the nonlinear Schrodinger equation, and compare the numerical simulation with the experimental results. In addition to passively adjusting cavity loss, the InP QD SA can balance the large walk off between different transverse modes, due to the spatial and spectral filtering effects. Our work has important implications for the design and application of all-fiber STML lasers.

Nonlinear internal gravity waves in the atmosphere: Rogue waves, breathers and dark solitons

We study nonlinear internal gravity waves (IGWs) in the atmosphere. The reductive perturbation method is used to derive a system of two-dimensional nonlinear equations for the envelope of velocity stream function and the mean flow. In the one-dimensional case, we obtain a nonlinear Schrödinger (NLS) equation corresponding to both horizontal and vertical propagation of IGWs. Depending on the characteristic wavelengths, the NLS equation is focusing or defocusing. In the focusing case, non-stationary solutions in the form of the Peregrine soliton, the Akhmediev breather and the Kuznetsov–Ma breather are considered as potential candidates for the modeling of rogue waves in the atmosphere. In the defocusing case, stationary nonlinear IGWs are considered in the form of nonlinear periodic waves and dark solitons.

Paul Harry Roberts. 13 September 1929—17 November 2022

Paul Roberts was a physicist and applied mathematician. He made important and often pioneering contributions in diverse research areas, but mainly with a fluid dynamic theme, which include rotating fluids, geophysical and astrophysical flows and superfluids. The cornerstone was magnetohydrodynamics with particular application to the geodynamo. His most notable achievement was the first three-dimensional self-consistent numerical geodynamo model, which built on the equations he had derived previously governing the Earth's core dynamics. His considerations included the role of thermal and compositional convection, and the inner core boundary layer (a mixed phase region). In addition to his remarkable illustration of magnetic field reversals, he applied his model to core–mantle coupling and variations in the length of the day. Paul investigated superfluid liquid helium on two fronts. He had a lifelong interest in Bose–Einstein condensates on the microscale governed by the nonlinear Schrodinger equation. His early work on the robust derivation of the mean-field Hall–Vinen–Bekharevic–Khalatnikov equations was far ahead of its time. Their importance and relevance was not appreciated until much later.

Applications of complete discrimination system approach to analyze the dynamic characteristics of the cubic–quintic nonlinear Schrodinger equation with optical soliton and bifurcation analysis

In this research, the complete discriminant system (CDS) of polynomial method (CDSPM) will be applied to analyze the dynamic characteristics of the cubic–quintic nonlinear Schrodinger equation (CQNLSE) with an additional anti-cubic nonlinear term (CQNLSE-AC) with a particular emphasis on the introduction of various optical solitons and wave structures. Our analysis of the CQNLSE-AC illustrates the importance of CDSPM, and we give dynamic results, such as critical conditions and bifurcation points for solutions. Additionally, we determine several types of optical soliton solutions, including Jacobian elliptic function (JEF), hyperbolic function, and trigonometric function solutions, and also convert JEF into solitary wave (SW) solutions.

Bright and Dark Soliton Pulse in Solid Core Photonic Crystal Fibers

The nonlinear Schrodinger (NLS) equation was utilized to conduct an analytical study on the soliton control in homogeneous photonic crystal fibers (PCFs) within both the normal and anomalous dispersion regimes. The split step Fourier method and MATLAB computation were used to generate the analytical soliton solutions for the NLS problem. The bright and dark soliton can be controlled by the group-velocity distribution (GVD). It is capable of demonstrating the fabrication of a fully coherent PCF. Moreover, dark soliton pulses have been seen in PCF in the typical dispersion domain. Here, we present the bound states of bright–dark soliton pairs that are mutually confined in a PCF. Solitons are produced when two modes with opposing dispersions are replanted. One laser operating in the anomalous dispersion domain produces the bright soliton, while the second laser running in the normal dispersion phase produces the dark soliton by normal dispersion cross-phase modulation with the light soliton. The results unequivocally point to a novel method of generating dark soliton pulses. Capturing both bright and dark solitons can produce light states that, interestingly, have a consistent power output and spectrally resemble a frequency comb. These results may have use in soliton states in atomic physics, ultrafast optics, frequency comb technologies, and telecommunications systems. A radically new approach to the stabilization of powerful wave packages in the negative and positive GVD regions of interacting waves is illustrated.

Unveiling optical solitons: Solving two forms of nonlinear Schrödinger equations with unified solver method

The motivation of this work is to attain optical solitons of the generalized third-order nonlinear Schrödinger equation (NLSE) and the integrable (2+1)-dimensional coupled nonlinear Schrödinger equation (CNLSE). These models have various applications, including ultra-short pulses in optical fibers, making them noteworthy in applied sciences and mathematical physics. To extract the optical solitons of these models, the unified solver method (USM) is applied. The USM is a mathematically modified version of the Riccati equation and its related solutions. A unique advantage of this method is the elimination of the need for a balancing constant, simplifying the analytical process and enhancing the utility of these soliton solutions in practical applications. The obtained solutions include dark, singular, and periodic singular soliton solutions, enriching the understanding of optical solitons. In addition to the analytical solutions, the article also includes graphical representations to visually depict the obtained soliton solutions. These visualizations enhance the understanding and provide a clear picture of how the solitons behave in the given system.

Phase portraits and new exact traveling wave solutions of the (2+1)-dimensional Hirota system

In this paper, we investigate the (2+1)-dimensional Hirota system of equations, which is essential for modeling physical events since it contains the NLS equation and the cmKdV equation. We apply the Jacobi elliptic function method to obtain new results. By using some ansatz in terms of finite Jacobi elliptic functions, we have obtained various exact solutions, such as dark solitons, bright solitons, and periodic solutions. Moreover, for some reductions as α=0,β=1 and α=1,β=0, we obtain the exact solutions for the (2+1)-dimensional NLS equation and the (2+1)-dimensional cmKdV equation. The dynamics of the obtained solutions are presented in the figures. We also analyze the phase portraits of the (2+1)-dimensional Hirota system.

Performance of affine-splitting pseudo-spectral methods for fractional complex Ginzburg-Landau equations

We evaluate the performance of novel numerical methods for solving one-dimensional nonlinear fractional dispersive and dissipative evolution equations. The methods are based on affine combinations of time-splitting integrators and pseudo-spectral discretizations using Hermite and Fourier expansions. We show the effectiveness of the proposed methods by numerically computing the dynamics of soliton solutions of the standard and fractional variants of the nonlinear Schrödinger equation (NLSE) and the complex Ginzburg-Landau equation (CGLE), and by comparing the results with those obtained by standard splitting integrators. An exhaustive numerical investigation shows that the new technique is competitive when compared to traditional composition-splitting schemes for the case of Hamiltonian problems both in terms accuracy and computational cost. Moreover, it is applicable straightforwardly to irreversible models, outperforming high-order symplectic integrators which could become unstable due to their need of negative time steps. Finally, we discuss potential improvements of the numerical methods aimed to increase their efficiency, and possible applications to the investigation of dissipative solitons that arise in nonlinear optical systems of contemporary interest. Overall, the method offers a promising alternative for solving a wide range of evolutionary partial differential equations.

Bifurcations, chaotic dynamics, sensitivity analysis and some novel optical solitons of the perturbed non-linear Schrödinger equation with Kerr law non-linearity

This study introduces an analysis of bifurcations, chaotic dynamics, and sensitivity using the Galilean transformation applied to the perturbed non-linear Schrödinger equation (NLSE) with Kerr-law nonlinearity. Additionally, we employ the efficient Sardar sub-equation (Sse) method to derive various optical soliton solutions. Initially, we outline the general framework of the Sse method proposed. Next, we utilize a traveling wave transform on the NLSE, transforming it into a system of nonlinear ordinary differential equations (NLODEs) which are further separated into their real and imaginary components. Furthermore, we apply this method to derive new optical solitons for the NLSE equation with Kerr law. We perform exact analytical simulations of these optical soliton solutions and investigate the effects of different parameters on the soliton waves.

Propagation of chirped optical solitons for perturbation higher order nonlinear Schrödinger equation with dual-power law nonlinearity by [formula omitted] expansion method

New exact solutions for the perturbation higher order nonlinear Schrödinger equation (NLSE) with dual-power law nonlinearity are investigated using the ϕ6 expansion method. This type of NLSE models the propagation of femtosecond pulses in fiber optics which has many applications such as infrared time-resolved spectroscopy, ultrahigh-bit-rate optical communication systems and ultrafast optics. The application of ϕ6 expansion technique provided many different types of solutions. Dark solitons, bright solitons, singular solitons, hyperbolic solutions, periodic and singular periodic solutions are obtained. The graphical representations of some selected solutions are provided to present both the dynamical behavior of solutions and the effect of changing the nonlinearity degree on their dynamical behavior. This study concludes that the nonlinearity degree controls the dynamical behavior of the light pulse.

Adaptive Absorbing Boundary Layer for the Nonlinear Schrödinger Equation

Abstract We present an adaptive absorbing boundary layer technique for the nonlinear Schrödinger equation that is used in combination with the Time-splitting Fourier spectral method (TSSP) as the discretization for the NLS equations. We propose a new complex absorbing potential (CAP) function based on high order polynomials, with the major improvement that an explicit formula for the coefficients in the potential function is employed for adaptive parameter selection. This formula is obtained by an extension of the analysis in [R. Kosloff and D. Kosloff, Absorbing boundaries for wave propagation problems, J. Comput. Phys. 63 1986, 2, 363–376]. We also show that our imaginary potential function is more efficient than what is used in the literature. Numerical examples show that our ansatz is significantly better than existing approaches. We show that our approach can very accurately compute the solutions of the NLS equations in one dimension, including in the case of multi-dominant wave number solutions.

Stability of ground states of nonlinear Schrodinger systems

In this article, we study existence and stability of ground states for a system of two coupled nonlinear Schrodinger equations with logarithmic nonlinearity. Moreover, global well-posedness is verified for the Cauchy problem in \(H^{1}(\mathbb{R})\times H^{1}(\mathbb{R})\) and in an appropriate Orlicz space.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/76/abstr.html

Differences between two methods to derive a nonlinear Schrödinger equation and their application scopes

The present paper chooses a dusty plasma as an example to numerically and analytically study the differences between two different methods of obtaining nonlinear Schrödinger equation (NLSE). The first method is to derive a Korteweg–de Vries (KdV)-type equation and then derive the NLSE from the KdV-type equation, while the second one is to directly derive the NLSE from the original equation. It is found that the envelope waves from the two methods have different dispersion relations, different group velocities. The results indicate that two envelope wave solutions from two different methods are completely different. The results also show that the application scope of the envelope wave obtained from the second method is wider than that of the first one, though both methods are valuable in the range of their corresponding application scopes. It is suggested that, for other systems, both methods to derive NLSE may be correct, but their nonlinear wave solutions are different and their application scopes are also different.

Dynamical forms of various optical soliton solutions and other solitons for the new Schrödinger equation in optical fibers using two distinct efficient approaches

In this work, the new nonlinear [Formula: see text]-dimensional Schrödinger (NLS) equation, which describes wave propagation in optical fibers, has been successfully studied using the [Formula: see text]-expansion method and the Unified method (UM). These techniques have been applied for the first time in extracting new analytic solutions for the mentioned NLS equation. The obtained solutions exhibit various types of optical solitons, including singular solitons, solitary waves, periodic solitons, and wave collisions. These solutions have wide-ranging applications in various fields such as fiber optics, telecommunication systems, plasma physics, hydrodynamics, and nonlinear optics. The behavior of the derived solutions has been visually represented using three-dimensional, two-dimensional, and contour plots, considering appropriate choices of the involved parameters within specific time intervals. These plots provide insights into the dynamics and characteristics of the solutions. The effectiveness, reliability, and ease of application of the [Formula: see text]-expansion method and the UM are demonstrated through the obtained solutions. These mathematical tools have proven to be powerful in solving nonlinear partial differential equations, offering valuable contributions to the field of mathematical physics, soliton theory and nonlinear sciences.

Numerical simulation of coherent mid-infrared supercontinuum generation based on carbon chloride core tapered photonic crystal fibers

We numerically studied coherent mid-infrared (MIR) supercontinuum (SC) generated by filling carbon chloride (CCl4, C2Cl4) into the tapered photonic crystal fiber (PCF). The mechanism involves nonlinear soliton compression, Raman soliton self-frequency shift and dispersive wave generated in a carbon chloride core tapered PCF pumped by a 1.98 μm femtosecond laser. Numerical solution of the generalized nonlinear Schrodinger equation shows that 1.15–5.5 μm (CCl4-core tapered PCF) and 1.15–5.7 μm (C2Cl4-core tapered PCF) coherent MIR-SC sources can be realized. Our results show that the above liquid-core tapered PCFs are promising nonlinear media for constructing broadband coherent MIR-SC light sources.

Obliquely propagating optical wave patterns to the (2 + 1)-dimensional chiral nonlinear Schrödinger equation in the absence and presence of Atangana derivative

The study aims to explore obliquely propagating optical wave solutions to the (2 + 1)-dimensional chiral nonlinear Schrödinger (NLS) equation in both the absence and presence of the Atangana derivative. In order to convert the classical-order chiral nonlinear Schrödinger equation to an ordinary differential equation, a transformation associated with wave obliqueness is applied. Hereafter, the unified method is applied to the reduced equation. As outcomes, the dark, periodic singular with unequal wave length, periodic with equal wavelength, periodic, unsmooth periodic, singular periodic soliton solutions are received to the ordinary differential equation. Later, the acquired solutions are then put to the applied transformation associated with obliqueness. Moreover, the fractional-order chiral NLS equation is solved by using the spatiotemporal Atangana derivative with oblique wave transformation and the unified method. In terms of wave obliqueness, fractionality, and applied technique sense, all generated wave solutions are revealed to be novel. Along with their physical explanations, the impacts of obliqueness and fractionality on the solutions are graphically illustrated. It is exposed that as obliqueness and fractionality increase, the optical wave phenomena change. Additionally, it is discovered that the employed method can be used to obtain novel optical soliton features to the chiral nonlinear Schrödinger equation with or without fractional and obliqueness constraints. It can be assured that the utilized method is more powerful than the other methods. As a result, the method can be used in future research to explain the many physical phenomena that arise in optical fiber communication networks.

Spectral, Scattering and Dynamics: Gelfand–Levitan–Marchenko–Krein Equations

In this paper, we consider the Gelfand–Levitan–Marchenko–Krein approach. It is used for solving a variety of inverse problems, like inverse scattering or inverse problems for wave-type equations in both spectral and dynamic formulations. The approach is based on a reduction of the problem to the set of integral equations. While it is used in a wide range of applications, one of the most famous parts of the approach is given via the inverse scattering method, which utilizes solving the inverse problem for integrating the nonlinear Schrodinger equation. In this work, we present a short historical review that reflects the development of the approach, provide the variations of the method for 1D and 2D problems and consider some aspects of numerical solutions of the corresponding integral equations.