Nonlinear Dispersion Research Articles

Abundant optical solitons to Biswas–Milovic model having parabolic law nonlinearity and spatio-temporal dispersion using improved modified extended tanh function scheme

The objective of this work is to examine the optical solitons of the Biswas–Milovic model, which has been the focus of many studies and continues to be important in this field of study of nonlinear optics. Besides the group velocity dispersion, spatio-temporal dispersion and parabolic law of nonlinearity effects are also added to this model. The improved modified extended tanh function scheme is applied to retrieve optical solitons and other exact wave solutions to the model. This method depends on the extended Riccati equation. So, various solutions can be extracted such as (bright, dark and singular) solitons, exponential, singular periodic, Weierstrass elliptic and rational type solutions. The graphical illustrations are depicted for some selected solutions to show the dynamics of the extracted solutions. The optical solitons produced in respect to this form have never been explored by the proposed technique before, and the results have never been published.

Interplay between intensity-dependent dispersion and Kerr nonlinearity on the soliton formation.

A generalized nonlinear Schrödinger equation is studied with the interplay between Kerr nonlinearity and intensity-dependent dispersion. The supported soliton solutions are characterized analytically in different families by the pseudo-potential method, in terms of Maimistov and Cuspon solitons for different ratio between the intensity-dependent dispersion and Kerr nonlinearity. Direct numerical simulations also agree with our analytical formulas. In addition to the well-studied Kerr-type nonlinearity, our results reveal an unexplored scenario with the introduction of the nonlinear corrections to wave dispersion.

New kink-periodic and convex–concave-periodic solutions to the modified regularized long wave equation by means of modified rational trigonometric–hyperbolic functions

Abstract The significance of different types of periodic solutions in nonlinear equations is vital across various practical applications. Our objective in this study was to uncover novel forms of periodic solutions for the modified regularized long wave equation. This particular model holds great importance in the realm of physics as it characterizes the propagation of weak nonlinearity and space-time dispersion waves, encompassing phenomena like nonlinear transverse waves in shallow water, ion-acoustic waves in plasma, and phonon waves in nonlinear crystals. By employing the methodology of modified rational sine-cosine and sinh–cosh functions, we successfully derived new kink-periodic and convex–concave-periodic solutions. To showcase the superiority of our proposed approach, we conducted a comparative analysis with the alternative Kudryashov-expansion technique. Furthermore, we visually depicted the diverse recovery solutions through 2D and 3D plots to enhance the understanding of our findings.

Nonlinearly dispersive KP equations with new compacton solutions

A complete classification of compacton solutions is carried out for a generalization of the Kadomtsev–Petviashvili (KP) equation involving nonlinear dispersion in two and higher spatial dimensions. In particular, precise conditions are given on the nonlinearity powers in this equation under which a travelling wave can be cut off to obtain a compacton. Numerous explicit examples having various wave profiles are derived, including a quadratic function, powers of a cosine, and powers of Jacobi cn functions, all of which are symmetric. The cosine and cn symmetric compactons have an anti-symmetric counterpart. In comparison, explicit solitary waves of the generalized KP equation are found to have profiles given by a power of a sech and a reciprocal quadratic function. Kinematic properties of all of the different types of compactons and solitary waves are discussed, along with conservation laws of the generalized KP equation.

Nonlinear Dispersion Relation and Out‐of‐Plane Second Harmonic Generation in MoSSe and WSSe Janus Monolayers

AbstractJanus transition metal dichalcogenides are an emerging class of atomically thin materials with engineered broken mirror symmetry that gives rise to long‐lived dipolar excitons, Rashba splitting, and topologically protected solitons. They hold great promise as a versatile nonlinear optical platform due to their broadband harmonic generation tunability, ease of integration on photonic structures, and nonlinearities beyond the basal crystal plane. Here, second and third harmonic generation in MoSSe and WSSe Janus monolayers is studied. Polarization‐resolved spectroscopy is used to map the full second‐order susceptibility tensor of MoSSe, including its out‐of‐plane components. In addition, the effective third‐order susceptibility and the second‐order nonlinear dispersion close to exciton resonances for both MoSSe and WSSe are measured at room and cryogenic temperatures. This work sets a bedrock for understanding the nonlinear optical properties of Janus transition metal dichalcogenides and probing their use in the next‐generation on‐chip multifaceted photonic devices.

Chirped optical solitons and stability analysis of the nonlinear Schrödinger equation with nonlinear chromatic dispersion

The present paper aims to investigate the chirped optical soliton solutions of the nonlinear Schrödinger equation with nonlinear chromatic dispersion and quadratic-cubic law of refractive index. The exquisite balance between the chromatic dispersion and the nonlinearity associated with the refractive index of a fiber gives rise to optical solitons, which can travel down the fiber for intercontinental distances. The effective technique, namely, the new extended auxiliary equation method is implemented as a solution method. Different types of chirped soliton solutions including dark, bright, singular and periodic soliton solutions are extracted from the Jacobi elliptic function solutions when the modulus of the Jacobi elliptic function approaches to one or zero. These obtained chirped optical soliton solutions might play an important role in optical communication links and optical signal processing systems. The stability of the system is examined in the framework of modulational instability analysis.

Effects of the higher-order dispersion on solitary waves and modulation instability in a monomode fiber

In this paper, we investigate the modulation instability and the dynamics of solitary waves in a higher-order nonlinear Schrödinger equation. We use a powerful and robust method known as the modified sub-equation method to secure the solitary waves and other numerous solutions to this nonlinear model. Several constraint conditions that guarantee the existence of these solutions are highlighted. Utilizing a linearizing technique, we establish the modulation instability gain. The influence of the higher nonlinear component on the modulation instability is also discussed. Furthermore, we perform the numerical simulations of this model by using the split-step Fourier method. Finally, several solutions are shown in two and three dimensions to better explain the behavior of the considered model. The two-dimensional plots depict Peregrine soliton under the influence of the nonlinear dispersion term. These graphs are useful for understanding the dynamic properties of the obtained results.

Numerical Solutions of the Nonlinear Dispersive Shallow Water Wave Equations Based on the Space–Time Coupled Generalized Finite Difference Scheme

This study applies the space–time generalized finite difference scheme to solve nonlinear dispersive shallow water waves described by the modified Camassa–Holm equation, the modified Degasperis–Procesi equation, the Fornberg–Whitham equation, and its modified form. The proposed meshless numerical scheme combines the space–time generalized finite difference method, the two-step Newton’s method, and the time-marching method. The space–time approach treats the temporal derivative as a spatial derivative. This enables the discretization of all partial derivatives using a spatial discretization method and efficiently handles mixed derivatives with the proposed mesh-less numerical scheme. The space–time generalized finite difference method is derived from Taylor series expansion and the moving least-squares method. The numerical discretization process only involves functional data and weighting coefficients on the central and neighboring nodes. This results in a sparse matrix system of nonlinear algebraic equations that can be efficiently solved using the two-step Newton’s method. Additionally, the time-marching method is employed to advance the space–time domain along the time axis. Several numerical examples are presented to validate the effectiveness of the proposed space–time generalized finite difference scheme.

Steady-state hydroelastic waves generated by a moving load in a uniform current

Steady-state hydroelastic waves generated by a moving load with a uniform current are investigated analytically in the framework of nonlinear potential theory. Considering the load with an exponential distribution in horizontal direction, we focus on the wave dynamics characteristics when the relative current speed or the nonlinearity increases. With the aid of the homotopy analysis method (HAM), appropriate solution expressions for the unknown variables are established by comparing the relative current speed U and the minimal phase speed cmin of the hydroelastic waves. Our results demonstrate that, the wave deflection is symmetric about the load if U is much smaller than cmin. The wave deflection becomes more oscillatory and its amplitudes increase in the vicinity of the load when U increasingly approaches cmin. If U is greater than cmin, the hydroelastic waves will be generated far away from the load, and the period of the gravity waves in the downstream region are relatively larger than those of the hydroelastic waves in the upstream region. As U increases continuously, the amplitudes of gravity and hydroelastic waves all first increase and then decrease. It should be noted that there is a larger cmin of the approximate nonlinear dispersion relation than that of the linear one, and the variation of the nonlinear wave deflection would be underestimated if the linear dispersion relation is used. Finally, the graphical representations show the effects of several important physical parameters including the uniform current speed, the amplitude of incident wave, and the plate thickness on the hydroelastic waves.

Derivation and calibration of spectral response for a channeled spectropolarimeter.

The channeled spectropolarimeter (CSP) measures the spectrally-resolved Stokes vector from a snapshot by employing spectral modulation. The spectral modulation transfer function (SMTF) of the spectrometer preferentially suppresses the high-frequency channel amplitude in CSP, resulting in reduced measurement accuracy. This paper rigorously derives the SMTF theory and proposes an efficient calibration method for SMTF via channel shifting in a CSP. The SMTF value, obtained by channel shifting, is used to correct the high-frequency channel amplitude. Moreover, alignment and phase errors, as well as nonlinear dispersion, are compensated in situ. Other than rotating the retarder twice, no additional instruments or algorithms are required in the proposed method. In simulations and experiments, the proposed method shows high accuracy, with a maximum root-mean-square error (RMSE) of the reconstructed Stokes spectrum below 0.01, demonstrating its potential for enhancing the simplicity and practicability of Stokes CSP.

Electron holes in a regularized kappa background

Abstract. The pseudopotential method is used to derive electron hole structures in a suprathermal plasma with a regularized κ probability distribution function background. The regularized character allows the exploration of small κ values beyond the standard suprathermal case for which κ>3/2 is a necessary condition. We found the nonlinear dispersion relation yielding the amplitude of the electrostatic potential in terms of the remaining parameters, in particular the drift velocity, the wavenumber and the spectral index. Periodic, solitary wave, drifting and non-drifting solutions have been identified. In the linear limit, the dispersion relation yields generalized Langmuir and electron acoustic plasma modes. Standard electron hole structures are regained in the κ≫1 limit.

Dark Korteweg–De Vrise System and Its Higher-Dimensional Deformations

The new dimensional deformation approach is proposed to generate higher-dimensional analogues of integrable systems. An arbitrary (K+1)-dimensional integrable Korteweg–de Vries (KdV) system, as an example, exhibiting symmetry, is illustrated to arise from a reconstructed deformation procedure, starting with a general symmetry integrable (1+1)-dimensional dark KdV system and its conservation laws. Physically, the dark equation systems may be related to dark matter physics. To describe nonlinear physics, both linear and nonlinear dispersions should be considered. In the original lower-dimensional integrable systems, only liner or nonlinear dispersion is included. The deformation algorithm naturally makes the model also include the linear dispersion and nonlinear dispersion.

Conservation laws, exact solutions and nonlinear dispersion: A lie symmetry approach

In this study, we investigated a set of equations that exhibit compact solutions and nonlinear dispersion. We used the classical lie symmetry approach to derive ordinary differential equations (ODEs) that are well suited for qualitative study. By examining the dynamic behavior of these ODEs, we gained insights into the intricate nature of the underlying system. We also used a powerful multiplier approach to establish nontrivial conservation laws and exact solutions for these equations. These conservation laws provide essential information regarding the underlying symmetries and invariants of the system, and shed light on its fundamental properties.

On the Evolution Equations of Nonlinearly Permissible, Coherent Hole Structures Propagating Persistently in Collisionless Plasmas

AbstractA fundamental study describing nonlinear plasma wave propagation is presented. Elementary linear wave theory describes small‐amplitude random waves, but lacks information about coherent structures. This improved wave model arises from the fact that structure formation is inevitably associated with particle trapping, which can only be properly addressed by the pseudo‐potential method instead of Bernstein, Greene, and Kruskal (BGK) ‐ likemethods. Only by using this method can legitimate nonlinear dispersion relations be obtained and reconciled with trapping scenarios. This privilege is used to derive evolution equations for five structures, the derivation being simplified by the acoustic nature of the permitted modes. The focus is on a special structure, the solitary electron hole of negative polarity, with which it can explain a spacecraft observation for the first time. Furthermore, it is shown that an intrinsically nonlinear structure can become macroscopically linear and thus harmonic by suitably adjusting the trapping scenario. An example is the monochromatic ion acoustic wave that propagates at ion sound velocity without dispersion. In this literature research, it also takes a critical look at a recently awarded work.

Role of topological charges in the nonlinear optical response from Weyl semimetals

The successful realization of the topological Weyl semimetals has revolutionized contemporary physics. In recent years, multi-Weyl semimetals, a class of topological Weyl semimetals, has attracted broad interest in condensed-matter physics. Multi-Weyl semimetals are emerging topological semimetals with nonlinear anisotropic energy dispersion, which is characterized by higher topological charges. In this study, we investigate how the topological charge affects the nonlinear optical response from multi-Weyl semimetals. It has been observed that the laser-driven electronic current is characteristic of the topological charge, and the laser polarization's direction influences the current's direction and amplitude. In addition, the anomalous current, perpendicular to the laser's polarization, carries a distinct signature of the topological charges and encodes the information about the parity and amplitude of the nontrivial Berry curvature. We show that the anomalous current associated with the anomalous Hall effect remains no longer proportional to the topological charge at higher laser intensity -- a significant deviation from the linear response theory. High-harmonic spectroscopy is employed to capture the distinct and interesting features of the currents in multi-Weyl semimetals where the topological charge drastically impacts the harmonics' yield and energy cutoff.

Qualitative analysis and stationary optical patterns of nonlinear Schrödinger equation including nonlinear chromatic dispersion

Qualitative analysis and stationary optical patterns of nonlinear Schrödinger equation including nonlinear chromatic dispersion

Bilinear form, bilinear auto-Bäcklund transformation, soliton and half-periodic kink solutions on the non-zero background of a (3+1)-dimensional time-dependent-coefficient Boiti-Leon-Manna-Pempinelli equation

Nonlinear evolution equations (NLEEs) are seen in such fields as fluid dynamics, plasma physics and optics. A (3+1)-dimensional time-dependent-coefficient Boiti-Leon-Manna-Pempinelli equation is investigated in this paper. Via the logarithmic transformation on non-zero background, a bilinear form is derived. Via the bilinear form, a bilinear-Bäcklund transformation with some solutions is acquired, while the one-soliton, two-soliton and multiple soliton solutions with two different nonlinear dispersion relations are worked out. On some non-zero backgrounds, multi-kink solutions are derived. Via the complex conjugation, half-periodic kink solutions are obtained.

Nonlinear diffusion in multi-patch logistic model.

We examine a multi-patch model of a population connected by nonlinear asymmetrical migration, where the population grows logistically on each patch. Utilizing the theory of cooperative differential systems, we prove the global stability of the model. In cases of perfect mixing, where migration rates approach infinity, the total population follows a logistic law with a carrying capacity that is distinct from the sum of carrying capacities and is influenced by migration terms. Furthermore, we establish conditions under which fragmentation and nonlinear asymmetrical migration can lead to a total equilibrium population that is either greater or smaller than the sum of carrying capacities. Finally, for the two-patch model, we classify the model parameter space to determine if nonlinear dispersal is beneficial or detrimental to the sum of two carrying capacities.

Kerr nonlinearity and group velocity dispersion of InGaAs/InP and GaAsSb/InP waveguides in the mid-infrared

We report measurements of Kerr nonlinearity and group velocity dispersion in In0.53Ga0.47As/InP and GaAs0.51Sb0.49/InP ridge waveguides in the mid-infrared using four-wave mixing at λ ≈ 5 µm. Measured values of Kerr nonlinearity are significantly higher compared to those reported for any other materials systems suitable for building dielectric waveguides with low losses and low group velocity dispersion in the mid-infrared (λ ≈ 3–15 μm). Our measurements establish both In0.53Ga0.47As/InP and GaAs0.51Sb0.49/InP materials as promising platforms for the development of on-chip mid-infrared frequency comb generation and supercontinuum light sources.

Wave-by-wave forecasts in directional seas using nonlinear dispersion corrections

We develop a new methodology for the deterministic forecasting of directional ocean surface waves based on nonlinear frequency corrections. These frequency corrections can be pre-computed based on measured energy density spectra and, therefore, come at no additional computational cost compared to linear theory. The nonlinear forecasting methodology is tested on highly nonlinear synthetically generated seas with a variety of values of average steepness and directional spreading and is shown to consistently outperform a linear forecast.