Solitary Waves Research Articles

Features of Radio Emission Propagation in the Ionosphere under Conditions of Threshold Nonlinearity

The well-known problem of nonlinear “wave – ionosphere” interaction under conditions of threshold nonlinearity is considered. It is believed that nonlinear effects arise only for high-power radiation, when the wave amplitude exceeds a certain threshold value. The possibility of the existence of concentrated wave fields under these conditions is shown. It is revealed that a certain ratio of nonlinearity parameters leads to an increase in the radio emission intensity, since the interaction of individual solitons can lead to their merging into a higher-power solitary wave. The presence of threshold nonlinearity can lead to the formation of an ordered structure of solitary waves.

On the three-dimensional structure of instabilities beneath shallow-shoaling internal waves

The stimulation of instability and transport in the bottom boundary layer by internal solitary waves has been documented for over twenty years. However, the challenge of shallow slopes and a disparity of scales between the large-scale wave and the small-scale boundary layer has proven challenging for simulations. We present laboratory scale simulations that resolve the three-dimensionalisation in the boundary layer during the entire shoaling process. We find that the late stage, in which the incoming wave fissions into boluses, provides the most consistent source of three-dimensionalisation. In the early stage of shoaling, three-dimensionalisation occurs not so much due to separation bubble instability, but due to the interaction of vortices shed from the separation bubble with the overlying pycnocline. This interaction overturns the pycnocline, and creates bursts in kinetic energy and viscous dissipation, suggesting that the shed vortices induce turbulent motion and sediment resuspension in the water column above and behind the separation bubble.

Solitary wave solutions and their limits to the fractional Schrödinger system

This study is concerned with solitary wave solutions and the dynamic behavior of the (2+1)-dimensional nonlinear fractional Schrödinger system. By exploring the dynamic properties of the equilibrium levels to the corresponding Hamiltonian, the expressions of exact solutions of the above system are obtained, including solitary wave solutions, periodic wave solutions, singular periodic wave solutions, singular wave solutions, kink wave solutions, and anti-kink wave solutions. Moreover, the linear stability, geometric characteristics, and limiting behavior of these solutions to the nonlinear fractional Schrödinger system were investigated.

Nonlinear dust ion acoustic solitary waves propagation in a magnetized plasma with Tsallis electron distribution

Abstract We have done a theoretical investigation into the propagation of nonlinear dust ion acoustic solitary waves and their soliton properties in a three-component magnetized collisionless plasma consisting of inertial positively charged ions, noninertial electrons following a Tsallis q–distribution, and stationary negatively charged dust grains. We consider a uniform external magnetic field along the z–direction, and the wave propagation occurs obliquely to the magnetic field direction. It is observed that the two types of wave modes namely slow and fast modes, are appears in the linear analysis. By employing the reductive perturbation method, the Korteweg-de Vries (KdV) and modified KdV equations are determined to describe the small amplitude dust ion acoustic soliton. The dependence of several physical plasma parameters including nonextensive q–parameter, magnetic field strength etc. of our plasma on the propagating dust ion acoustic solitary waves potential are numerically examined. This study shows the simultaneous existence of compressive and rarefactive solitons due to the variation of first order nonlinearity coefficient represented by the KdV equation and it is found that there is a critical point for the plasma parameters where the amplitude of the soliton of KdV equation become diverges. The mKdV equation with second order nonlinearity coefficient is derived from there and observed the solitons only with finite amplitude. The present study could be helpful for understanding the nonlinear travelling waves propagating in laboratory and space plasma environments.

Ferroelectric frontiers: Navigating phase portraits, chaos, multistability and sensitivity in thin-film dynamics

This research includes the study of the non-linear dynamics of thin-film ferroelectric materials governed by an equation of wave dynamics within the material. This equation plays a key role in both physics and the study of aqueous flow. The work is planned as examining the symmetries group analysis drops, studying the dynamical system features through bifurcation phase portraits, and carrying dynamic phenomena in chaos theory. Diverse techniques are taken, such as Lyapunov exponent, 2D, and 3D phase portraits, Poincaré maps, time series analysis, and sensitivity to multistability under the different conditions of the initial state. In addition, the study involves using the extended hyperbolic function method to obtain the general analytical solutions via which various kinds of solitary wave solutions are produced including trigonometric and hyperbolic functions and periodic, bright, and singular soliton solutions. These solutions are followed by a list of constraint conditions in the form of equations. Visual data of 2D, 3D, and contour plots are presented, with parameters carefully set to reflect various scenarios. Sensitivity analysis is performed using alternative initial conditions, and stability analysis is demonstrated graphically. To fully grasp the dynamic features of these systems and accurately predict outcomes, it is essential to advance new technologies and methodologies that can further enhance our understanding and predictive capabilities in complex systems.

Solitary and traveling wave solutions to nematic liquid crystal equations using Jacobi elliptic functions

In our paper we apply the Jacobi elliptic function (JEF) expansion method to obtain exact solutions to the system of equations governing nematic liquid crystals, a system of high importance in nonlinear optics with numerous physical applications. We obtain solutions that are second-order polynomials in terms of JEFs for both the wave function and the tilt angle of molecular orientation. The solutions differ from previously obtained solutions in including both traveling and solitary wave solutions, with and without chirp. They also include the longitudinal dependence of coefficients in the equations, allowing for the management of both the dispersion and diffraction. Only two parameters of the differential equation need to be defined in terms of other coefficients, providing a wide range of flexibility when it comes to constructing solutions.

A plethora of novel solitary wave solutions related to van der Waals equation: a comparative study

In this article, we explore exact solitary wave solutions to the van der Waals equation which is crucial for numerous applications involving a variety of physical occurrences. This system is used to define the behavior of real gases taking into consideration finite size of molecules and also has some applications in industry for granular materials. The model is studied under the effect of fractional derivatives by employing two different definitions: β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}, and M-truncated. Further, new extended direct algebraic method is employed to construct the solitary wave solutions for the model. The solutions transmit several novel solutions, such as dark-singular, dark–bright, singular-periodic and dark solutions, and this method establishes the conditions required for the formation of these structures. To show the comparative analysis between two different fractional operators, results are graphically represented in the form of 2-dimensional and 3-dimensional visualizations.

Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives

Abstract This research explores new soliton solutions to the Atangana–Baleanu derivative (ABD) fractional system of equations for ion sound and Langmuir waves (FISLW). We utilize the fractional ABD operator to convert our system into an ordinary differential equations. In recent years, machine learning (ML) evolves significantly in the context of data analysis and computing different solutions, which typically enables systems to operate wisely. Now, we are going to use numerous ML tools including matplotlib. pyplot as plt, scipy.integrate, mpl toolkits.mplot3d, and Axes3D to generate various types of optical solutions by using complete discriminant of the polynomial method. We will also analyze solutions for the hyperbolic function, trigonometric function, Jacobian elliptic function (JEF), and other solitary wave solutions. Solitons have extensive uses in pure and applied mathematics, including nonlinear partial differential equations: the Boussinesq equation, the nonlinear Schrödinger equation, and the sine-Gordon equation, Lie groups, Lie algebras, and differential and algebraic geometry. In addition, we study the chaotic behaviour, i.e., 2D, 3D, time series, Poincarè maps, and sensitivity analysis of our governing model. Sensitivity analysis explores how changes in a system’s variables affects its behaviour.

Nonlinear Jeans instability analysis of gravitating astrofluids in Eddington-inspired Born-Infeld gravity framework

The nonlinear Jeans instability dynamics in viscoelastic astrofluids within the Eddington-inspired Born-Infeld (EiBI) gravitational framework excitable in star-forming interstellar cloud fluids is semi-analytically investigated. Application of multiple scaling techniques here results in a unique form of the Korteweg-de Vries (KdV) equation moderated with a peculiar set of distinctive fluid polyparametric coefficients. A numerical illustrative platform portrays the excitation of diversified compressive solitary-chain wave patterns. The evolutionary existence of such structural patterns is further confirmed geometrically with the help of illustrative phase-plane analyses. The various parametric stabilizers and destabilizers, alongside accelerating and decelerating factors on the clouds, are illustratively analysed. It is seen that the composite cloud stability is significantly influenced by the strength polarity of the gravitational EiBI-parameter (χ). The dependency of the obtained solitary wave patterns on the EiBI-polarity is further confirmed in a numerical simulation platform illustratively. Our analysis also highlights that the influence of material density on the system stability is dependent on the EiBI-polarity. The nonlinear spatiotemporal evolution of the instability reveals a unique type of soliton fission phenomenon, indicative of inherently produced additional disturbances within the astrofluid. Our obtained results could be widely helpful in application to the EiBI-centric diverse astrocosmic phenomena, such as bounded structure formation and evolution in diverse complex astroenvirons.

Signature of conservation laws and solitary wave solution with different dynamics in Thomas–Fermi plasma: Lie theory

We propose a Lie group method to discuss the modified KP equation appearing in Thomas–Fermi (TM) plasma, characterised by cold and hot electrons. The Lie method facilitates the identification of similarity reductions, infinitesimal symmetries, group-invariant solutions, and novel analytical solutions for nonlinear models. The similarity reduction method is carried out to transform the nonlinear partial differential equations (NLPDE) into the nonlinear ordinary differential equations (NLODE). This study focuses on solitary wave profiles due to their usefulness in various engineering applications, including monitoring public transportation systems, managing coastlines, and mitigating disaster risks. It also addresses the conservation laws associated with the modified KP equation. The generalised auxiliary equation (GAEM) scheme is used to compute the new solitary wave patterns of the modified KP equation, which explains the dynamics of nonlinear waves in Thomas–Fermi plasma. The idea of nonlinear self-adjointness is used to compute the conservation laws of the examined model. The graphical behaviour of some solutions is represented by adjusting the suitable value of the parameters involved.

The fractional solitary wave profiles and dynamical insights with chaos analysis and sensitivity demonstration

This work investigates the dynamics of solitary waves in a thin-film ferroelectric material model, an essential component of optical systems. The study recognizes the limitations of the inverse scattering transform in solving the Cauchy problem for this conformable thin-film ferroelectric material model and applies a new Kudryashov approach and Bernoulli’s equation technique to ordinary differential equations. We apply these methods to the present model and obtain closed form analytical solutions. A variety of traveling wave and soliton solutions are constructed, including kink, bell, bright, and dark soliton solutions. It is acknowledged that the nonlinear evolution equation can be transformed into an ordinary differential equation by applying traveling wave transformations. Suitable physical parameters are chosen to create 2D, 3D, and contour plots that graphically depict the graphical dispersion of obtained fractional soliton solutions. The impact of the conformable fractional parameter is further demonstrated by the graphical representation of solitons propagation. Furthermore, the study develops the Hamiltonian function of the dynamical system to discuss the system’s total energy in terms of momentum and position. While a chaos analysis describes chaotic, quasi-periodic, and periodic behaviors, a sensitivity analysis demonstrates how responsive the model is to various initial conditions. Due to their dual-purpose nature as nonlinear ferroelectric and dielectric materials, thin ferroelectric films are widely used in modern electrical devices.

Acceleration of ion phase-space holes due to interactions with ion solitons in a wave-guided plasma

Ion phase-space holes are solitary kinetic structures found in the ion phase-space of collision-less plasmas, and are nonlinear solutions to the Vlasov-Poisson equations, identified as Bernstein-Greene-Kruskal (BGK) modes. In this study, interactions between an ion phase-space hole and a travelling ion KdV soliton is presented. This interaction, which is simulated in a fully ionised highly magnetised plasma within a cylindrical wave-guide, exhibits acceleration and deacceleration of the ion hole, depending on its mode of collision with the travelling ion soliton. We present these interactions and discuss the mechanism of this interaction between the two solitary waves.

Classical and nonclassical Lie symmetries, bifurcation analysis, and Jacobi elliptic function solutions to a 3D-modified nonlinear wave equation in liquid involving gas bubbles

The current paper undertakes an in-depth exploration of the dynamics of nonlinear waves governed by a 3D-modified nonlinear wave equation, a significant model in the study of complex wave phenomena. To this end, the study employs both classical and nonclassical Lie symmetries for rigorously deriving invariant solutions of the governing equation. These symmetries enable the formal construction of exact solutions, which are crucial for understanding the complex behavior of the model. Furthermore, the research extends into the realm of bifurcation analysis through the application of planar dynamical system theory. Such an analysis reveals the conditions under which the 3D-modified nonlinear wave equation admits Jacobi elliptic function solutions. The study also delves into the impact of the nonlinear parameter on the physical characteristics of bright and kink solitary waves as well as continuous periodic waves using Maple. Overall, the comprehensive analysis presented not only enhances the understanding of complex nonlinear wave dynamics but also sets the stage for future advancements in vast areas of fluid dynamics and plasma physics.

Rigid-foldable spiral origami with compression-torsion coupled motion mode

Rigid foldable origami enables smooth and precise folding without stretching or bending its constituent panels and is promising for applications such as reprogrammable matter, self-folding machines, reconfigurable antennas, and deployable spacecraft. The diverse range of potential applications necessitates the need for the design and detailed analysis of different rigid-foldable origami structures, especially those with intricate motion modes. In this paper, we introduce a rigid-foldable spiral origami design that features a compression-torsion coupled motion mode. This design exhibits rich static and dynamic properties. Under static conditions, the compression-torsion coupled motion mode creates multiple self-locking positions and allows for the development of mechanical static diodes. Under dynamic conditions, the compression-torsion coupling effect in the spiral origami facilitates precise control of wave modes within the origami chain when impacted by a ball with a moderate initial velocity. In the case of large initial velocities of the ball, the spiral origami can function as a wave generator, producing rarefaction solitary waves or compressive solitary waves. The proposed spiral origami design provides an opportunity to explore new applications of rigid-foldable origami with compression-torsion coupling effects.

Investigation of rogue wave and dynamic solitary wave propagations of the $$\mathbf{M}$$-fractional (1 + 1)-dimensional longitudinal wave equation in a magnetic-electro-elastic circular rod

Investigation of rogue wave and dynamic solitary wave propagations of the $$\mathbf{M}$$-fractional (1 + 1)-dimensional longitudinal wave equation in a magnetic-electro-elastic circular rod

Algebraic solitons in the massive Thirring model.

We present exact solutions describing dynamics of two algebraic solitons in the massive Thirring model. Each algebraic soliton corresponds to a simple embedded eigenvalue in the Kaup-Newell spectral problem and attains the maximal mass among the family of solitary waves traveling with the same speed. By coalescence of speeds of the two algebraic solitons, we find a new solution for an algebraic double-soliton which corresponds to a double embedded eigenvalue. We show that the double-soliton attains the double mass of a single soliton and describes a slow interaction of two identical algebraic solitons.

Command of three-dimensional solitary waves via photopatterning.

Multidimensional solitons are prevalent in numerous research fields. In orientationally ordered soft matter system, three-dimensional director solitons exemplify the localized distortion of molecular orientation. However, their precise manipulation remains challenging due to unpredictable and uncontrolled generation. Here, we utilize preimposed programmable photopatterning in nematics to control the kinetics of director solitons. This enables both unidirectional and bidirectional generation at specific locations and times, confinement within micron-scaled patterns of diverse shapes, and directed propagation along predefined trajectories. A focused dynamical model provides insight into the origins of these solitons and aligns closely with experimental observations, underscoring the pivotal role of anchoring conditions in soliton manipulation. Our findings pave the way for diverse fundamental research avenues and promising applications, including microcargo transportation and optical information processing.

Exploring Nonlinear Fractional (2+1)-Dimensional KP–Burgers Model: Derivation and Ion Acoustic Solitary Wave Solution

The purpose of this study is to conduct a complete examination into the fractional (2+1)-dimensional nonlinear KP-Burgers (KP-B) model in the setting of quantum plasma. The main goal is to present a mathematical explanation and derivation for the fractional (2+1)-dimensional nonlinear KP-B model. We used the reductive perturbation technique to obtain the fractional (2+1)-dimensional ion acoustic solitary wave, which leads to the nonlinear KP-B model. To solve the fractional space-time KP-B model, we used the modified sub-equation technique and the extended hyperbolic function methodology. This methodology covers rational, periodic wave, and hyperbolic function solutions. Ion acoustic waves were explored in relation to the effects of ion pressure and an external electric field. The effects of density and fractional order on the properties of a single solution are examined. The plasma system is defined as an unmagnetized epi plasma system composed of relativistic ions, positrons, and nonextensive electrons that can be found in a range of astrophysical and cosmological environments. The solution variables include electron-positron and ion-electron temperature ratios, electron and positron nonextensivity strength, ion kinematic viscosity, positron concentration, and the weakly relativistic streaming factor. The fractional order and associated plasma properties have a considerable influence on the phase velocity of ion acoustic waves. When the fractional order equals one, the obtained results are consistent with known outcomes. This work makes a substantial contribution to our understanding of nonlinear events in quantum plasmas, particularly ion acoustic waves. It provides a solid approach for solution development and analysis, with implications for a variety of astrophysical and cosmological scenarios.

Dust–ion–acoustic damped solitary waves and shocks in laboratory and Saturn's E-ring magnetized nonthermal dusty plasmas with anisotropic ion pressure and dust-charge fluctuation

We study the oblique propagation of weakly nonlinear dust–ion–acoustic (DIA) solitary waves (SWs) and shocks in collisional magnetized nonthermal dusty plasmas that are relevant in laboratory and space (Saturn's E-ring) environments. We consider plasmas to be composed of q-nonextensive hot electrons, thermal positive ions, and immobile negatively charged dust grains immersed in a static magnetic field and take into account the effects of ion creation (source), ion loss (sink), ion–neutral and ion–dust collisions, anisotropic ion pressure, and dust-charge fluctuations on the evolution of small-amplitude SWs and shocks. The ion–neutral collision enhancement equilibrium dust-charge number is self-consistently determined using Newton–Raphson method. We found that in laboratory dusty plasmas with adiabatic dust-charge variation [i.e., when the dust charging frequency (νch) is much higher than the dust–plasma oscillation frequency (ωpd)], the DIA solitary waves (DIASWs) get damped by the effects of the ion–dust and ion–neutral collisions, whereas the ion creation and ion loss lead to the amplification of solitary waves, and they appear as only compressive types with positive potential. On the other hand, in Saturn's E-ring plasmas, where the collisional and ion creation or ion loss effects are insignificant, the non-adiabaticity of dust-charge variation can give rise to the evolution of either damped DIASWs or DIA shocks, depending on the smallness of the ratios νch/ωpd or ωpd/νch, respectively. Furthermore, two critical values of the nonextensive parameter q exist, below (or above) which, the DIASWs and shocks can appear as rarefactive (or compressive) types. The characteristics of DIASWs and shocks are also analyzed numerically for parameters relevant to the laboratory and Saturn's E-ring plasmas.

Fuzzy uncertainty modeling of generalized Hirota–Satsuma coupled Korteweg–de Vries equation

This article explores the solitary wave solutions of a generalized Hirota–Satsuma Coupled Korteweg–de Vries (HSCKdV) equation. The HSCKdV equation is a mathematical model that describes certain types of long waves, particularly those found in shallow water. The generalized HSCKdV equation is solved exactly using the Homotopy Perturbation Transform Method (HPTM). By applying this technique, the authors obtain solutions in the form of a convergent power series. These solutions offer an understanding of the characteristics of solitary waves within the domain of shallow water waves. The HSCKdV equation has been solved using the adomian decomposition method, and the results have been compared with those obtained from the HPTM. This comparison demonstrates the effectiveness of the HPTM in solving such nonlinear equations. Further, the HSCKdV equation is extended to a fuzzy version considering the initial condition as a fuzzy parameter. Uncertainty in the initial condition is addressed by representing it using triangular and trapezoidal fuzzy numbers. The generalized fuzzy HSCKdV equation is subsequently tackled using the fuzzy HPTM (FHPTM) providing fuzzy bound solutions. Using the FHPTM, we explain the fuzzy results, highlighting how the solitary wave splits into two solitary waves and noting that the lower and upper bound solutions are interchanged due to negative fuzzy results.