Solitary Waves Research Articles

Dynamics of the breather and solitary waves in plasma using the generalized variable coefficient Gardner equation

This paper presents an analytical approach to study the generalized variable coefficient Gardner equation, aiming to investigate the behaviour of plasma phenomena, which presents challenges due to the non-linear nature of the Gardner equation. The point symmetry group associated with the proposed equation is determined by Lie symmetry analysis. The Exp-function method is employed to construct exact travelling wave solutions and to obtain anomalous solutions with non-zero amplitude. Additionally, conservation laws are derived and analysed from the equation in the frame of Noether symmetry. Through simulation, breather, periodic, and localized solitary wave solutions are also presented in graphical form. Our results are expected to elucidate the impact of the equation in the study of the propagation of Alfvén waves with non-uniform density, temperature, and wave-particle interactions in a magnetized plasma.

Analytical investigation of space–time shifted nonlocal stochastic Sasa–Satsuma equation using the enhanced modified extended tanh-expansion method for stochastic solitary waves Solutions

In this paper, discover exact solutions of the space–time shifted nonlocal stochastic Sasa–Satsuma equations using the enhanced modified extended tanh-expansion method. This method is a sophisticated mathematical approach for analytical and approximate solution of a nonlinear partial differential equations. The space–time shifted nonlocal stochastic Sasa–Satsuma equations define the behavior of two complex-valued functions, u and v, in a nonlinear system. These equations are complicated to solve analytically because they contain a number of components that describe space–time shifting, nonlocal interactions, and stochastic problems. To increase reliability and effectiveness of the solutions, this method couples the extended tanh-function method with an enhanced method. In order to demonstrate how white noise affects the solutions to the space–time shifted nonlocal stochastic Sasa–Satsuma equations, we also use Matlab to create 3D surfaces and contour graphs of exact solutions.

Spatiotemporal chirped solitary waves to the generalized ($${\textbf {3}}+{\textbf {1}}$$)-dimensional nonlinear Schrödinger equation with varying sources under different diffraction and potential functions

Spatiotemporal chirped solitary waves to the generalized ($${\textbf {3}}+{\textbf {1}}$$)-dimensional nonlinear Schrödinger equation with varying sources under different diffraction and potential functions

Peakon, Periodic Peakons, Compactons and Bifurcations of nonlinear Schrödinger’s Equation with Kudryashov’s Law of Refractive Index

In this paper, we consider the nonlinear Schrödinger’s equation with Kudryashov’s law of refractive index. By using the method of dynamical systems, we obtain bifurcations of the phase portraits of the traveling wave system under different parameter conditions. Corresponding to some special level curves, we derive possible exact explicit parametric representations of solutions (including peakon, periodic peakon, solitary wave solutions and compactons) under different parameter conditions.

Abundant dynamical solitary waves solutions of M -fractional Oskolkov model

This work uses a truncated M-fractional derivative variant of the Oskolkov model to investigate the dynamic behavior of solitary wavefronts. The methods used in this framework produce a variety of solitary waveforms, such as bright and dark solitons. A suitable choice of the free parameters is used to investigate the geometrical structures for the wave solutions, which are further characterized by stable bright periodic and soliton waves. The coefficient of the highest-order derivative and the effects of fractionality are shown in the figures. Moreover, the graphics are arranged to highlight the characteristics of novel soliton wave propagation. The findings of this research demonstrate that the fractional Oskolkov model may accommodate fundamental and higher-order soliton behaviors, each of which has unique characteristics. The fractional form of the several dynamical solitary waves seen in the study represents their practical ramifications. These waves can be seen as transmission waves via a Kelvin-Voigt fluid.

تخامد لانداو للأمواج السوليتونية الصوتية الغبارية في بلازما غبارية غير شاملة

تم دراسة الأمواج السوليتونية الصوتية الغبارية تحت تأثير تخامد لانداو للبلازما الغبارية الفضائية تحوي أيونات خاضعة لتوزع سرعة غير شامل. قمنا بالتحقيق في تأثير تخامد لانداو والبارمتر غير الشامل على البنى السوليتونية الصوتية الغبارية من خلال الحل العددي لمعادلة كورديفيك- دي فريس المعدلة بحد تخامد لانداو. باستخدام طريقة الاضطراب الاختزالية قمنا باستنتاج معادلة كورتيفيك – دي فريس مع إضافة حد تخامد لانداو. أظهرت هذه الدراسة أنً كثافة حبيبات الغبار تلعب دوراَ أساسياً في ظهور أو عدم ظهور الأمواج السوليتونية DA في البلازما الغبارية، والتأثير الواضح للخاصية غير الشاملة للأيونات على ظاهرة تخامد لانداو للبنى اللاخطية المتشكلة في البلازما الغبارية. تفيد نتائج هذه الدراسة في محاولة فهم الآلية الفيزيائية للانتشار الأمواج السوليتونية تحت تأثير تخامد لانداو في بيئات البلازما المغبرة غير الشاملة كالمناطق الكوكبية، والأغلفة المغناطيسية، وبيئات البلازما الفضائية.

Dynamic nature of analytical soliton solutions of the nonlinear ZKBBM and GZKBBM equations

A solitary wave, characterized as a localized perturbation in a medium, emerges as a result of a delicate equilibrium between nonlinear and dispersive phenomena. Solitons, a subtype of solitary waves, exhibit persistent shape and velocity during propagation, representing a fundamental phenomenon observed widely in natural systems and possessing various applications in nonlinear dynamics. This investigation focuses on two nonlinear evolution equations (NEEs), specifically the Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation and the generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony (GZKBBM) equation, which find relevance in domains such as fluid dynamics and ocean engineering. Utilizing an ansatz-based methodology, soliton solutions of both bright and dark characteristics are derived, alongside exploration of rogue wave-type solutions. Notably, the manifestation of dark, bright, and rogue waves aligns with the physical interpretation of the generated solitons. Computational simulations conducted using Wolfram Mathematica aim to provide a comprehensive description of the physical phenomena. The novelty of this study lies in its unreported investigation, contributing new insights into the solitonic dynamics within the considered models.

Dynamics of multi-mode nonlinear internal waves in fluids of great depth

This paper is mainly concerned with the dynamics of strongly nonlinear internal waves in two-dimensional fluids of great depth. A fully nonlinear model for a three-layer fluid of great depth containing topography, a Miyata–Choi–Camassa (MCC)-type system, is developed based on the generalization of the Ablowitz–Fokas–Musslimani global-relation formulation for free-surface water waves. Mode-1 internal solitary waves are numerically found in the MCC-type equation using the Petviashvili iteration technique and in the full Euler equations based on the boundary integral equation method. The comparison between the results validates the broad applicability of the MCC-type system. Mode-2 internal solitary waves are found in the MCC-type equations and are shown to be a type of gap solitary wave. Using the multi-mode internal solitary-wave solutions in the MCC-type equations, we apply the conformal mapping technique to calculate streamlines and particle trajectories. For unsteady simulations, we propose a pseudo-spectral algorithm to handle the time-coupled equations and investigate the generation mechanism of multi-mode internal solitary waves due to the resonance effect of local protrusion on the rigid wall, as well as their collisions when the wall is flat. For flows past protrusion on the rigid wall, the flow speed can be divided into subcritical, transcritical, and supercritical. The shedding of multi-mode internal solitary waves can occur only in the transcritical region of flow speed. We implement the carving of bifurcation curves with inflection to achieve the regional delimitation and, based on this, the interpretation of the generation mechanism from a physical point of view.

Uncertainties in regularized long-wave equation and its modified form: A triangular fuzzy-based approach

This article explores the solution of the regularized long-wave equation (RLWE) and modified RLWE (MRLWE) using a semi-analytical approach known as the homotopy perturbation transform method (HPTM), revealing the characteristics of shallow water waves and ion-acoustic plasma waves. The effectiveness and accuracy of the technique are demonstrated by solving scenarios involving a single solitary wave (SSW) and two solitary waves (TSW) presented and compared with the exact solution of the RLWE. Furthermore, we introduced a fuzzy model for both RLWE and MRLWE, considering uncertainties in the coefficients related to the wave amplitude, and to understand the behavior of both fuzzy RLWE (FRLWE) and fuzzy MRLWE (FMRLWE) in the SSW by examining various numerical results using MATLAB.

Intermittency, bursty turbulence, and ion and electron phase-space holes formation in collisionless current-carrying plasmas

In the previous studies of nonlinear saturation of the Buneman instability caused by high electron drift velocity relative to ions, the phase-space holes and the plateau on the electron velocity distribution function were identified as features of the saturation stage of instability [notably in the paper by Omura et al., J. Geophys. Res. 108, 1197 (2003)]. We have performed a much longer simulation of the Buneman instability and observed a secondary instability. This secondary instability generates fast electron-acoustic waves. By analyzing the phase-space plot of ions and electrons, we show that the fast electron heating and the formation of the plateau of electron velocity distribution function are not due to the quasi-linear diffusion but due to the nonlinear interaction of ion- and electron-acoustic solitary waves (phase-space holes) by exchange of trapped electrons in each wave. We also report the details on the intermittent and bursty nature of turbulence driven by this instability.

Examining the Effect of Kappa Index on Solitary Wave Potential at plasma environment of Mars

Examining the Effect of Kappa Index on Solitary Wave Potential at plasma environment of Mars

Dust acoustic solitons in a positively charged dust plasma with regularized-κ distributed electrons in the presence of generalized polarization force

The propagation features of dust acoustic waves in a three-component plasma system composed of regularized Kappa distributed electrons, Maxwellian ions, and dust grains carrying positive charges are investigated. The reductive perturbation technique is employed to derive the KdV equation. A generalized expression for the polarization force is derived and the effect of the polarization force is taken into consideration as well. The bifurcation analysis is used, and the solitary wave solution was investigated. The critical value of the superthermal spectral index κ is introduced at which the solitonic structure turns up from rarefactive to compressive. It is found that in the range 0<κ<2.23, a rarefactive structure is obtained while the compressive structure appears for κ>2.23. In addition, it is found that by increasing the value of cutoff parameter α, the polarization strength increases too. All the obtained results are helpful to investigate the characteristics of the nonlinear wave propagating in the mesosphere region.

Bifurcations and Exact Solutions of Optical Soliton Models in Fifth-Order Weakly Nonlocal Nonlinear Media

For the optical soliton model in fifth-order weakly nonlocal nonlinear media, to find its exact explicit solutions, the corresponding traveling wave system is formulated as a planar dynamical system with a singular straight line. Then, by using techniques from dynamical systems and singular traveling wave theory developed by [Li & Chen, 2007] to analyze the planar system and find the corresponding phase portraits, the dynamical behavior of the amplitude component can be assessed. Under different parameter conditions, exact explicit solitary wave solutions, periodic wave solutions, kink, and anti-kink wave solutions, compacton solutions, as well as peakons and periodic peakons are found with precise formulations.

Propagation of solitary wave solutions to (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili equation arise in mathematical physics and stability analysis

This innovative study simulates the (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili (DSKP) equation, which has anomalous applications in the field of mathematical physics. The current research has two basic pillars. Firstly, numerous novel soliton solutions are synthesised in distinct formats, such as dark, bright, periodic, combo, W-shape, mixed trigonometric, exponential, and rational, based on the modified sardar sub-equation (MSSE) method and the improved F-expansion method. Secondly, the stability analysis of the selected model is manifested to study modulation instability (MI) gain. Furthermore, for the physical demonstration of the acquired solutions in 3D and 2D, contour and density plots are depicted. The discovered results have a distinctive feature that has not been developed before and indicate a good balance between the nonlinear physical components. Also, the resulting structure of the acquired results can enrich the nonlinear dynamical behaviours of the given system and may be useful in many domains, such as mathematical physics and fluid dynamics, as well as demonstrate that the approaches used are effective and worthy of validation.

Dynamics of solitary waves, chaotic behaviors, and Jacobi elliptic wave solutions in telecommunication systems

This research explores bifurcation of a nonlinear model concerning with the telecommunication strait. Possible all phase plane diagrams due to various parametric conditions are found. Derivation of analytical tangled wave propagating solutions following all phase orbits of the corresponding phase portraits are established. As a results, the soliton, shock wave, singular soliton, periodic wave and singular periodic solutions are obtained by direct integration from Hamiltonian energy function. The periodic solutions of the model are formulated in the form of generalized Jacobi elliptic functions, which also provide the solitonic solution setting the value of beta is unity. Additionally, chaotic and quasi-periodic behaviors have been found for a range of parameter values after adding the perturbed term. The perturbed system’s quasi-periodic and chaotic behavior have been demonstrated using sensitivity analysis. Finally, picturesque explorations are delivered exposing effects of exist parameters of the gained wave solutions. The majority of the achieved results are derived for the first time. Moreover, the solutions show that the novel schemes are very simple, outright, fruitful and successful and that they can be used in wide range of other nonlinear partial differential equations (NLPDEs), which create different kinds of dynamical features of other wave model.

Propagation of compression solitary waves on tensegrity-like lattices made of truncated octahedrons

This work studies the properties of compression solitary waves propagating through one-dimensional mass–spring lattices, whose unit cells show an interaction law mimicking that of a truncated tensegrity octahedron. Analytic methods based on the Weierstrass criterion for compact solitary waves, and the long wave approximation theory are employed to demonstrate that the analyzed systems support compression solitary waves in a suitable range of wave speeds. These systems exhibit an initially soft response, which progressively turns into an elastically hardening behavior ending with a linear branch in the pre-buckling regime of the units cells. Explicit formulae are given for the lower and upper bounds of the wave speed interval that produce compact solitary waves, together with analytic laws of the shape of the traveling wave. Numerical simulations provide a validation of the presented theory.

Solitary waves for the delayed shallow-water wave equations

Solitary waves for the delayed shallow-water wave equations

Investigations into Motion Responses of Suspended Submersible in Internal Solitary Wave Field

When the underwater submersible encounters an internal solitary wave (ISW), its loadings and motions are significantly disturbed. To investigate the interaction mechanism between the suspended submersible and the ISW, a three-dimensional ISW–submersible-interaction numerical model was established, based on the computational fluid dynamics (CFD) method. The generation and propagation of the ISW was simulated in a two-layer fluid numerical wave tank, according to the eKdV theory. The standard operation equation of the submersible was introduced to simulate the six degree of freedom (6DoF) motions of the submersible combined with the overset dynamic mesh method. The motion simulation method was effectively validated by comparing it with published experimental results on the motion responses of a slender body under the ISW. Based on the constructed numerical model, the dynamic mechanisms between the suspended submersible and the ISW were studied, and the effects of the initial submerged depths and the ISW amplitudes on the dynamic responses of the submersible were revealed. According to the numerical results, the motions of the submersible have been significantly determined by its initial submerged depths. The submersible located above the ISW interface has a significant motion along the propagation direction of the ISW and its motion trajectory resembles a counterclockwise semi ellipse. The motion of the submersible located below the ISW interface follows the trace of the lower layer of fluid, which presents as an unclosed clockwise ellipse. The corresponding motions of the submersible would be increased with the increase in the ISW amplitudes.

Solitary wave solutions of the fractional Peyrard Bishop DNA model

Solitary wave solutions of the fractional Peyrard Bishop DNA model

Multi-dimensional phase portraits of stochastic fractional derivatives for nonlinear dynamical systems with solitary wave formation

This manuscript delves into the examination of the stochastic fractional derivative of Drinfel’d-Sokolov-Wilson equation, a mathematical model applicable in the fields of electromagnetism and fluid mechanics. In our study, the proposed equation is through examined through various viewpoints, encompassing soliton dynamics, bifurcation analysis, chaotic behaviors, and sensitivity analysis. A few dark and bright shaped soliton solutions, including the unperturbed term, are also examined, and the various 2D and 3D solitonic structures are computed using the Tanh-method. It is found that a saddle point bifurcation causes the transition from periodic behavior to quasi-periodic behavior in a sensitive area. Further analysis reveals favorable conditions for the multidimensional bifurcation of dynamic behavioral solutions. Different types of wave solutions are identified in certain solutions by entering numerous values for the parameters, demonstrating the effectiveness and precision of Tanh-methods. A planar dynamical system is then created using the Galilean transformation, with the actual model serving as a starting point. It is observed that a few physical criteria in the discussed equation exhibit more multi-stable properties, as many multi-stability structures are employed by some individuals. Moreover, sensitivity behavior is employed to examine perturbed dynamical systems across diverse initial conditions. The techniques and findings presented in this paper can be extended to investigate a broader spectrum of nonlinear wave phenomena.