Amplitude Solitary Waves Research Articles

Ginzburg–Landau equations involving different effects and their solitary waves

Ginzburg–Landau equations describe a wide range of phenomena involving superconductivity, superfluidity, etc. In the present paper, Ginzburg–Landau equations involving distinct laws are considered, and as a consequence, their solitary waves in the presence of perturbation terms are formally derived using the Kudryashov method. The effect of Kerr and parabolic laws on the dynamics of solitary waves is examined in detail. The outcomes of the current paper present suitable ways to control the width and amplitude of solitary waves. The authors believe that the results reported in the current study will contribute significantly to studies related to Ginzburg–Landau equations with distinct laws.

The effect of dust on the propagation of a hydromagnetic solitary wave along the magnetic field in a cold collision-free plasma

The effect of dust is studied on the parallel propagation of a solitary hydromagnetic wave in a magnetized, cold, collision-free plasma. The fully nonlinear analysis is based on the Adlam–Allen method. The system of nonlinear differential equations was derived using the assumption of massive dust, namely, that the dust velocity is constant and parallel to the background magnetic field. Self-consistent solutions obtained from the model for the lighter particles dynamics and for the resulting electric and magnetic fields are presented, together with the validity range of Mach numbers as a function of dust parameters. In particular, supersonic solutions are possible in scenarios tending to two-component plasmas (positively charged ions and negatively charged dust). The stationary solitary pulse in terms of particle densities, magnitude of transverse magnetic field, transverse electron–ion velocities, and bipolar electric field is presented. The magnitude of these amplitudes is maximum at the center of wave and minimum at the boundary. Furthermore, it is shown that all the solitary wave amplitudes are decreases with dust parameter.

Uniform solitary wave theory for viscous flow over topography

The flow of a density stratified fluid over obstacles has been intensively explored from a natural and scientific point of view. This flow has been successfully governed by using the forced Korteweg–de Vries-Burgers equation that generated solitons in a viscous flow. This is done by adding the viscous term beyond the Korteweg–de Vries approximation. It is based on the conservation laws of the Korteweg–de Vries-Burgers equation for mass and energy, and assumes that the upstream wavetrains are composed of solitary waves. Our results show that the influence of viscosity plays a key role in determining the upstream solitary wave amplitude of the bore. A good comparison is obtained between the numerical and analytical solutions.

Numerical modelling of ship-generated solitary waves

A ship sailing at the wave speed in shallow water produces a complex wave pattern including a downstream disturbance in addition to a periodically generated upstream disturbance. The upstream component consists of solitary waves which are generated at the ship bow and emitted forward as soon as the local water depth is sufficiently modified enabling them to bypass the shallow water wave speed limit. Reynolds Averaged Navier-Stokes numerical simulations are carried out to explore this phenomenon in a fully non-linear and viscous virtual towing tank in cases where a ship sails at the waterway centreline as well as off-centreline conditions. Results indicate that friction attains no more than approximately 25 % of the total resistance coefficient, justifying the use of inviscid methods by previous studies in the literature. Water depth may have a significant impact on the frequency and amplitude of ship-generated solitary waves, but the manner in which this occurs is highly sensitive to the width of the waterway.

Nonlinear interaction of head-on solitary waves in integrable and nonintegrable systems

This study numerically investigates the nonlinear interaction of head-on solitary waves in a granular chain (a nonintegrable system) and compares the simulation results with the theoretical results in fluid (an integrable system). Three stages (the pre-in-phase traveling stage, the central-collision stage, and the post-in-phase traveling stage) are identified to describe the nonlinear interaction processes in the granular chain. The nonlinear scattering effect occurs in the central-collision stage, which decreases the amplitude of the incident solitary waves. Compared with the leading-time phase in the incident and separation collision processes, the lagging-time phase in the separation collision process is smaller. This asymmetrical nonlinear collision results in an occurrence of leading phase shifts of time and space in the post-in-phase traveling stage. We next find that the solitary wave amplitude does not influence the immediate space-phase shift in the granular chain. The space-phase shift of the post-in-phase traveling stage is only determined by the measurement position rather than the wave amplitude. The results are reversed in the fluid. An increase in solitary wave amplitude leads to decreased attachment, detachment, and residence times for granular chains and fluid. For the immediate time-phase shift, leading and lagging phenomena appear in the granular chain and the fluid, respectively. These results offer new knowledge for designing mechanical metamaterials and energy-mitigating systems.

Propagation characteristics of nonlinear dust acoustic solitary waves in complex plasma with nonthermal electrons and trapped ions

The propagation characteristics of nonlinear dust acoustic solitary waves in a complex plasma system with nonthermal electrons and trapped ions are investigate in this work. The nonlinear dispersion relation of dust acoustic waves is obtained by using the linear method, and the two-dimensional autonomous system governing the motion of nonlinear dust acoustic waves is derived by using the Sagdeev potential method. At the same time, the specific expression of the Sagdeev potential function is obtained based on the Sagdeev potential equation. The numerical simulations are used to analyze the phase portraits of the two-dimensional autonomous system, revealing the linear periodic wave orbits, nonlinear periodic wave orbits, and homoclinic orbits co-existing in the complex dusty plasma system with nonthermal electrons and trapped ions. Furthermore, from the variations of the Sagdeev potential function with different system parameters it follows that only the compressive solitary waves exist in this complex plasma system. The significant influences of various system parameters on the amplitude, width, and waveform of the nonlinear dust acoustic solitary wave in the complex plasma system are discussed in detail. The results demonstrate that the Mach number, the nonthermal electrons and trapped ions, undisturbed dust particle number density, temperature, and charge have important effects on the propagating characteristics of the nonlinear dust acoustic solitary waves in a complex plasma with nonthermal electrons and trapped ions.

Nonplanar ion-acoustic solitary and cnoidal waves in a non-Maxwellian plasma: Study on nonplanar (modified) Kawahara equation

This study aims to examine the properties of the nonplanar (cylindrical and spherical) ion-acoustic solitary waves (SWs) and cnoidal waves (CWs) in a collisionless, unmagnetized electron-ion (EI) plasma having a Cairns–Tsallis distribution for the electrons. This study is structured around two main lines. The first trend involves deriving the nonplanar Korteweg-de Vries (KdV) equation by utilizing the method of reductive perturbation (MRP). This equation describes small-amplitude (non)planar acoustic waves (AWs). Furthermore, the nonplanar Kawahara equation (KE) is formulated to examine the significant magnitude of planar and nonplanar SWs and CWs. The current plasma model supports compressive and rarefactive IA solitary and cnoidal structures, depending upon the associated physical factors such as the nonextensive parameter (nonextensivity) and nonthermal parameter (nonthermality). When the plasma compositions reach some critical values, such as the critical value of nonthermality, the coefficient of the quadratic nonlinear term vanishes. Hence, both nonplanar modified KdV (mKdV) equation and modified KE (mKE) with cubic nonlinearity are derived to accurately depict the dynamics of both small and large amplitudes of nonplanar SWs and CWs and any other structures related to this family of evolution equations. The influences of the nonextensivity and nonthermality on the profile of (non)planar KdV soliton and the (non)planar Kawahara SWs and CWs are numerically examined using some semi-analytical and numerical approximations. Also, the impact of the nonextensivity on the profile of (non)planar mKdV soliton and the (non)planar modified Kawahara SWs and CWs is reported. It is tracked down that the variety of different plasma parameters significantly alters the characteristic properties of the small and large amplitude ion-acoustic waves (IAWs) discussed by the nonplanar KdV-type equations.

Dynamics and “falling deep” mechanism of submerged floating body under internal solitary waves

Internal solitary waves (ISWs) can result in notable alterations to the hydrodynamic characteristics of submerged bodies, affecting their motion and even leading to serious “falling deep” accidents. This study utilizes a combination of Computational Fluid Dynamics (CFD) and the modified Korteweg-de Vries (mKdV) theory. By combining techniques such as the Volume of Fluid (VOF) model and overset grid, a numerical method for simulating the interaction between ISWs and submerged bodies is established. The dynamics of submerged floating bodies under internal solitary waves are systematically investigated. The results demonstrate that the primary cause of changes in motion and force is the rotating velocity field induced by ISWs. Additionally, the influences of different locations and internal solitary wave amplitudes on the hydrodynamic characteristics of the submerged body are examined. The submerged body is more likely to experience “falling deep” due to the influence of ISWs when it is situated in the lower layer of the fluid.

Modulation theory for solitary waves generated by viscous flow over a step

The effect of dissipation on the flow of a fluid over a step or jump is examined. This flow has been described theoretically and numerically by using the framework of the forced Korteweg–de Vries–Burgers equation. This model includes the influence of the viscosity of the fluid beyond the Korteweg–de Vries approximation. Modulation theory solutions for the undular bores generated upstream and downstream of the forcing are found. Predictions show that the effect of viscosity plays a key role in determining the properties of waves such as the upstream and downstream solitary wave amplitudes and the widths of the bores. The numerical simulations have shown that the unsteady flow consists of upstream and downstream undular bores, which are connected by a locally steady solution over a step. These numerical solutions are compared with modulation theory results with the excellent agreement obtained. Data from the undular bores are all tabulated. Also, the numerical solutions of the Euler equations are compared with the results of the model, which gave good comparisons. Moreover, the comparisons with the experimental measurements of Lee et al. (1989) together with the Navier–Stokes equations show evidence of the significance of this model.

Effects of Nonextensive Electrons on Dust-Ion Acoustic Waves in a Collisional Dusty Plasma with Negative Ions.

The effects of nonextensive electrons on nonlinear ion acoustic waves in dusty negative ion plasmas with ion-dust collisions are investigated. Analytical results show that both solitary and shock waves are supported in this system. The wave propagation is governed by a Korteweg-de Vries Burgers-type equation. The coefficients of this equation are modified by the nonextensive parameter q. Numerical calculations indicate that the amplitude of solitary wave and oscillatory shock can be obviously modified by the nonextensive electrons, but the monotonic shock is little affected.

Investigating overtaking collisions of solitary waves in the Schamel equation

This article presents a numerical investigation of overtaking collisions between two solitary waves in the context of the Schamel equation. Our study reveals different regimes characterized by the behavior of the wave interactions. In certain regimes, the collisions maintain two well-separated crests consistently over time, while in other regimes, the number of local maxima undergoes variations following the patterns of 2→1→2→1→2 or 2→1→2. These findings demonstrate that the geometric Lax-categorization observed in the Korteweg–de Vries equation (KdV) for two-soliton collisions remains applicable to the Schamel equation. However, in contrast to the KdV, we demonstrate that an algebraic Lax-categorization based on the ratio of the initial solitary wave amplitudes is not feasible for the Schamel equation. Additionally, we show that the statistical moments for two-solitary wave collisions are qualitatively similar to the KdV equation and the phase shifts after soliton interactions are close to ones in integrable KdV and modified KdV models.

Interaction of Interfacial Waves with an External Force: The Benjamin-Ono Equation Framework

This study aims to explore the complex interactions between an internal solitary wave and an external force using the Benjamin-Ono equation as the theoretical framework. The investigation encompasses both asymptotic and numerical approaches. By assuming a small amplitude for the external force, we derive a dynamical system that describes the behavior of the solitary wave amplitude and the position of its crest. Our findings reveal three distinct scenarios: (i) resonance between the solitary wave and the external force, (ii) oscillatory motion with closed orbits, and (iii) displacement from the initial position while maintaining the wave direction. However, through numerical simulations, we observe a different relationship between the amplitude of the solitary wave and its crest position. Specifically, for external forces of small amplitude, the simulations indicate the presence of an unstable spiral pattern. Conversely, when subjected to external forces of larger amplitudes, the solitary wave exhibits a stable spiral trajectory which resembles the classical damped mass-spring system.

Serre Equations in Channels and Rivers of Arbitrary Cross Section

A variational approach was used to derive a set of Serre equations for fully nonlinear, dispersive waves in channels of arbitrary cross section. A family of travelling waves was found, as well as the relation between amplitude and celerity of solitary waves. An upper bound is proposed for the solitary wave amplitude as a function of the Froude number in trapezoidal cross-sectional canals, and it showed good agreement with existing theory. For waves of moderate amplitude, cnoidal waves result with a soliton limit; these waves and their properties (celerity and wave number) are written as functions of the channel bank slope and channel bank curvature. The theoretical findings are in agreement with well-established results in the literature, in particular with more-recent Boussinesq-type theories. A validation is proposed against existing experimental data.

Application scope of the reductive perturbation method to derive the KdV equation and CKdV equation in dusty plasma

The application scopes of two different reductive perturbation methods to derive the Korteweg–de Vries (KdV) equation and coupled KdV (CKdV) equation in two-temperature-ion dusty plasma are given by using the particle-in-cell (PIC) numerical method in the present paper. It suggests that the reductive perturbation method (RPM) is valid if the amplitude of the CKdV solitary wave is small enough. However, for the KdV solitary wave, RPM is valid not only if the amplitude of the KdV solitary wave is small enough, but also if the nonlinear coefficient of the KdV equation is not tending to zero.

Nonlinear Dynamics of Ion-Acoustic Waves in a Magnetized Plasma With Trapped Electrons

The dynamics of obliquely propagating ion-acoustic waves (IAWs) in a collisionless, magnetized plasma consisting of free and trapped electrons, having a vortex-like distribution and fluid ions immersed in an external static magnetic field, has been investigated. The reductive perturbation technique (RPT) has been used to derive a Schamel’s modified Laedke–Spatschek (S–LS) equation. Several nonlinear structures have been obtained, like solitary waves, periodic and quasi-periodic oscillations, as well as chaotic structures. The effects of the magnetic field strength, the wave propagation obliqueness, the Mach number, and the electrons trapping on these nonlinear structures have been explored. The increase in the magnetic field strength has been found to reduce the soliton width and enhance its amplitude, while the role of rising the trapping parameter is only to increase the solitary wave amplitude. For a relatively weak magnetic field, the increase in the Mach number may change the solitonic structure to a periodic oscillation and the latter to a quasi-periodic one, while the decrease in the propagation angle to the magnetic field direction may bring about the second transition (periodic <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\rightarrow$</tex-math> </inline-formula> quasi-periodic). Furthermore, depending on the plasma parameters, the intensification of the magnetic field can lead to the transition from a quasi-periodic to a chaotic behavior of the electrostatic wave.

Analysis of solitons structure of the damped KdV equation arising in superthermal plasmas: Application of homotopy analysis method

AbstractThe aim of the proposed work is to analyze the soliton structures of dust‐ion acoustic waves obtained in the framework of the Korteg‐de Vries (KdV) equation with the presence of a damping term. The concept of electron acoustic solitary wave in an unmagnetized plasma consisting of superthermal electrons has been taken into consideration. The KdV equation with the presence of a damping term has been derived with the help of the reductive perturbation technique and solved by using the well‐known homotopy analysis method. The considered method approximates all problems in a straightforward and simplified manner. The method computes the series solution efficiently and provides a simple way to ensure its convergence. The approximate analytical solution obtained from the present analysis is compared with available results in the literature for a different choice of pertinent parameters. The upshots specified that the amplitude of solitary waves increases for increasing values of the damping parameter. This study would in a way to demonstrate the potential and effectiveness of the homotopy analysis method to evaluate the various kinds of nonlinear equations arising in the soliton theory.

Propagation of periodic and solitary waves in a highly dispersive cubic–quintic medium with self-frequency shift and self-steepening nonlinearity

We study the dynamics of femtosecond light pulse propagation in a cubic–quintic medium exhibiting dispersive effect up to the fourth order as well as self-frequency shift and self-steepening nonlinearity. A rich variety of periodic and solitary wave solutions are derived for the governing generalized higher-order nonlinear Schrödinger equation in the presence of self-frequency shift and self-steepening effects. It is found that the frequency shift, inverse velocity, amplitude and wave number of both periodic and solitary waves depend on dispersion coefficients and nonlinearity parameters as well. The conditions on optical fiber parameters for the existence of these structures are presented. The stability of these periodic and solitary wave solutions is studied numerically by adding white noise. It is proved by using the numerical split-step Fourier method that the derived exact solutions are stable by Lyapunov criteria.

Linear and nonlinear analysis of Ion-Temperature-Gradient (ITG) driven mode in the asymmetric pair-ion magnetoplasma

We have investigated linear and nonlinear dynamics of ion-temperature-gradient (ITG) driven drift mode for Maxwellian and non Maxwellian pair-ion plasma embedded in an inhomogeneous magnetic field having gradients in the temperature and number density of ions. Linear dispersion relations are derived and analyzed analytically as well as numerically for different cases. It has been found that growth rate of instability increases with increasing η. By using the transport equations of Braginskii, model, a set of nonlinear equations are derived. In the nonlinear regime, soliton structures are found to exist. Our numerical analysis shows that amplitude of solitary waves increases by increasing ion to electron number density ratio. These solitary structures are also found to be sensitive to asymmetries in pair plasma and non thermal kappa and Cairns distributed electrons. Our present work may contribute a good illustration of the observation of nonlinear solitary waves driven by the ITG mode in magnetically confined pair-ion plasmas and space pair-ion plasmas as the formation of localized structures along drift modes is one of the striking reasons for L-H transition in the region of improved confinements in magnetically confined devices like tokamaks.

Instability of opposite polarity charged dusty plasma with vortex-like distributed electrons

Electrons in space plasma may follow vortex-like distribution, that is why we have derived the three-dimentional (3D) nonlinear modified Zakharov–Kuznetsov (mZK), describing the propagation of dust acoustic solitary to the plasma system composed of four components, magnetized charged dust plasma with electrons modelled by vortex-like distribution. By using the reductive perturbation method, it has been found that both solitary wave amplitude and width are affected by the plasma parameters: graim charge ratio ρ, population parameter β, and temperature ratio σ. The small-k perturbation technique has also been applied to study the instability criterion and growth rate of this instability. Cyclotron frequency Ω, temperature ratio , direction cosine ( and ), and grain charge ratio ρ are found to modify the instability growth rate Γ.

The Solitary Wave in Advanced Nuclear Energy System

The solitary wave naturally arises in many areas of mathematical physics, including in nonlinear optics, plasma physics, quantum field theory, and fluid mechanics. In the past few years, for an advanced nuclear energy system, a particular class of traveling wave reactor called the Constant Axial shape of Neutron flux, nuclide number densities and power shape During Life of Energy production (CANDLE) reactor has been proposed, and an analytical solution has been desired since it could reveal the global characters of the solution. In this study, from the perspective of the solitary wave, the analytical solution of this advanced nuclear energy system is demonstrated through coupling the one-group neutron diffusion equation with the burnup equation. The tanh-function method is applied to solve that nonlinear partial differential equation. The relationship between the velocity of the solitary wave, wave amplitude, or neutron flux and the evolution of the nuclide is revealed by the analytical method. The results demonstrate that the neutron flux is proportional to the wave velocity. The results also imply that the amplitude of the neutron flux is proportional to the square root of the diffusion coefficient but is inversely proportional to the initial 238U density.