Formation Of Solitary Waves Research Articles

The Influence of Lateral Spreading upon Solitary Wave Formation by Internal Tides

The Influence of Lateral Spreading upon Solitary Wave Formation by Internal Tides

Solitary wave form of reaction rate in graphite diffusive medium using different neutron absorbers

Abstract Graphite nuclear properties such as moderating power and absorption cross-section, are not as good as those of heavy water. But its pure form can be prepared. Its structural and thermal properties are good and it has a high thermal conductivity. The thermal neutron in graphite performs an average of 1,200 scattering collisions before it is absorbed. This very low absorption cross section makes graphite as an ideal material for applications in nuclear reactors. In the current research, graphite is assumed as a diffusive medium due to its low absorption cross-section (0.0035 b) and having a low mass close to the neutron mass. In this medium: Boron (10B), Cadmium (113Cd), Samarium (149Sm), Europium (151Eu), Hafnium (177Hf) and Gadolinium (157Gd), separately are also considered as neutron absorbers. The aim of this paper is obtaining the solitary wave form of reaction rate in graphite diffusive medium using these neutron absorbers.

EFFECT OF EXTERNAL HARMONIC VIBRATION ON THE FORMATION OF SOLITARY WAVES IN FALLING TWO-LAYER LIQUID FILMS

Abstract We study the influence of a low-frequency harmonic vibration on the formation of the two-dimensional rolling solitary waves in vertically co-flowing two-layer liquid films. The system consists of two adjacent layers of immiscible fluids with the first layer being sandwiched between a vertical solid plate and the second fluid layer. The solid plate oscillates harmonically in the horizontal direction inducing Faraday waves at the liquid–liquid and liquid–air interfaces. We use a reduced hydrodynamic model derived from the Navier–Stokes equations in the long-wave approximation. Linear stability of the base flow in a flat two-layer film is determined semi-analytically using Floquet theory. We consider sub-millimetre-thick films and focus on the competition between the long-wavelength gravity-driven and finite wavelength Faraday instabilities. In the linear regime, the range of unstable wave vectors associated with the gravity-driven instability broadens at low and shrinks at high vibration frequencies. In nonlinear regimes, we find multiple metastable states characterized by solitary-like travelling waves and short pulsating waves. In particular, we find the range of the vibration parameters at which the system is multistable. In this regime, depending on the initial conditions, the long-time dynamics is dominated either by the fully developed solitary-like waves or by the shorter pulsating Faraday waves.

Painlevé analysis of the resonant third-order nonlinear Schrödinger equation

The resonant Schrödinger equation of the third order is studied. The Painlevé test for nonlinear partial differential equations is used to determine integrability of equation. It is shown that the necessary condition for integrability of partial differential equations by the inverse scattering transform is fulfilled at certain parameter restriction. Analytical solutions in the form of periodic and solitary wave are presented.

Effect of squared, cubic and quartic on-site potentials on heat conduction in a nonlinear silicon lattice

This paper explains the process of heat conduction in a nonlinear silicon lattice governed by squared, cubic and quartic on-site potentials along with squared and quartic interaction potentials. The tanh method is employed to construct solutions to the resulting nonlinear equations. The results show the transfer of heat in the form of solitary waves which are presented graphically.

Traveling wave solutions of the derivative nonlinear Schrödinger hierarchy

The first three members of the nonlinear ordinary differential equations for the derivative nonlinear Schrödinger hierarchy are considered. Reduction to nonlinear ordinary differential equations is utilized to obtain traveling wave solutions. Lax pairs for these nonlinear ordinary differential equations are presented. The first integrals of nonlinear ordinary differential equations are obtained by taking Lax pairs into account. Exact solutions in the form of solitary and periodic waves are found by means of the generalized method of auxiliary equations.

Generation and Propagation of Flexoelectricity-Induced Solitons in Nematic Liquid Crystals.

Solitons in nematic liquid crystals facilitate the rapid transport and sensing in microfluidic systems. Little is known about the elementary conditions needed to create solitons in nematic materials. In this study, we apply a combination of theory, computational simulations, and experiments to examine the formation and propagation of solitary waves, or "solitons", in nematic liquid crystals under the influence of an alternating current (AC) electric field. We find that these solitary waves exhibit "butterfly"-like or "bullet"-like structures that travel in the direction perpendicular to the applied electric field. Such structures propagate over long distances without losing their initial shape. The theoretical framework adopted here helps identify several key factors leading to the formation of solitons in the absence of electrostatic interactions. These factors include surface irregularities, flexoelectric polarization, unequal elastic constants, and negative anisotropic dielectric permittivity. The results of simulations are shown to be in good agreement with our own experimental observations, serving to establish the validity of the theoretical concepts and ideas advanced in this work.

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.

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.

Invariance properties of the microstrain wave equation arising in microstructured solids

The Lie group method stands as a significant, powerful, and straightforward mathematical tool for discovering precise invariant solutions and traveling wave solutions in prevalent nonlinear evolution equations across engineering, applied mathematics, and physics. The interplay of dispersive effects arising from material microstructure, combined with nonlinearities, results in the formation of solitary waves. In this article, we apply the Lie group method to derive invariant and comprehensive traveling wave solutions for the strain wave equation in microstructured solids. This yields a diverse array of exact solutions, including traveling wave solutions, solitons, shock waves, periodic, singular, and rational solutions, all of which bear potential significance in various engineering applications.

On the dynamics of soliton interactions in the stellar environments

The effects of trapping of relativistically degenerate electrons are studied on the formation and interaction of nonlinear ion-acoustic solitary waves (IASWs) in quantum plasmas. These plasmas are detected in high-density astrophysical entities and can be created in the laboratory by interacting powerful lasers with matter. The formula for the number density of electrons in a state of relativistic degeneracy is provided, along with an analysis of the non-relativistic and ultra-relativistic scenarios. While previous studies have delved into specific aspects of relativistic effects, there needs to be a more detailed and systematic examination of the fully relativistic limit, which is essential for gaining a holistic perspective on the behavior of solitons in these extreme conditions. The aim of this work is to comprehensively investigate the fully relativistic limit of the system to fill this gap. The reductive perturbation technique is utilized to deduce the Korteweg–de Vries (KdV) equation, which is used to analyze the properties of the IASWs. Hirota bilinear formalism is applied to obtain single- and multi-soliton solutions for the KdV equation. The numerical analysis is focused on the plasma properties of the white dwarf in the ongoing investigation. The amplitude of the IASWs is found to be maximum for the non-relativistic, intermediate for the ultra-relativistic, and minimum for the fully relativistic limit. Most importantly, it is found that the fastest interaction occurs in the non-relativistic limit and the slowest in the fully relativistic limit.

Modeling Study of Factors Determining Efficacy of Biological Control of Adventive Weeds

We model the spatiotemporal dynamics of a community consisting of competing weed and cultivated plant species and a population of specialized phytophagous insects used as the weed biocontrol agent. The model is formulated as a PDE system of taxis–diffusion–reaction type and computer-implemented for one-dimensional and two-dimensional cases of spatial habitat for the Neumann zero-flux boundary condition. In order to discretize the original continuous system, we applied the method of lines. The obtained system of ODEs is integrated using the Runge–Kutta method with a variable time step and control of the integration accuracy. The numerical simulations provide insights into the mechanism of formation of solitary population waves (SPWs) of the phytophage, revealing the factors that determine the efficacy of combined application of the phytophagous insect (classical biological method) and cultivated plant (phytocenotic method) to suppress weed foci. In particular, the presented results illustrate the stabilizing action of cultivated plants, which fix the SPW effect by occupying the free area behind the wave front so that the weed remains suppressed in the absence of a phytophage.

Symmetry reductions and qualitative analysis of time fractional K(m,1) equation

This paper studies symmetry reductions of time fractional K(m,1) equation with generalized evolution which is one of the important equation in fractional partial differential equations (FPDEs). Here, the classical Lie symmetries are obtained and utilized for the reduction of time fractional K(m,1) equation into ordinary differential equations. Some new exact solutions are obtained which are in the form of solitary waves by using G′G-expansion method and one of the obtained solutions is acceptable for qualitative analysis. By showing the phase portraits, its dynamical behaviour is further completely explained.

On novel analytical solutions to a generalized Schrödinger’s equation using a logarithmic transformation-based approach

Investigating analytical solutions of nonlinear Schrd̈inger equations in the form of solitary waves can be quite a challenge, but it holds the promise of uncovering fascinating discoveries. In this paper we discover novel solitary wave solutions to the Schrödinger equation, taking into account important aspects such as dispersion, diffraction, and Kerr nonlinearity. To achieve this goal, we utilized an efficient technique that involved applying logarithm transformations. This approach effectively converts the original equation into an ordinary differential equation, presenting symbolic solutions. The resulting solutions exhibit diverse forms such as trigonometric, hyperbolic, and rational functions, finding applications in various fields including plasma physics, nonlinear optics, optical fibers, and nonlinear sciences. The presence of selectively chosen constants in these solutions results in rich and complex dynamical behavior. Furthermore, the article showcases several numerical simulations that correspond to these findings. An important aspect to note is that the methodologies employed in this research have not been previously utilized to solve the model under investigation. Additionally, these techniques can be easily adapted to address a diverse range of other nonlinear problems.

Collisions of ion-acoustic solitary waves in a plasma with negative ions

Head-on collisions of rarefactive ion-acoustic solitary waves and head-on collisions of compressive ion-acoustic solitary waves in a plasma with negative ions are considered. It is shown that such collisions at large amplitudes of the colliding waves lead to a decrease in their amplitudes after the collision. Physical phenomena leading to a loss of energy and a corresponding decrease in the amplitude of the wave after a collision are considered. The phenomenon of wave breaking is studied and the breaking amplitude is determined. The formation of ion-acoustic solitary waves with trapped ions is described. Such a wave, unlike ordinary ion-acoustic solitary waves, transfers not only momentum and energy, but also matter. It is shown that ion-acoustic solitary waves of large amplitudes in a plasma with negative ions cannot be classified as ion-acoustic solitons.

Archipelagos, islands, necklaces, and other exotic structures in external force-driven chaotic dusty plasmas

In this paper, the modified Kadomtsev Petviashvili (mKP) equation is derived by considering the dynamics of the dust-acoustic waves (DAWs) with kappa-distributed hot and cold ions and Boltzmannian electrons. The reductive perturbation technique is applied for deriving mKP. The bifurcation theory of the planar dynamical system is used to obtain the phase portrait of the DAWs in the framework of mKP equation. The formation of dust-acoustic solitary waves (DASWs) is analyzed for different plasma parameters in the Saturn’s magnetosphere. In the current plasma model, the external periodic force is introduced to study the quasiperiodic and chaotic behavior of the DAWs. It is noted that periodic initial conditions lead to the emergence of many outlandish features in the system by comparison with the solitary initial conditions. The frequency of the external periodic force ω plays an important role in the transition from quasiperiodic to chaotic behavior. Moreover, the strength of the external periodic force, the value of the nonlinear coefficient of the mKP equation, and the nonlinear acoustic speed are also found to have a significant effect on the chaotic behavior of the system.

Classical Solutions for the Generalized Korteweg-de Vries Equation

The Korteweg-de Vries equation models the formation of solitary waves in the context of shallow water in a channel. In our system, f or p=2 and p=3 (Korteweg-de Vries equations (KdV)) and (modified Korteweg-de Vries (mKdV) respectively), these equations have many applications in Physics. (gKdV) is a Hamiltonian system. In this article we investigate the generalized Korteweg-de Vries (gKdV) equation. A new topological approach is applied to prove the existence of at least one classical solution. The arguments are based upon recent theoretical results.

On logarithmic transformation-based approaches for retrieving traveling wave solutions in nonlinear optics

Finding solutions in the solitary wave-form for nonlinear Schrödinger’s equations with dispersive and dispersion terms is challenging but would yield interesting results. The basic idea of this paper is to use the idea of logarithmic transformation-based approaches in combination with symbolic structures of exponential functions. The article discusses a nonlinear equation that serves as a universal version of Schrödinger’s equations and has numerous applications in the field of nonlinear optics. We obtain a diverse set of solutions involving trigonometric, hyperbolic, and rational forms. These forms have a broad application spectrum in fields such as plasma physics, nonlinear optics, optical fibers, and nonlinear sciences. Due to the presence of various arbitrarily chosen constants, these solutions exhibit extensive and rich dynamical behavior. In this direction, several numerical simulations corresponding to these results are presented in the article. One notable aspect to consider is that the methodologies utilized in this paper have not been previously employed to solve the model being studied. Furthermore, these techniques may be readily adaptable for solving a diverse range of other nonlinear partial differential equations.

Dependence of the kinetic energy absorption capacity of bistable mechanical metamaterials on impactor mass and velocity

Using an alternative mechanism to dissipation or scattering, bistable structures and mechanical metamaterials have shown promise for mitigating the detrimental effects of impact by reversibly locking energy into strained material. Herein, we extend prior works on impact absorption via bistable metamaterials to computationally explore the dependence of kinetic energy transmission on the velocity and mass of the impactor, with strain rates exceeding 102 s−1. We observe a large dependence on both impactor parameters, ranging from significantly better to worse performance than a comparative linear material. We then correlate the variability in performance to solitary wave formation in the system and give analytical estimates of idealized energy absorption capacity under dynamic loading. In addition, we find a significant dependence on damping accompanied by a qualitative difference in solitary wave propagation within the system. The complex dynamics revealed in this study offer potential future guidance for the application of bistable metamaterials to applications including human and engineered system shock and impact protection devices.

Quasi-parallel propagating solitons in magnetised relativistic electron–positron plasmas

In this article, we study nonlinear waves propagating along the background magnetic field in relativistic electron–positron plasmas. Using the reductive perturbation method, we derive a three-dimensional equation describing these waves. When the perturbations do not vary in the directions orthogonal to the background magnetic field this equation reduces to the vector modified Kortewed–de Vries equation. We present solutions of the obtained equation in the form of planar solitary waves and describe the results of study of their stability with respect to transverse perturbations. We also study numerically non-planar solitary waves.