Nonlinear Solitary Waves Research Articles

Two-component nonlinear waves

The generalized equation for the study of two-component nonlinear waves in different fields of physics is considered. In special cases, this equation is reduced to a set of the various well-known equations describing nonlinear solitary waves in the different areas of physics. Using the generalized perturbation reduction method the explicit analytical expressions for the shape and parameters of two-component nonlinear wave is presented. This nonlinear wave solution coincides with the vector 0π pulse of the self-induced transparency.

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.

Three-dimensional nonlinear ion-acoustic solitary waves in the Venusian ionosphere at high altitude

Three-dimensional nonlinear ion-acoustic solitary waves in the Venusian ionosphere at high altitude

Obliquely nonlinear solitary waves in magnetized electron–positron–ion plasma

Obliquely nonlinear solitary waves in magnetized electron–positron–ion plasma

Oblique Electrostatic Solitary and Supersolitary Waves in Earth’s Magnetosheath

Nonlinear electrostatic solitary waves (ESWs) are routinely detected at various regions of Earth’s magnetosphere using the Wideband Data plasma wave receivers mounted on board the Cluster satellite(s). This mission has facilitated the observation and analysis of ESW characteristics, such as amplitude and temporal duration within the magnetosheath, while concurrently determining the density and temperature profiles of energetic electrons. These electron parameters, in conjunction with data from ion experiments, have served as input for the ion-acoustic solitary wave model developed in this article, with the ambition to contribute to the understanding of ESW generation mechanisms. Assuming as the starting point a plasma system comprising inertial ion fluid and kappa-distributed electron populations of different temperatures (i.e., “cold” and “hot” electrons), a nonperturbative approach has been adopted to investigate the existence and properties of solitary waves. A thorough parametric investigation has scrutinized the existence conditions for such localized structures in terms of the plasma configuration parameters. An interesting aspect emerges from the analysis, namely, the possibility for the coexistence of positive and negative polarity structures associated with ion-acoustic modes, in fact, manifested as simultaneously occurring positive polarity supersolitary waves and negative polarity regular solitary waves. Furthermore, our study has investigated the combined effect of the magnetic field strength, electron density, and suprathermal electron statistics on wave dynamics. The outcomes of this research are in agreement with observed electrostatic wave phenomena in the magnetosheath region, thus underscoring the intrinsic relevance of electrostatic supersolitary structures in data obtained by Cluster and other satellite missions.

THE NUMERICAL APPROXIMATION OF STATIONARY WAVE SOLUTIONS FOR TWO-COMPONENT SYSTEM OF NONLINEAR SCHRÖDINGER EQUATIONS BY USING GENERALIZATION PETVIASHVILI METHOD

The Petviashvili method is a numerical method for obtaining fundamental solitary wave solutions of stationary scalar nonlinear wave equations with power-law nonlinearity: , where is a positive definite and self-adjoint operator and is constant. Due to the case being a system of solitary nonlinear wave equations, we generalize the Petviashvili method. We apply this generalized method for a two-component system of Nonlinear Schr dinger Equations (NLSE) for 2-D.

Seasonal and Fortnight Variations in Internal Solitary Waves in the Indonesian Seas From the SWOT Measurements

AbstractUsing the wide‐swath sea surface height (SSH) data from the 1‐day repeat calibration/validation phase of the Surface Water and Ocean Topography (SWOT) mission, we explored fine‐scale evolution of nonlinear internal solitary waves (ISWs) in the Banda/Molucca Seas in the Indonesian Archipelagos. Generated in the Ombai Strait and Lifamatola Passage through semidiurnal tide‐topography interaction, respectively, ISWs in the Banda/Molucca Seas have a SSH amplitude of 10–20 cm and exhibit multiple wave packets with the leading wave crest followed by a series of rank‐ordered secondary wave crests. Due to nonlinearity, the ISWs are observed to propagate northward at a speed faster than the regional mode‐1 internal gravity waves and their amplitudes are modulated fortnightly following the regional spring‐neap tidal cycle. On longer timescale, the observed ISW amplitudes are controlled by seasonal upper ocean stratification changes with a weaker stratification favoring a larger amplitude. By converting the SWOT‐measured SSH data to the interior ocean pressure signals, we quantified the depth‐integrated energy flux associated with the northward‐propagating ISWs to be 5 and 2 kW m−1 in the central Banda and Molucca Seas, respectively. Comparisons with past studies indicate that the ISWs contribute to close to 100% and 40% of the tidally‐induced, northward energy fluxes, respectively, across the Banda and Molucca Seas.

Nonlinear wave localization in an acoustic metamaterial with attached masses through one element of main chain

The nonlinear strain solitary waves in a metamaterial mass-in-mass lattice model with a di-atomic symmetry is studied when attached masses tare located through one element of main chain are studied in the continuum limit. The reasonable variables are introduced to account for strains. An asymptotic procedure is developed to reduce coupled continuum equations to a single equation for longitudinal strains. A correspondence with the classic model with a di-atomic symmetry is established. An improvement of the model is suggested to account for a switch-on/off of the parts of the metamaterial model arising due to the studied kind of symmetry. It allows us to change the behavior of the localized nonlinear strain waves. This improvement is important for description of a durability of the metamaterial. Also it helps to understand new heat processes, like thermal diode, in complex symmetric lattices.

Nonlinear ion-acoustic waves with Landau damping in non-Maxwellian space plasmas

The dynamics of nonlinear ion-acoustic solitary waves in the presence of kinetic (Landau type) damping have been investigated in a collisionless, non-magnetized electron-ion plasma. A cold ion fluid model, coupled to a Vlasov-type kinetic equation for the electron dynamics, has been adopted as a starting point. The electron population was assumed to be in a kappa-distributed state, in account of the non-Maxwellian behavior of energetic (suprathermal) electrons often observed in Space. A multiscale perturbation technique has led to an evolution equation for the electrostatic potential, in the form of a modified Korteweg-de Vries (KdV) equation, incorporating a non-local term accounting for Landau damping (associated with the electron statistics). Exact analytical solutions have been obtained, representing solitary waves undergoing amplitude decay over time. The combined effect of Landau damping and non-Maxwellian electron statistics (via the kappa parameter) on the characteristics of IASWs has been examined. Numerical integration of the evolution equation has been undertaken, to elucidate the importance of kinetic Landau damping on a shock-shaped initial condition. The results of this investigation aim to improve our understanding of the dynamics of nonlinear electrostatic waves under the influence of Landau damping in various space plasma environments.

Observation of dense collisional soliton complexes in a two-component Bose-Einstein condensate

Solitons are nonlinear solitary waves which maintain their shape over time and through collisions, occurring in a variety of nonlinear media from plasmas to optics. We present an experimental and theoretical study of hydrodynamic phenomena in a two-component atomic Bose-Einstein condensate where a soliton array emerges from the imprinting of a periodic spin pattern by a microwave pulse-based winding technique. We observe the ensuing dynamics which include shape deformations, the emergence of dark-antidark solitons, apparent spatial frequency tripling, and decay and revival of contrast related to soliton collisions. For the densest arrays, we obtain soliton complexes where solitons undergo continued collisions for long evolution times providing an avenue towards the investigation of soliton gases in atomic condensates.

Transcending classical diffusion models: nonlinear dynamics and solitary waves in the fractional Chaffee–Infante equation

Transcending classical diffusion models: nonlinear dynamics and solitary waves in the fractional Chaffee–Infante equation

Propagation of solitary waves in origami-inspired metamaterials

We propose a design strategy for creating origami-like mechanical metamaterials with diverse non-linear mechanical properties and capable of remote actuation. The proposed triangulated cylindrical origami (TCO)-inspired metamaterials enable the highly desirable strain-softening/hardening and snap-through behaviors via a multi-material and highly deformable hinge design. Moreover, we couple these novel non-linear mechanical properties of the TCO origami-inspired metamaterials with the transformative ability of hard-magnetic active materials, allowing for untethered shape- and property-actuation in the developed metamaterials. We develop a mathematical modeling framework for the proposed TCO origami-inspired metamaterials, building on approximating the highly deformable hinges as a combination of longitudinal and rotational springs. We validate the accuracy of the developed mathematical modeling approach by comparing the analytically predicted compressive response of a unit cell structure with the corresponding numerical and experimental results. Using the developed mathematical modeling framework, we investigate the magnetic field-induced large deformation and superimposed solitary wave propagation in the TCO origami-inspired metamaterial system. We show that the proposed metamaterial allows us to tune the key characteristics of the enabled non-linear solitary waves, including their characteristic width and amplitude. The proposed design strategy for readily manufacturable origami-inspired metamaterial systems paves a novel path for practical engineering applications. Our studies also underscore the potential of magneto-mechanical interaction in the design of reconfigurable metamaterial systems with superior non-linear mechanical and elastic wave properties.

Linear limit continuation: Theory and an application to two-dimensional Bose–Einstein condensates

We present a coherent and effective theoretical framework to systematically construct numerically exact nonlinear solitary waves from their respective linear limits. First, all possible linear degenerate sets are classified for a harmonic potential using lattice planes. For a generic linear degenerate set, distinct wave patterns are identified in the near-linear regime using a random searching algorithm by suitably mixing the linear degenerate states, followed by a numerical continuation in the chemical potential extending the waves into the Thomas–Fermi regime. The method is applied to the two-dimensional, one-component Bose–Einstein condensates, yielding a spectacular set of waveforms. Our method opens a remarkably large program, and many more solitary waves are expected. Finally, the method can be readily generalized to three dimensions, and also multi-component condensates, providing a highly powerful technique for investigating solitary waves in future works.

Lie symmetries, solitary wave solutions, and conservation laws of coupled KdV‐mKdV equation for internal gravity waves

The propagation of nonlinear solitary and shallow water waves can be examined by nonlinear evolution equations. The dynamics of nonlinear behavior are described by exact solutions having parametric functions. The integrability of nonlinear equation discloses various characteristics of real phenomenon with continuous and fluctuating background. This work aims to generate symmetry reductions and exact solutions of KdV‐mKdV equation, which is a completely integrable system of nonlinear equations and describes combined solitary wave effects in inhomogeneous medium. The infinitesimal generators and their commutative relations are deduced under invariance property of Lie groups. The invariance property provides a symmetry reduction of governing system. Therefore, a continuous process of reductions transmutes the KdV‐mKdV equation to an equivalent system of ODEs. This system of ODEs leads to the exact solutions involving several arbitrary constants and functions. Such obtained results are generalized than earlier works and never reported in literature. Furthermore, adjoint equation and conserved quantities associated with Lagrangian formulation are constructed under Lie symmetries. To reveal the significance of governing phenomenon, the exact solutions are traced via numerical simulation and thus the graphical structures demonstrate line multisoliton, single soliton, lumps, parabolic and stationary wave profiles.

Nonlinear dust-ion acoustic solitary waves for collisional electronegative dusty plasma in the presence of trapped electron distribution

We have investigated the characteristics of nonlinear propagation of dust-ion acoustic solitary waves in collisional electronegative unmagnetized dusty plasma, which consists of trapped electrons, Boltzmann negative ions, mobile positive ions, mobile negative dust particulates, and a uniform background of neutral particles. In account of ion-neutral collisions, the modified Korteweg–de Vries relation has been derived by employing the standard reductive perturbation method. Analytical and numerical solutions of the damped Korteweg–de Vries equation has been presented in which finite difference method is used for numerical solution. On the other hand, the dust charging equation has been solved by using Newton’s Raphson method. It is found that the temperature ratio of free to trapped electrons, ion-neutral collision, concentration of negative ions, dust number density, and dust density perturbation modify the basic properties of the dust-ion acoustic solitary waves. The temporal evolution of dust-ion acoustic solitary waves is crucial as it affects the amplitude and width of wave structure. In addition, the analytical and numerical solutions are compared, and their deviation is graphically illustrated.

Nonlinear complex dynamical analysis and solitary waves for the (3+1)-D nonlinear extended Quantum Zakharov–Kuznetsov equation

This manuscript is a captivating exploration of the (3+1)-D nonlinear extended Quantum Zakharov–Kuznetsov (NLEQZK) equation, revealing its complex dynamics and solitary wave solutions. We unveil the general method behind this equation and transform it into an ordinary differential equation (ODE), then venture further by using the Galilean transformation to create a system of ODEs. Further, we journey through bifurcations, chaos, and other fascinating dynamical properties, culminating in the visualization and analysis of solitary wave solutions. From the elegant W-shaped solitons to singular solitons with unconventional characteristics and hybrid singular and periodic solitons, each is discussed in vivid detail. This work presents a significant advancement in understanding the unpredictable and intricate behavior of the model, inviting readers to explore the captivating world of non-linear waves and dynamical systems.

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.

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.

Solitary deformation waves in two coaxial shells made of material with combined nonlinearity and forming the walls of annular and circular cross-section channels filled with viscous fluid

The aim of the paper is to obtain a system of nonlinear evolution equations for two coaxial cylindrical shells containing viscous fluid between them and in the inner shell, as well as numerical modeling of the propagation processes for nonlinear solitary longitudinal strain waves in these shells. The case when the stress-strain coupling law for the shell material has a hardening combined nonlinearity in the form of a function with fractional exponent and a quadratic function is considered. Methods. To formulate the problem of shell hydroelasticity, the Lagrangian–Eulerian approach for recording the equations of dynamics and boundary conditions is used. The multiscale perturbation method is applied to analyze the formulated problem. As a result of asymptotic analysis, a system of two evolution equations, which are generalized Schamel– Korteweg– de Vries equations, is obtained, and it is shown that, in general, the system requires numerical investigation. The new difference scheme obtained using the Grobner basis technique is proposed to discretize the system of evolution equations. Results. The exact solution of the system of evolution equations for the special case of no fluid in the inner shell is found. Numerical modeling has shown that in the absence of fluid in the inner shell, the solitary deformation waves have supersonic velocity. In addition, for the above case, it was found that the strain waves in the shells retain their velocity and amplitude after interaction, i.e., they are solitons. On the other hand, calculations have shown that in the presence of a viscous fluid in the inner shell, attenuation of strain solitons is observed, and their propagation velocity becomes subsonic.

On the Method of Transformations: Obtaining Solutions of Nonlinear Differential Equations by Means of the Solutions of Simpler Linear or Nonlinear Differential Equations

Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear differential equations. In such a way, we can obtain numerous exact solutions of nonlinear differential equations. We apply this methodology to the classical parabolic differential equation (the wave equation), to the classical hyperbolic differential equation (the heat equation), and to the classical elliptic differential equation (Laplace equation). In addition, we use the methodology to obtain exact solutions of nonlinear ordinary differential equations by means of the solutions of linear differential equations and by means of the solutions of the nonlinear differential equations of Bernoulli and Riccati. Finally, we demonstrate the capacity of the methodology to lead to exact solutions of nonlinear partial differential equations on the basis of known solutions of other nonlinear partial differential equations. As an example of this, we use the Korteweg–de Vries equation and its solutions. Traveling wave solutions of nonlinear differential equations are of special interest in this article. We demonstrate the existence of the following phenomena described by some of the obtained solutions: (i) occurrence of the solitary wave–solitary antiwave from the solution, which is zero at the initial moment (analogy of an occurrence of particle and antiparticle from the vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves (analogy of splitting of a particle into two particles); (iii) soliton behavior of some of the obtained waves; (iv) existence of solitons which move with the same velocity despite the different shape and amplitude of the solitons.