Types Of Solitary Waves Research Articles

Electron‐acoustic solitons and double layers in the inner magnetosphere

AbstractThe Van Allen Probes observe generally two types of electrostatic solitary waves (ESW) contributing to the broadband electrostatic wave activity in the nightside inner magnetosphere. ESW with symmetric bipolar parallel electric field are electron phase space holes. The nature of ESW with asymmetric bipolar (and almost unipolar) parallel electric field has remained puzzling. To address their nature, we consider a particular event observed by Van Allen Probes to argue that during the broadband wave activity electrons with energy above 200 eV provide the dominant contribution to the total electron density, while the density of cold electrons (below a few eV) is less than a few tenths of the total electron density. We show that velocities of the asymmetric ESW are close to velocity of electron‐acoustic waves (existing due to the presence of cold and hot electrons) and follow the Korteweg‐de Vries (KdV) dispersion relation derived for the observed plasma conditions (electron energy spectrum is a power law between about 100 eV and 10 keV and Maxwellian above 10 keV). The ESW spatial scales are in general agreement with the KdV theory. We interpret the asymmetric ESW in terms of electron‐acoustic solitons and double layers (shocks waves).

Weak bond detection in composites using highly nonlinear solitary waves

We experimentally investigate a diagnostic technique for identifying a weak bond in composites using highly nonlinear solitary waves (HNSWs). We set up a one-dimensional chain of granular crystals, consisting of spherical particles with nonlinear interactions, to generate HNSWs. These solitary wave packets are transmitted into an inspection area of composites by making a direct contact with the chain. We demonstrate that a strong type of solitary waves injected to the weak bond area can break the weak bond of laminates, thereby causing delamination. Then, to identify the creation of the delamination, we transmit a weak type of solitary waves by employing the same apparatus, and measure the solitary waves reflected from the specimens. By analyzing these reflected solitary waves, we differentiate the weak bond samples with the pristine bond ones in an efficient and fast manner. The diagnostic results based on the proposed method are compared with the strength and energy release rate at bond interfaces, which are measured via standard testing methods such as three point bending and end notched flexure tests. This study shows the potential of solitary wave-based detection of weak bonds for hot spot monitoring of composite-based structures.

From solitons to rogue waves in nonlinear left-handed metamaterials.

In the present work, we explore soliton and roguelike wave solutions in the transmission line analog of a nonlinear left-handed metamaterial. The nonlinearity is expressed through a voltage-dependent, symmetric capacitance motivated by recently developed ferroelectric barium strontium titanate thin-film capacitor designs. We develop both the corresponding nonlinear dynamical lattice and its reduction via a multiple scales expansion to a nonlinear Schrödinger (NLS) model for the envelope of a given carrier wave. The reduced model can feature either a focusing or a defocusing nonlinearity depending on the frequency (wave number) of the carrier. We then consider the robustness of different types of solitary waves of the reduced model within the original nonlinear left-handed medium. We find that both bright and dark solitons persist in a suitable parametric regime, where the reduction to the NLS model is valid. Additionally, for suitable initial conditions, we observe a rogue wave type of behavior that differs significantly from the classic Peregrine rogue wave evolution, including most notably the breakup of a single Peregrine-like pattern into solutions with multiple wave peaks. Finally, we touch upon the behavior of generalized members of the family of the Peregrine solitons, namely, Akhmediev breathers and Kuznetsov-Ma solitons, and explore how these evolve in the left-handed transmission line.

CTE method and interaction solutions for the Kadomtsev-Petviashvili equation

The consistent tanh expansion method is applied to the Kadomtsev-Petviashvili equation. The interaction solutions among one soliton and other types of solitary waves, such as multiple resonant soliton solutions and cnoidal waves, are explicitly given. Some special concrete interaction solutions are discussed both in analytical and graphical ways.

Scattering of bulk strain solitary waves in bi-layers with delamination

We study the scattering of longitudinal bulk strain solitary waves in delaminated bi-layers with different types of bonding. The direct numerical modelling of these problems is challenging and has natural limitations. We develop a semi-analytical approach, based on the use of several matched asymptotic multiple-scale expansions and the Integrability Theory of the Korteweg - de Vries equation by the Inverse Scattering Transform. We show that the semi-analytical approach agrees well with the direct numerical simulations and use it to study the scattering of different types of longitudinal bulk strain solitary waves in a wide range of bi-layers with delamination. In particular, we model the dynamics of a long longitudinal strain solitary wave in a symmetric perfectly bonded bi-layer with delamination. The numerical modelling confirms that delamination causes fission of an incident solitary wave and, thus, can be used to detect the defect. We then extend our approaches to the modelling of the waves in bi-layers with soft (“imperfect”) bonding, described by a system of coupled Boussinesq equations and supporting radiating solitary waves. The results may help us to control the integrity of layered structures.

Localized traveling wave solution for a logarithmic nonlinear Schrödinger equation

We investigate localized traveling wave solutions for a Schrödinger equation with two logarithmic nonlinear terms under no external potential. It is shown that it can have solitary wave type solutions whose envelope profile depends on the two types of nonlinearity. Remarkably, the profile has cutoffs in the coordinate of propagation. We argue also some fundamental properties that discriminate it from power law type nonlinear Schrödinger equations.

Wave propagation in periodic buckled beams. Part I: Analytical models and numerical simulations

Periodic buckled beams possess a geometrically nonlinear, load–deformation relationship and intrinsic length scales such that stable, nonlinear waves are possible. Modeling buckled beams as a chain of masses and nonlinear springs which account for transverse and coupling effects, homogenization of the discretized system leads to the Boussinesq equation. Since the sign of the dispersive and nonlinear terms depends on the level of buckling and support type (guided or pinned), compressive supersonic, rarefaction supersonic, compressive subsonic and rarefaction subsonic solitary waves are predicted, and their existence is validated using finite element simulations of the structure. Large dynamic deformations, which cannot be approximated with a polynomial of degree two, lead to strongly nonlinear equations for which closed-form solutions are proposed.

An Approximate Method for Analysis of Solitary Waves in Nonlinear Elastic Materials

Two types of solitary elastic waves are considered: a longitudinal plane displacement wave (longitudinal displacements along the abscissa axis of a Cartesian coordinate system) and a radial cylindrical displacement wave (displacements in the radial direction of a cylindrical coordinate system). The basic innovation is the use of nonlinear wave equations similar in form to describe these waves and the use of the same approximate method to analyze these equations. The distortion of the wave profile described by Whittaker (plane wave) or Macdonald (cylindrical wave) functions is described theoretically

Numerical simulation of interaction between internal solitary waves and submerged ridges

Direct numerical simulations are performed to study the transformation of internal solitary waves (ISWs) of depression type propagating over an underwater ridge in a two-layer fluid system. Bottom ridges with relatively smooth vertex are employed to represent sills in natural lakes and oceans. Consistent with previous experiments, three interaction types (weak, moderate and strong) are observed to be based on the energy loss. In addition, the moderate interaction are found to be categorized into transmitted and reflected type according to their distinct transformation process. General flow characteristics for ISW–ridge interaction in the benthic boundary layer and in the pycnocline is monitored and analysed. A modified degree of blocking Bm considering both the nonlinear effect of incident ISWs and the blockage effect of the submerged ridge is proposed. Different ISW–ridge interactions are discovered to be linked with Bm. Maximum wave-induced velocities, wave energy losses, reflected and transmitted wave amplitudes are found to have a self-similar feature with Bm. The maximum energy loss is up to 35% and the maximum wave-induced velocity can reach 1.8 times of the phase speed of the incident ISW. Empirical equations are obtained based on the data fitting to predict some useful physical parameters during ISW–ridge interaction.

On solitary patterns in Lotka–Volterra chains

We present and study a class of Lotka–Volterra chains with symmetric -neighbors interactions. To identify the types of solitary waves which may propagate along the chain, we study their quasi-continuum approximations which, depending on the coupling between neighbors, reduce into a large variety of partial differential equations. Notable among the emerging equations is a bi-cubic equation which we study in some detail. It begets remarkably stable topological and non-topological solitary compactons that interact almost elastically. They are used to identify discretons, their solitary discrete antecedents on the lattice, which decay at a doubly exponential rate. Many of the discrete modes are robust while others either decompose or evolve into breathers.

Existence and stability results on a class of nonlinear Schrödinger equations in bounded domains with Dirichlet boundary conditions

Existence of solution, and \(L^{2}\) and \(H^{1}\) localization results on a class of nonlinear Schrodinger type equations with a bounded nonlinearity are obtained, for a bounded domain and with Dirichlet boundary conditions. The kind of stability under discussion shows that the corresponding solution exhibits features of a solitary wave type.

Steady free-surface flow over spatially periodic topography

Two-dimensional free-surface flow over a spatially periodic channel bed topography is examined using a steady periodically forced Korteweg–de Vries equation. The existence of new forced solitary-type waves with periodic tails is demonstrated using recently developed non-autonomous dynamical-systems theory. Bound states with two or more co-existing solitary waves are also identified. The solution space for varying amplitude of forcing is explored using a numerical method. A rich bifurcation structure is uncovered and shown to be consistent with an asymptotic theory based on small forcing amplitude.

Dynamics of modulated waves in a lossy modified Noguchi electrical transmission line.

We study analytically the dynamics of modulated waves in a dissipative modified Noguchi nonlinear electrical network. In the continuum limit, we use the reductive perturbation method in the semidiscrete limit to establish that the propagation of modulated waves in the network is governed by a dissipative nonlinear Schrödinger (NLS) equation. Motivated with a solitary wave type of solution to the NLS equation, we use both the direct method and the Weierstrass's elliptic function method to present classes of bright, kink, and dark solitary wavelike solutions to the dissipative NLS equation of the network. Through the exact solitary wavelike solutions to the dissipative NLS equation, we investigate the effects of the dissipative elements of the network on wave propagation. We show that the wave amplitude decreases and its width increases when the dissipative element of the network increases. It has been also found that the dissipative element of the network can be used to manipulate the motion of solitary waves through the network. This work presents a good analytical approach of investigating the propagation of solitary waves through discrete electrical transmission lines and is very important for studying modulational instability.

Kinetic behavior of subsonic solitary wave in graphene nanoribbon

In this paper we investigate the kinetic behavior of subsonic solitary waves in graphene nanoribbons by means of molecular dynamics simulations. Unlike generating the supersonic solitary waves by a strong excitation, we generate three types of subsonic solitary waves by absorbing the thermal fluctuations in the armchair and zigzag graphene nanoribbons. They are localized in longitudinal, transverse, or coupled in both velocity directions with propagation speeds lower than the sound speeds. Their typical width is about 20–80 nm, which is much longer than the width of the supersonic solitary wave. More interestingly, they correspond to energy cavities rather than energy summits in the energy distribution due to the deformation in the density distribution. The observation of subsonic solitary waves with energy cavities implies the numerical evidence of dark solitary waves in graphene. Furthermore, the collisions between two solitary waves are investigated. The nonlinear phase shift only occurs during the collision of two solitary waves localized in the same velocity direction. We hope our results shed light on understanding the particular nonlinear properties of graphene.

Nonlinear Exact Solutions of the 2-Dimensional Rotational Euler Equations for the Incompressible Fluid* *Supported by the National Natural Science Foundation of China under Grant No. 11301269, Jiangsu Provincial Natural Science Foundation of China under Grant No. BK20130665, the Fundamental Research Funds KJ2013036 for the Central Universities, Student Research Training under Grant No. 1423A02 of Nanjing Agricultural University, and the Research

In this paper, the Clarkson-Kruskal direct approach is employed to investigate the exact solutions of the 2-dimensional rotational Euler equations for the incompressible fluid. The application of the method leads to a system of completely solvable ordinary differential equations. Several special cases are discussed and novel nonlinear exact solutions with respect to variables x and y are obtained. It is of interest to notice that the pressure p is obtained by the second kind of curvilinear integral and the coefficients of the nonlinear solutions are solitary wave type functions like tanh(kt/2) and sech (kt/2) due to the rotational parameter k ≠ 0. Such phenomenon never appear in the classical Euler equations wherein the Coriolis force arising from the gravity and Earth's rotation is ignored. Finally, illustrative numerical figures are attached to show the behaviors that the exact solutions may exhibit.

Modulation of Electron-Acoustic Waves in a Plasma with Vortex Electron Distribution

Abstract In the present work, employing a one-dimensional model of a plasma composed of a cold electron fluid, hot electrons obeying a trapped/vortex-like distribution and stationary ions, we study the amplitude modulation of electron-acoustic waves by use of the conventional reductive perturbation method. Employing the field equations with fractional power type of nonlinearity, we obtained the nonlinear Schrödinger equation as the evolution equation of the same order of nonlinearity. Seeking a harmonic wave solution with progressive wave amplitude to the evolution equation it is found that the NLS equation with fractional power assumes envelope type of solitary waves.

Nonlinear internal solitary waves in the northeast South China Sea near Dongsha Atoll using seismic oceanography

We studied nonlinear internal solitary waves in the northeast South China Sea near Dongsha Atoll, mainly using seismic oceanography. The type of internal solitary wave examined was the first mode depression wave, with maximum amplitude 85.5 m. From the depth of this amplitude and below, the amplitude of solitary waves generally decreased linearly, and the rate of decrease with depth was about 0.2. The seafloor can influence the waveforms of these waves. Specifically, the seafloor can cut the bottom of solitary waves, making them discontinuous, and the seafloor friction can induce many short waves near it. The apparent width of solitary waves directly from seismic sections should be corrected to obtain true width. Apparent widths of two solitary waves studied were 4.52 and 3.36 km; these were about 0.5-3 km after correction. There was vertical noise at solitary wave locations, reflecting strong vertical shear forces there. The contrast between horizontal wavenumber spectra of the solitary waves and GM (Garrett Munk) model spectra indicate that their wave energy was about two orders of magnitude larger than the GM spectra at low wave numbers.

Experimental investigation of thermal structures in regular three-dimensional falling films

Interfacial waves on the surface of a falling liquid film are known to modify heat and mass transfer. Under non-isothermal conditions, the wave topology is strongly influenced by the presence of thermocapillary (Marangoni) forces at the interface which leads to a destabilization of the film flow and potentially to critical film thinning. In this context, the present study investigates the evolution of the surface topology and the evolution of the surface temperature for the case of regularly excited solitary-type waves on a falling liquid film under the influence of a wall-side heat flux. Combining film thickness (chromatic confocal imaging) and surface temperature information (infrared thermography), interactions between hydrodynamics and thermocapillary forces are revealed. These include the formation of rivulets, film thinning and wave number doubling in spanwise direction. Distinct thermal structures on the films’ surface can be associated to characteristics of the surface topology.

An exponential expansion method and its application to the strain wave equation in microstructured solids

Abstract The modeling of wave propagation in microstructured materials should be able to account for various scales of microstructure. Based on the proposed new exponential expansion method, we obtained the multiple explicit and exact traveling wave solutions of the strain wave equation for describing different types of wave propagation in microstructured solids. The solutions obtained in this paper include the solitary wave solutions of topological kink, singular kink, non-topological bell type solutions, solitons, compacton, cuspon, periodic solutions, and solitary wave solutions of rational functions. It is shown that the new exponential method, with the help of symbolic computation, provides an effective and straightforward mathematical tool for solving nonlinear evolution equations arising in mathematical physics and engineering.

Study on hydrodynamics and heat transfer of solitary wave on film flow with CFD method

In this paper, a capacitance probe and high speed camera are adopted to study the solitary wave of falling film on large flat plate experimentally. Two types of solitary wave are studied with a Computational Fluid Dynamic (CFD) method based on the measuring data. The water–air shear stress on the interface of the solitary wave is modelled in the numerical calculation. The results indicate that the vortices inside the solitary wave have a great effect on the heat transfer distribution. Heat transfer in the solitary waves is dominated by either conduction or convection, which depends on the Reynolds number. In a reasonable air velocity range, the HTC of the solitary wave is enhanced by downward current air action while it is weakened by counter current air action, which is significant to Passive Containment Cooling System (PCCS) in nuclear engineering application.