Formation Of Solitary Waves Research Articles

Existence and stability of dispersive solutions to the Kadomtsev–Petviashvili equation in the presence of dispersion effect

The paper deals with Kadomtsev–Petviashvili (KP) equation in presence of a small dispersion effect. The nature of solutions are examined under the dispersion effect by using Lyapunov function and dynamical system theory. We prove that when dispersion is added to the KP equation, in certain regions, yet there exist bounded traveling wave solutions in the form of solitary waves, periodic and elliptic functions. The general solution of the equation with or without the dispersion effect are obtained in terms of Weirstrass ℘ functions and Jacobi elliptic functions. New form of kink-type solutions are established by exploring a new technique based on factorization method, use of functional transformation and the Abel’s first order nonlinear equation. Furthermore, the stability analysis of the dispersive solutions are examined which shows that the traveling wave velocity is a bifurcation parameter which governs between different classes of waves. We use the phase plane analysis and show that at a critical velocity, the solution has a transcritical bifurcation.

Nonlinear propagation of electrostatic travelling waves in degenerate dense magnetoplasmas

The electrostatic travelling wave structures have been investigated in degenerate dense magnetoplasmas consisting of relativistic degenerate inertialess electrons and positrons, as well as nondegenerate inertial cold ions. The well known reductive perturbation method has been applied in obtaining the nonlinear Zakharov-Kuznetsov equation. The formation of the electrostatic solitary waves and periodic travelling waves is carried out by the bifurcation theory of planar dynamical systems. The results demonstrate that the nonlinear electrostatic travelling waves strongly depend on the concentration of positrons, the ion cyclotron frequency, and the direction cosines of the wave vector. The obtained results here may be useful for better understanding of the structure of nonlinear electrostatic travelling wave in dense plasmas, such as the interior of massive white dwarfs.

Few-cycle solitons in supercontinuum generation dynamics

We review several propagation models that do not rely on the slowly-varying-envelope approximation (SVEA), and can thus be considered as fundamental models addressing the formation and propagation of few-cycle pulsed field structures and solitary waves arising in the course of intense ultrashort optical pulse evolution in nonlinear media and beyond octave-bandwidth optical spectrum broadening. These generic models are: the modified-Korteweg-de Vries (mKdV), the sine-Gordon (sG), and the mixed mKdV-sG equations. To include wave polarization dynamics, the vector extensions of both mKdV and sG equations are introduced. Multi-octave-spanning supercontinuum generation and few-cycle soliton structures are highlighted from numerical simulations.

Conservation laws and conserved quantities for (1+1)D linearized Boussinesq equations

Conservation laws and physical conserved quantities for the (1+1)D linearized Boussinesq equations at a constant water depth are presented. These equations describe incompressible, inviscid, irrotational fluid flow in the form of a non steady solitary wave. A systematic multiplier approach is used to obtain the conservation laws of the system of third order partial differential equations (PDEs) in dimensional form. Physical conserved quantities are derived by integrating the conservation laws in the direction of wave propagation and imposing decaying boundary conditions in the horizontal direction. One of these is a newly discovered conserved quantity which relates to an energy flux density.

Dust ion acoustic solitary and shock waves in strongly coupled dusty plasma in presence of attractive interparticle interaction potential

The streaming of ions through the dust particles significantly modifies the repulsive Debye-Hückel type of interaction in the sheath and pre-sheath regions and leads to the appearance of oscillatory and attractive wake type interaction. In this paper, we investigate the formation of dust-ion acoustic solitary and shock waves in the presence of streaming ions in a strongly coupled dusty plasma. The generalized hydrodynamic equation is used to describe the dust dynamics, and the dust-dust interaction has been taken into account via the electrostatic pressure term. The novel feature of the present study is the incorporation of both the repulsive and attractive interactions in the electrostatic pressure term. The flow of ions along the vertical direction in the sheath/presheath region of the plasma brings asymmetry to the dust dynamics, which has been carefully handled while considering the effect of strong coupling in the formation of nonlinear structures in such plasma. The study reveals that the ion streaming velocity has a crucial role in controlling the nonlinear coherent structures.

Solitary Waves in Chains of High-Index Dielectric Nanoparticles

We study both linear and nonlinear propagation of pulses in waveguides composed of high-index dielectric nanoparticles supporting both electric and magnetic resonances. We reveal that short pulses (∼100 fs) broaden significantly in the linear regime after propagating only several tens of micrometers, due to a strong waveguide dispersion. In the nonlinear regime, the pulses propagating in the chains of spherical nanoparticles broaden even more strongly than in the linear regime due to defocusing nonlinearity. However, for the chains of nanodisks the pulse broadening can be compensated by the nonlinear effect, due to the interplay of the electric and magnetic resonances that can change the sign of the group-velocity dispersion for some frequencies, making possible the formation and propagation of solitary waves and effective generation of the new frequencies. Our results demonstrate that considered systems can serve as a promising platform for nonlinear and ultrafast nanophotonics, allowing the observation ...

Solitary wave generation from continuum in asymmetric oppositely directed nonlinear waveguide coupler

In a model which describes asymmetric oppositely directed nonlinear waveguide coupler it was observed in the numerical simulation a phenomenon of solitary wave formation from the input constant continuous wave set at the entrance of a waveguide with negative index of refraction. Threshold value of the amplitude of the constant continuous wave, which defines the condition of appearance of the first solitary wave, decreases with increasing of the parameter of nonlinearity. The period of solitary wave formation decreases with increasing of the continuum wave amplitude.

Solitary waves in diatomic chains.

We consider the mechanism of formation of isolated localized wave structures in the diatomic Fermi-Pasta-Ulam (FPU) model. Using a singular multiscale asymptotic analysis in the limit of high mass mismatch between the alternating elements, we obtain the typical slow-fast time scale separation and formulate the Fredholm orthogonality condition approximating a sequence of mass ratios supporting the formation of solitary waves in the general type of diatomic FPU models. This condition is made explicit in the case of a diatomic Toda lattice. Results of numerical integration of the full diatomic Toda lattice equations confirm the formation of these genuinely localized wave structures at special values of the mass ratio that are close to the analytical predictions when the ratio is sufficiently small.

On the nonlinear internal waves propagating in an inhomogeneous shallow sea

Summary A concept of conservation of energy flux for the internal waves propagating in an inhomogeneous shallow water is examined. The emphasis is put on an application of solution of the Korteweg–de Vries (KdV) equation in a prescribed form of the cnoidal and solitary waves. Numerical simulations were applied for the southern Baltic Sea, along a transect from the Bornholm Basin, through the Slupsk Sill and Slupsk Furrow to the Gdansk Basin. Three-layer density structure typical for the Baltic Sea has been considered. An increase of wave height and decrease of phase speed with shallowing water depth was clearly demonstrated. The internal wave dynamics on both sides of the Slupsk Sill was found to be different due to different vertical density stratification in these areas. The bottom friction has only negligible influence on dynamics of internal waves, while shearing instability may be important only for very high waves. Area of possible instability, expressed in terms of the Richardson number Ri , is very small, and located within the non-uniform density layer, close to the interface with upper uniform layer. Kinematic breaking criteria have been examined and critical internal wave heights have been determined.

Solitary Wave Formation from a Generalized Rosenau Equation

A generalized viscous Rosenau equation containing linear and nonlinear advective terms and mixed third- and fifth-order derivatives is studied numerically by means of an implicit second-order accurate method in time that treats the first-, second-, and fourth-order spatial derivatives as unknown and discretizes them by means of three-point, fourth-order accurate, compact finite differences. It is shown that the effect of the viscosity is to decrease the amplitude, curve the wave trajectory, and increase the number and width of the waves that emerge from an initial Gaussian condition, whereas the linear convective term pushes the wave front towards the downstream boundary. It is also shown that the effect of the nonlinear convective term is to increase the steepness of the leading wave front and the number of sawtooth waves that are generated behind it, while that of the first dispersive term is to increase the number of waves that break up from the initial condition as the coefficient that characterizes this term is decreased. It is also shown that, for reasons of stability, the second dispersion coefficient must be much smaller than the first one and its effects on wave propagation are relatively small.

Generation of solitary waves from continuous radiation in a nonlinear oppositely directed coupler

We consider a nonlinear coupler formed by two tunnel-coupled waveguides, one waveguide being made of a conventional dielectric and the other – of a negative-index material. The possibility of the formation of solitary waves from continuous radiation having a constant intensity is shown provided that the radiation is coupled into the input of a negative-index coupler channel (on the back side of the waveguide system). With increasing intensity of the input light, the speed and amplitude of the generated solitary waves increase and the period of their formation is reduced.

On the possibility of the existence of shock waves in nonisothermic plasma

The possibility of the existence of shock waves in nonisothermic plasma with Te » Ti, in which isothermal ion sound exists, Te = const, is theoretically studied. It is shown that propagation of only a density perturbation in the form of a nonlinear solitary wave (soliton) is possible in such plasma. Electron and ion charges are separated in the density solitary wave with the formation of a double electric layer with a strong electric field. The field potential shape, amplitude, and front width of the wave are found.

Electron-scale dissipative electrostatic solitons in multi-species plasmas

The linear and nonlinear properties of small-amplitude electron-acoustic solitary waves are investigated via the fluid dynamical approach. A three-component plasma is considered, composed of hot electrons, cold electrons, and ions (considered stationary at the scale of interest). A dissipative (wave damping) effect is assumed due to electron-neutral collisions. The background (hot) electrons are characterized by an energetic (excessively superthermal) population and are thus modeled via a κ-type nonthermal distribution. The linear characteristics of electron-acoustic excitations are discussed, for different values of the plasma parameters (superthermality index κ and cold versus hot electron population concentration β). Large wavelengths (beyond a threshold value) are shown to be overdamped. The reductive perturbation technique is used to derive a dissipative Korteweg de-Vries (KdV) equation for small-amplitude electrostatic potential disturbances. These are expressed by exact solutions in the form of dissipative solitary waves, whose dynamics is investigated analytically and numerically. Our results should be useful in elucidating the behavior of space and experimental plasmas characterized by a coexistence of electron populations at different temperatures, where electron-neutral collisions are of relevance.

Dust Acoustic Compressive Waves in a Warm Dusty Plasma Having Non-Thermal Ions and Non-Isothermal Electrons

In this article an investigation is presented on the properties of dust acoustic (DA) compressive solitary wave propagation in an adiabatic dusty plasma, including the effect of non-thermal positive and negative ions and non-isothermal electrons. The reductive perturbation method has been employed to derive the lower degree modified Kadomtsev-Petviashivili (mK-P), 3D Schamel-Korteweg-de-Vries equation or modified Kadomtsev-Petviashivili (mK-P) equations for dust acoustic solitary waves in a homogeneous, unmagnetized and collisionless plasma whose constituents are non-isothermal electrons, singly charged positive and negative non-thermal ions and massive charged dust particles. The stationary analytical solutions of the lower degree mK-P and mK-P equations are numerically analyzed, where the effect of various dusty plasma constituents on DA solitary wave propagation is taken into account. It is observed that both the ions in dusty plasma play a key role in the formation of DA compressive solitary waves, and also the ion concentration and non-isothermal electrons control the transformation of the compressive potentials of the waves.

Alternating current-driven graphene superlattices: Kinks, dissipative solitons, dynamic chaotization.

The possibility of the solitary electromagnetic wave formation in graphene superlattice subjected to the electromagnetic radiation is discussed. The chaotic behavior of the electron subsystem in graphene superlattice is studied by Melnikov method. Dynamic chaos of electrons is shown to appear for certain intervals of frequencies of incident electromagnetic radiation. The frequency dependence of the radiation critical amplitude which determines the bound of chaos appearance is investigated. The values of radiation frequency at which the critical amplitude increases indefinitely were found.

Effect of nonthermal electrons and positrons on ion-acoustic solitary waves in a plasma with warm drifting ions

Using Sagdeev potential technique, ion-acoustic solitary waves have been theoretically studied in collisionless unmagnetized plasma consisting of nonthermal electrons, nonthermal positrons and warm drifting positive ions. Ion-acoustic solitary wave solutions have been obtained and its profiles have been numerically analyzed with respect to plasma parameters. The nonthermal parameters and temperatures are shown to play significant role on the formation and nature of solitary waves. It is shown that the nature, amplitude and width of lower- and higher-order solitons depend sensitively on these plasma parameters. The dependence of Sagdeev potential profile on different plasma parameters has also been examined. We have found that stronger nonthermality leads to shorter and wider solitons.

Rogue Waves and New Multi-wave Solutions of the (2+1)-Dimensional Ito Equation

AbstractA three-soliton limit method (TSLM) for seeking rogue wave solutions to nonlinear evolution equation (NEE) is proposed. The (2+1)-dimensional Ito equation is used as an example to illustrate the effectiveness of the method. As a result, two rogue waves and a family of new multi-wave solutions are obtained. The result shows that rogue wave can be obtained not only from extreme form of breather solitary wave but also from extreme form of double-breather solitary wave. This is a new and interesting discovery.

Granular chains with soft boundaries: Slowing the transition to quasiequilibrium.

We present here a detailed numerical study of the dynamical behavior of "soft" uncompressed grains in a granular chain where the grains interact via the intrinsically nonlinear Hertz force. It is well known that such a chain supports the formation of solitary waves (SWs). Here, however, the system response to the material properties of the grains and boundaries is explored further. In particular, we examine the details of the transition of the system from a SW phase to an equilibrium-like (or quasiequilibrium) phase, and for this reason we ignore the effects of dissipation in this study. We find that the soft walls slow the reflection of SWs at the boundaries of the system, which in turn slows the journey to quasiequilibrium. Moreover, the increased grain-wall compression as the boundaries are softened results in fewer average grain-grain contacts at any given time in the quasiequilibrium phase. These effects lead to increased kinetic energy fluctuations in the short term in softer systems. We conclude with a toy model that exploits the results of soft-wall systems. This toy model supports the formation of breather-like entities and may therefore be useful for localizing energy in desired places in the granular chain.

Solitary waves generated by bell-shaped initial conditions in the inviscid and viscous GRLW equations

The objective of this paper is three-fold. First, the accuracy of three methods of lines that employ either piecewise analytical integration and yield exponential methods or three-point, fourth-order accurate finite difference approximations for the spatial derivatives, is assessed by comparing the numerical solutions with the exact one of the generalized regularized long wave (GRLW) equation. Second, the three-point, fourth-order accurate, compact method is used to determine the disintegration or breakup of bell-shaped initial conditions for the GRLW and viscous GRLW equations. Third, the dynamics of the GRLW and viscous GRLW equations is studied as a function of the amplitude and width of Gaussian and triangular initial conditions. It is shown that the compact operator method yields slightly more accurate results than an exponential method that treats the source terms with fourth-order accuracy. It is also shown that the disintegration of the bell-shaped initial conditions is characterized by an initial transient, steepening of the leading front and the formation of solitary waves in the absence of viscosity. The number, speed and distance between successive solitary waves are shown to increase as the dispersion parameter is decreased in accord with a linear stability analysis. The number and speed of solitary waves are also functions of the amplitude and width of the bell-shaped initial conditions, whereas the effect of the linear advective terms is to push the solitary waves towards the downstream boundary. In the presence of viscosity, it is shown that the amplitude of the leading solitary wave that results from the disintegration of the initial conditions, decreases as a function of time and the waves exhibit a curvature which is a function of the linear and nonlinear advective and dispersion terms, and the viscosity coefficient. For large viscosities, it is shown that the initial flow adjustment results in a steep leading front whose slope decreases as time increases and waves analogous to the ones observed in the viscous Burgers’ equation are formed. It is also shown that, for the GRLW equation, the number and speed of solitary waves that are formed after the break-up of the initial profile increase as the amplitude and width of the initial conditions is increased, and that Gaussian initial conditions result in faster and larger amplitude solitary waves than triangular ones. The long-term dynamics of the viscous GRLW equation is found to be nearly independent of the initial conditions.

Fluid pressure waves trigger earthquakes

AbstractFluids—essentially meteoric water—are present everywhere in the Earth's crust, occasionally also with pressures higher than hydrostatic due to the tectonic strain imposed on impermeable undrained layers, to the impoundment of artificial lakes or to the forced injections required by oil and gas exploration and production. Experimental evidence suggests that such fluids flow along preferred paths of high diffusivity, provided by rock joints and faults. Studying the coupled poroelastic problem, we find that such flow is ruled by a nonlinear partial differential equation amenable to a Barenblatt-type solution, implying that it takes place in form of solitary pressure waves propagating at a velocity which decreases with time as v ∝ t [1/(n − 1) − 1] with n ≳ 7. According to Tresca-Von Mises criterion, these waves appear to play a major role in earthquake triggering, being also capable to account for aftershock delay without any further assumption. The measure of stress and fluid pressure inside active faults may therefore provide direct information about fault potential instability.