Damped Korteweg-de Vries Equation Research Articles

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.

Dissipative Kawahara ion-acoustic solitary and cnoidal waves in a degenerate magnetorotating plasma

The two-dimensional fluid quantum hydrodynamic (QHD) is adopted as the basis for a discussion of the effects of Landau quantization magnetic field, the Coriolis force, and collisional frequency on the (non)linear properties of the dissipative ion-acoustic waves (IAWs). By employing the reductive perturbation technique (RPT), the damped Korteweg–de Vries (KdV) equation which contains the lowest perturbation order actions is derived. It was found that with an increase in amplitude, the soliton width and the velocity diverge from the prediction of the damped KdV equation as observed in some laboratory experiments, i.e. the damped KdV approximation becomes invalid to describe the system. Therefore, it is necessary to investigate the effect of higher-order which leads to the damped Kawahara equation. This equation is a completely non-integrable differential equation. Thus, a new approximate solution which is called a semi-analytical solution is derived in detail. The obtained results can help in understanding the features of quantum IAWs in dense and slowly rotating astrophysical plasmas and maybe understand the quantum Hall effect of novel materials like graphene and topological insulators.

Learning a reduced basis of dynamical systems using an autoencoder.

Machine learning models have emerged as powerful tools in physics and engineering. In this work, we use an autoencoder with latent space penalization to discover approximate finite-dimensional manifolds of two canonical partial differential equations. We test this method on the Kuramoto-Sivashinsky (K-S), Korteweg-de Vries (KdV), and damped KdV equations. We show that the resulting optimal latent space of the K-S equation is consistent with the dimension of the inertial manifold. We then uncover a nonlinear basis representing the manifold of the latent space for the K-S equation. The results for the KdV equation show that it is more difficult to recover a reduced latent space, which is consistent with the truly infinite-dimensional dynamics of the KdV equation. In the case of the damped KdV equation, we find that the number of active dimensions decreases with increasing damping coefficient.

Effect of non adiabatic dust charge fluctuation on nonplanar dust acoustic waves in superthermal polarized plasma

Some properties of cylindrical/spherical dust acoustic solitary waves have been studied in collisionless dusty plasma with variable charges under the influence of polarization force by reductive perturbation method. It is shown that the collisionless damping due to dust charge fluctuation causes the dust acoustic wave propagation to be described by the nonplanar damped Korteweg de Vries equation. By use of weighted residual method, a progressive wave solution to nonplanar damped KdV equation is presented for the first time. It is shown that dust charge fluctuation produce damping which is responsible for wave attenuation. The solution shows that as one goes towards the axis of the cylinder or centre of the sphere, the wave attenuates faster and the effects are shown in some 2D figures. Within a small amplitude limit, the influence of dust charge fluctuation, polarisation power, superthermality and ion temperature on the time evolution of dust acoustic solitons is also demonstrated. In laboratory experiments and space regions, findings obtained here may shed light on the salient characteristics of DA waves.

Nonlinear propagation of fast and slow magnetosonic waves in collisional plasmas

AbstractLow‐frequency fast and slow magnetosonic waves propagating in electron ion plasmas with damping effects through ions and neutral atoms collisions are investigated. Linear wave analysis is performed to obtain dispersion relation. The reductive perturbation method is applied and it is shown that fast and slow modes of nonlinear magnetosonic wave are governed by damped Korteweg‐de Vries (DKdV) equation in the presence of ion neutral collisions in plasmas. The analytical solution of DKdV soliton is presented under the assumption of weak collisional effects and numerical solutions of DKdV equation are also obtained using two‐level finite difference scheme with the help of Runge–Kutta method at different plasma parameters. The damping of nonlinear fast and slow magnetosonic wave structures at different times are discussed in the context of space plasma situations where ions and neutral atoms collisions exist.

Dynamical properties of dust-ion-acoustic wave solutions in a nonextensive collisional dusty plasma

Dynamical properties of dust-ion-acoustic waves (DIAWs) are analysed in a collisional nonextensive dusty plasma composing of mobile ions, q-nonextensive electrons and stationary dust grains with slight collisions between dusts and ions. Reductive perturbation technique (RPT) is practiced to acquire the damped Korteweg-de Vries (DKdV) equation. In absence of collision between dusts and ions, the damping term vanishes and the DKdV equation reduces to the KdV equation. Bifurcation analysis of the dynamical system obtained from the KdV equation is carried out. Analytical solitary wave solution and numerical periodic wave solution of the KdV equation are presented. Approximate analytical DIAW solutions for the DKdV equation are furnished using the novel -expansion technique. As an outcome, novel approximate analytical solutions of DIAWs are obtained for the DKdV equation. The application of our work is significant to explore nonextensive plasma environment, such as, ionosphere of Earth.

Weak dissipation of electrostatic solitary structures in warm collisional pair-ion plasmas with non-Maxwellian electron population

The localized electrostatic structures with dissipation due to ion-neutral collisions in a symmetric warm pair-ion plasma in the presence of non-Maxwellian population of electrons are studied. The analytical model for ion dynamics is based on fluid equations and the evolution equation is derived by using the reductive perturbation scheme in the form of a damped Korteweg-de Vries equation. The parameter regime relevant to space-based observations and laboratory plasmas is considered and time evolution of the propagating ion-acoustic soliton is discussed. The energetic-particles-driven properties of soliton for various spectral indices, dissipation, ion temperature, and density are illustrated with comparison to the thermal mode for Boltzmann distribution of electrons.

Propagation of the nonlinear damped Korteweg‐de Vries equation in an unmagnetized collisional dusty plasma via analytical mathematical methods

In this article, we consider the problem formulation of dust plasmas with positively charge, cold dust fluid with negatively charge, thermal electrons, ionized electrons, and immovable background neutral particles. We obtain the dust‐ion‐acoustic solitary waves (DIASWs) under nonmagnetized collision dusty plasma. By using the reductive perturbation technique, the nonlinear damped Korteweg‐de Vries (D‐KdV) equation is formulated. We found the solutions for nonlinear D‐KdV equation. The constructed solutions represent as bright solitons, dark solitons, kink wave and antikinks wave solitons, and periodic traveling waves. The physical interpretation of constructed solutions is represented by two‐ and three‐dimensional graphically models to understand the physical aspects of various behavior for DIASWs. These investigation prove that proposed techniques are more helpful, fruitful, powerful, and efficient to study analytically the other nonlinear nonlinear partial differential equations (PDEs) that arise in engineering, plasma physics, mathematical physics, and many other branches of applied sciences.

Detection of self-organized criticality behavior in an electronic circuit designed to solve a third order non-linear ODE (NL-ODE) for a damped KdV equation.

In this work, we present an electronic implementation of a damped Korteweg-de Vries equation modeled as a third order nonlinear autonomous ordinary differential equation (jerk equation). The circuit has been realized using operational amplifiers, multipliers, and passive electronic components which provides the time series solution of the equation in agreement with the numerical simulation results. Using nonlinear time series analysis on the acquired waveform data, we have obtained different types of phase space portraits and further analysis reflected long range correlation in the chaotic time series. Important findings include hysteresis induced bifurcation and self-organized criticality behavior in the system which is mentioned in this work.

Second order conformal multi-symplectic method for the damped Korteweg–de Vries equation**Project supported by the Program for Innovative Research Team in Science and Technology in Fujian Province University, China, the Quanzhou High Level Talents Support Plan, China (Grant No. 2017ZT012), and the Promotion Program for Young and Middle-Aged Teacher in Science and Technology Research of Huaqiao University, China (Grant No. ZQN-YX502).

A conformal multi-symplectic method has been proposed for the damped Korteweg–de Vries (DKdV) equation, which is based on the conformal multi-symplectic structure. By using the Strang-splitting method and the Preissmann box scheme, we obtain a conformal multi-symplectic scheme for multi-symplectic partial differential equations (PDEs) with added dissipation. Applying it to the DKdV equation, we construct a conformal multi-symplectic algorithm for it, which is of second order accuracy in time. Numerical experiments demonstrate that the proposed method not only preserves the dissipation rate of mass exactly with periodic boundary conditions, but also has excellent long-time numerical behavior.

Electromagnetic dip and hump solitary structures in oxygen-hydrogen dissipative plasmas

The excitation of low frequency magnetosonic waves in O+−H+−e− and O+−H−−e− collisional plasmas is studied. The light ions (hydrogen) may be positive as well as negative and are warm, and the heavy ions (oxygen) are considered as the cold species. The inertia of isothermal electrons is also considered. The collisions of ions and electrons with neutrals are taken into account. The hydrodynamic equations represent the dynamics of positive ions, negative ions, and isothermal electrons along with Maxwell's equations. The damped Korteweg de Vries equation is derived by employing the reductive perturbation technique and its time dependent solution is presented. Dip magnetosonic solitary structures are observed when both ions are positive and hump structures are seen in the presence of negative ions. The effects of variations of different plasma parameters on magnetosonic solitary structures in the presence of collisions are discussed.

Weakly dissipative dust acoustic solitons in the presence of superthermal particles

An investigation of the linear and non‐linear properties of low‐frequency electrostatic (dust acoustic) waves in a collisional dusty plasma with negative dust grains, Maxwellian electrons, and κ‐distributed ions is carried out. Low dust–neutral collisions accounting for dissipation (wave damping effect) is considered. The linear properties of dust acoustic excitations are discussed for varying values of relevant plasma parameters. It is shown that large wavelengths (beyond a critical value) are overdamped. In the limit of low dust–neutral collision rate, we have derived a damped Korteweg de Vries (KdV) equation by using the reductive perturbation technique. Supplemented by vanishing boundary conditions, the time‐varying solution of damped KdV equation leads to a weakly dissipative negative potential soliton. The soliton evolution with the damping parameter and other physical plasma parameters (superthermality, dust concentration, ion temperature) is delineated.

Forced Oscillations of a Damped Korteweg-de Vries Equation in a Quarter Plane

Laboratory experiments have shown that when nonlinear, dispersive waves are forced periodically from one end of an undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. It is our purpose here to establish this as a fact at least in the context of a damped Korteweg-de Vries equation. Thus, consideration is given to the initial-boundary-value problem [Formula: see text] For this problem, it is shown that if the small amplitude, boundary forcing h is periodic of period T, say, then the solution u of (*) is eventually periodic of period T. More precisely, we show for each x > 0, that u(x, t + T) - u(x, t) converges to zero exponentially as t → ∞. Viewing (*) (without the initial condition) as an infinite dimensional dynamical system in the Hilbert space Hs(R+) for suitable values of s, we also demonstrate that for a given, small amplitude time-periodic boundary forcing, the system (*) admits a unique limit cycle, or forced oscillation (a solution of (*) without the initial condition that is exactly periodic of period T). Furthermore, we show that this limit cycle is globally exponentially stable. In other words, it comprises an exponentially stable attractor for the infinite-dimensional dynamical system described by (*).

Collisionless damping of nonlinear dust ion acoustic wave due to dust charge fluctuation.

A dissipation mechanism for the damping of the nonlinear dust ion acoustic wave in a collisionless dusty plasma consisting of nonthermal electrons, ions, and variable charge dust grains has been investigated. It is shown that the collisionless damping due to dust charge fluctuation causes the nonlinear dust ion acoustic wave propagation to be described by the damped Korteweg-de Vries equation. Due to the presence of nonthermal electrons, the dust ion acoustic wave admits both positive and negative potential and it suffers less damping than the dust acoustic wave, which admits only negative potential.

The Amplification and Capture of Atmospheric Solitons by Topography: A Theory of the Onset of Regional Blocking

Abstract The problem of long, quasi-geostrophic baroclinic waves interacting with topography is shown to reduce to an inhomogeneous, damped Korteweg-deVries equation. The soliton perturbation theory of Karpman and Maslov (1977, 1978) and Kaup and Newell (1978) is then used to examine the qualitative aspects of the interaction of travelling solitons with topography. The perturbation theory suggests and numerical experiments confirm that under certain conditions incident solitans can be simultaneously amplified and stalled by topography-a process which we interpret as a transition to a regional blocking configuration.