Damped Korteweg-de Vries Research Articles

Landau quantization effects on damping Kawahara solitons in electron–positron–ion plasma in rotating ionized medium

For the dynamics of three-dimensional electron–positron–ion plasmas, a fluid quantum hydrodynamic model is proposed by considering Landau quantization effects in dense plasma. Ion–neutral collisions in the presence of the Coriolis force are also considered. The application of the reductive perturbation technique produces a wave evolution equation represented by a damped Korteweg–de Vries equation. This equation, however, is insufficient for describing waves in our system at very low dispersion coefficients. As a result, we considered the highest-order perturbation, which resulted in the damped Kawahara equation. The effects of the magnetic field, Landau quantization, the ratio of positron density to electron density, the ratio of positron density to ion density, and the direction cosine on linear dispersion laws as well as soliton and conoidal solutions of the damped Kawahara equation are explored. The understanding from this research can contribute to the broader field of astrophysics and aid in the interpretation of observational data from white dwarfs.

Extreme events in a damped Korteweg–de Vries (KdV) autonomous system: A comprehensive analysis

This manuscript explores extreme events in a three-dimensional, damped Korteweg–de Vries (KdV) autonomous system that is derived from the jerk system with a modified basin. Through rigorous stability analysis, we identify the extreme events, confirmed using phase portraits, Poincaré return maps, time series, and probability distribution functions. The Dragon-king phenomenon is obtained and self-organised criticality is discussed. The regime of extreme events in a wide parameter range was obtained through two-parameter scanning. The analog circuit was constructed to mimicking dynamics of damped KdV equation and demonstrated in real-time hardware experiment as well as PSpice simulation. The experimental results show excellent agreement with the numerically obtained results. This study is the first comprehensive examination of extreme events in the damped KdV system, highlighting their nature and potential applications in chaos-based dynamical systems.

Extraction of newly soliton wave structure to the nonlinear damped Korteweg–de Vries dynamical equation through a computational technique

Extraction of newly soliton wave structure to the nonlinear damped Korteweg–de Vries dynamical equation through a computational technique

Ion acoustic solitary waves in an adiabatic dusty plasma: Roles of superthermal electrons, ion loss and ionization

Abstract The propagation of dust ion acoustic solitary wave (DIASW) is investigated in the multicomponent dusty plasma with adiabatic ions, superthermal electrons and stationary dust. The reductive perturbation method is employed to derive the damped Korteweg-de Vries (DKdV) equation which describes DIASW. The result reveals that, the adiabaticity of ions significantly modifies the basic features of the DIASW. The ionization effect makes the solitary wave growing, while collisions reduce the growth rate and even lead to the damping. With the increase in ionization cross section $\frac{{\Delta \sigma }}{{{\sigma _0}}}$, ion-to-electron density ratio ${{\delta}_{\rm{ie}}}$ and superthermal electrons parameter $\kappa $, the effect of ionization on DIASW enhances.

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.

Consequence of head‐on collision and double‐layer soliton with linear analysis in multi‐component dusty plasmas

AbstractThe unmagnetized multicomponent collisional dusty plasmas have been considered to investigate the consequence of head‐on collision of dust‐ion‐acoustic waves (DIAWs) with nonlinear characteristic analysis of DIAWs and collisionless plasmas have been considered for studying the production of double layer (DL). The plasma system consists of positively charged ion fluid, negatively charged cold dust fluid, thermal electrons, q‐nonextensive electrons, and immobile background neutral particles. The extended Poincaré‐Lighthill‐Kuo (ePLK) method has been used to derive two‐sided damped Korteweg–De Vries (dKdV) equations and the reductive perturbation method has been used to derive modified KdV (mKdV) equation and the standard Gardner equation (SGE). Linear dispersion relation is derived using linear perturbation technique. The nonlinear propagation of DIA Gardner solitons (DIAGSs) and double layers (DLs) as well as phase shifts have been investigated for the effects of the concerned parameters. It is found that the DIA soliton structures are significantly modified by the species densities, dust mass, and superextensive parameter. The rarefactive dip shape solitary Gardner solitons become produced for the effects of concerned parameters and the positive (negative) DLs become produced in the range of () for the effect of temperatures on the species. The superextensive and the subextensive parameters of electrons significantly affect the DL soliton structures.

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.

Terahertz plasma oscillations and dissipating solitary pulses in electron–hole semiconductor plasmas

Abstract Linearized modes of oscillation in terahertz range and weakly dissipating electrostatic solitary pulses are studied in semiconductor plasmas in the framework of quantum hydrodynamics. Many-particle effect is expressed in the local density approximation with the help of exchange-correlation potential and the species space charge effect is included through Poisson’s equation. Using reductive perturbation technique, damped Korteweg de-Vries equations is derived with a linear damping term arising due to electron (hole)–phonon collisions. Time evolution of solitary acoustic pulses is presented analytically and numerically. Typical values corresponding to GaAs, GaSb, InP and GaN semiconductors are used for parametric analysis and pulse profile with collision-induced dissipation and quantum effects of statistical pressure, Bohm potential and exchange-correlation potential. The pulses are stable and can withstand perturbations for a considerable time before damping.

The long-time behavior of solitary waves for the weakly damped KdV equation

In this paper, we first introduce the long-time behavior stability of solitary waves for the weakly damped Korteweg–de Vries equation. More concretely, solutions of the dissipative system with the initial values near a c_{0}-speed solitary wave, are approximated by a long curve on the family of solitary waves with the time-varying speed |c(t)-c_{0}| being small, in the long-time period (i.e., 0leqslant tleqslant O(frac{1}{epsilon ^{tau}})). Meanwhile, the approximation difference in a suitably weighted space H^{1}_{a}(mathbb{R}) is of the order of the damping coefficient and of some kind of exponential weight form. As a comparison, we also study the long-time behavior stability, i.e., for 0leqslant t<+infty , the solutions are approximated by a long curve on the family of solitary waves with the exponential decay speed c(t)= c_{0}e^{-beta t} (0<beta leqslant 1), when the initial values are near a c_{0}-speed solitary wave. However, here, the approximation difference merely defined in H^{1}(mathbb{R}) depends on the damping coefficient ϵ and the exponential decay coefficient β.

Head-on collision of magnetosonic solitary waves at low latitudes ionosphere plasma

The dynamics of head-on collision of two magnetosonic solitary waves in ionosphere plasma of the Earth is investigated. A geomagnetic field model standing for the relation between a magnetic field and latitude is used to study nonlinear magnetosonic waves in the ionosphere of Earth. The linear and weakly nonlinear properties of magnetosonic waves are studied by the dispersion relation and the extended Poincaré–Lighthill–Kuo (PLK) method, respectively. Two coupled damped Korteweg–de Vries equations (dKdV) are derived for oppositely propagating magnetosonic solitary waves. The explicit solitary wave solutions are obtained in the weak collision limit, and the trajectories and phase shifts of two magnetosonic solitary waves are derived, which show that collisional dynamics and their phase shifts are dependent on the collision caused by neutral particles and the properties of the ionosphere. This study is applied to investigate the two-counterpropagating magnetosonic solitary waves located in the F2-region of the Earth's ionosphere at low latitude. This study should be beneficial to understand the interaction dynamics of the head-on nonlinear magnetosonic waves located in the F2 layer of the ionosphere when collision effects caused by neutral particles and geomagnetic field distribution are considered.

Solitary waves dissipation in pair-ion plasmas for (r, q)-distributed electrons

Solitary waves in a symmetric pair-ion plasma are studied as affected by various plasma parameters, e.g., electron density, collision frequency, ion temperatures, (r, q)-distributed electrons. In this regard, plasma fluid equations have been used to simulate the plasma system for which a reductive perturbation technique is applied to derive the relevant damped Korteweg–de Vries equation. The effects of spectral indices, collision frequency, ion-electron temperature ratios, and density on the time evolution of perturbed potential profile have been illustrated and benchmarked with Boltzmann distributed electrons. It is observed that the nonthermal population significantly affects the soliton amplitude, which increases with r and q. Furthermore, it also grows with electron density and ion temperature; however, there is a in the slowing down rate in the case of the latter. An increase in the collision frequency leads to faster damping in both the amplitude and speed of the solitary potential profiles. This work will be useful in the determination of plasma dynamics for pair-ion plasma systems containing nonthermal electrons, especially with flat-topped distribution functions, e.g., as found in Earth's magnetosheath and magnetotail, as well as in laboratory experiments with the fullerene plasma. To the best of our knowledge, a general description for pair ion plasmas using (r, q) distribution has never been reported. This study can explore a huge variety of velocity distribution functions—via the two index nonthermal parameters, in contrast to single index counterparts.

Linear analysis and head-on collision of dust ion-acoustic shock waves in un-magnetized electronegative collisional plasmas

The linear properties and head-on collision of dust ion-acoustic shock waves (DIASWs) are investigated in un-magnetized electronegative collisional plasmas comprising non-inertial thermal electron and light negative ion, inertial positive ion, and stationary dust grain with negative charges. In this regard, the dispersion relation is derived using the standard normal mode analysis and the two-sided damped Korteweg–deVries (dKdV) equations are derived employing the extended Poincaré Lighthill Kuo (ePLK) method. The effects of physical parameters on linear dispersion relation, formation of shocks, phase shift due to head-on collision, and collision process are studied. It is found that the effect of is more dominant to the wave frequency than that of and in a certain range of the group velocity of the DIASWs is increasing and then decreasing for the effect of the ratio of negative ion to electron density. The amplitude and width of the DIASWs increase with the enhancement of electron temperature and the ion-neutral collision plays a significant role in the formation of shock. The consequence of () the phase shift is increasing (decreasing). The results of this study may be useful to the investigation of space and astrophysical environment where the electronegative collisional plasma exists.

Dissipative solitons in magnetized anisotropic plasma

The propagation of ion-acoustic (IA) dissipative solitons (DSs) is investigated in magnetized plasma with the presence of pressure anisotropy and Maxwellian distributed electrons. A nonlinear damped Korteweg-de Vries (d-KdV) type equation is derived for the potential by employing the reductive perturbation method (RPM) and its dissipative soliton solution is analyzed. The effect of plasma parameters like parallel anisotropic ion pressure p 1, perpendicular anisotropic ion pressure p 2, obliqueness l 3, ion-neutral collisional parameter γ and magnetic field strength Ω on the characteristics of DSs are investigated. The present investigation could be useful in space and astrophysical plasma systems.

The phase shift analysis of the colliding dissipative KdV solitons

In this work, the head-on collisions of the non-stationary dissipative soliton in ultracold neutral plasmas (UNPs) are investigated. The extended Poincare-Lighthill-Kuo (PLK) approach is adopted for reducing the fluid equations of the UNPs to two-counterpropagating damped Korteweg-de Vries (dKdV) equations. The dKdV equation is not an integrable Hamiltonian system, i.e., does not have an exact solution. Thus, one of the main goal of this paper is to find a new general approximate analytical solution to the dKdV equation for investigating the mechanism of the propagation and interaction of the non-stationary dissipative solitons. The residual error is estimated for checking the accuracy of the new obtained solution. The approximate analytical soliton solutions are adopted for deriving the temporal phase shifts after the collision. The impact of physical parameters on the nonstationary dissipative soliton profile and the temporal phase shifts is discussed. The obtained results will contribute to understand the mechanism of propagation and interaction of many nonlinear phenomena in different nonlinear mediums such as ocean, sea, optical fiber, plasma physics, etc.

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.

Collisionless damping of nonplanar dust acoustic waves due to dust charge fluctuation in nonextensive polarized plasma

A rigorous theoretical investigation has been made of the cylindrical/spherical dust acoustic waves 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, nonextensivity 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.

Projected exponential Runge–Kutta methods for preserving dissipative properties of perturbed constrained Hamiltonian systems

Preserving conservative and dissipative properties of dynamical systems is desirable in numerical integration. To this end, we develop and implement numerical methods that preserve the exact rate of dissipation in certain qualitative properties of dissipatively perturbed constrained Hamiltonian systems, which are shown to be conformal symplectic. Projection methods based on exponential Runge–Kutta methods are proposed for such systems. These numerical schemes are shown to be constraint preserving, conformal invariants preserving, symmetric, and second-order accurate. It is shown that these structure-preserving methods can be further composed to obtain higher-order structure-preserving methods. Linear stability analysis is used to derive stability properties and conformal preservation of the phase-space area. Numerical experiments, including constrained oscillators and a damped Korteweg–de Vries partial differential equation, demonstrate the advantages of geometric integration and verify theoretical results.

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.