A Direct Construction of Solitary Waves for a Fractional Korteweg–de Vries Equation With an Inhomogeneous Symbol

In this article, we applied the method of multiple scales for Korteweg–de Vries (KdV) type equations and we derived nonlinear Schrödinger (NLS) type equations. So we get a relation between KdV type equations and NLS type equations. In addition, exact solutions were found for KdV type equations. The ( G ′ G ) $\left( {{{G'} \over G}} \right)$ -expansion methods and the ( G ′ G , 1 G ) $\left( {{{G'} \over G},{\rm{ }}{1 \over G}} \right)$ -expansion methods were proposed to establish new exact solutions for KdV type differential equations. We obtained periodic and hyperbolic function solutions for these equations. These methods are very effective for getting travelling wave solutions of nonlinear evolution equations (NEEs).

The Sylvester Equation and Integrable Equations: The Ablowitz—Kaup—Newell—Segur System

Dust-ion-acoustic solitary waves in a two-temperature electrons with charge fluctuations in a dusty plasma

El Naschie’s Cantorian space time, Toda lattices and Cooper–Agop pairs

In this paper, we develop and analyze a series of conservative and dissipative local discontinuous Galerkin (LDG) methods for the dispersive system of Korteweg–de Vries (KdV) type equations. Based on a cardinal conservative quantity of this system, we design and discuss two different types of numerical fluxes, including the conservative and dissipative ones for the linear and nonlinear terms respectively. Thus, one conservative together with three dissipative LDG schemes for the KdV-type system are developed in our paper. The invariant preserving property for the conservative scheme and corresponding dissipative properties for the other three dissipative schemes are all presented and proven in this paper. The error estimates for two schemes are given, whose numerical fluxes for linear terms are chosen as the dissipative type. Assuming that the discontinuous piecewise polynomials of degree less than or equal to k are adopted, and conservative numerical fluxes are employed to discretize the nonlinear terms, we obtain a suboptimal a priori bound of order k; yet in the case of dissipative fluxes, we obtain a slightly better bound of order $$k+\frac{1}{2}$$ . Numerical experiments for this system in different circumstances are provided, including accuracy tests for two kinds of traveling waves, long-time simulations for solitary waves and interactions of multi-solitary waves, to illustrate the accuracy and capability of these schemes.

Nonlinear evolution equations (NLEEs) form the basis for mathematical models of problems arising in numerous areas. Over the past decades, evolution equations have earned a significant place in applied mathematics. In this report, the multiple scales method was applied for the analysis of Manakov equations. And (1 + 1) dimensional fifth-order nonlinear Korteweg–de Vries (fKdV) type equations were obtained. So, we have demonstrated the relationship between the KdV equations and the Manakov equation.

In the present article, we studied the effect of nonthermal electrons on the formation and existence of double‐layer structures in a three‐species plasma consisting of positive ions, nonthermal electrons, and immobile negative dust‐charged grains. Employing the reductive perturbations, a modified Korteweg–de Vries (mKdV) type equation is derived for the dust‐ion‐acoustic waves (DIAWs) bearing nonthermality. We found that both positive and negative polarity shock structures (double layer) can exist such that it switches polarity while changing the dust charge concentrations. However, strong nonthemality favours only rarefactive structures irrespective of the ion temperature. It is also found that increasing the nonthermal electron in the system the width of the double layer is increased; furthermore, the shock structure forms with small dust charge concentration. For small ionic temperature, increasing the nonthermal electrons in the system makes the double layer potential to increase; however, for σ = 1 reverse phenomena occurs. Our results are relevant to the shock observations in Q machine experiments and in the ionospheric regime of the earth.

The nonlinear propagation of ion-acoustic (IA) electrostatic solitary waves (SWs) is studied in a magnetized electron–ion (e–i) plasma in the presence of pressure anisotropy with electrons following Tsallis distribution. The Korteweg–de Vries (KdV) type equation is derived by employing the reductive perturbation method (RPM) and its solitary wave (SW) solution is determined and analyzed. The effect of nonextensive parameter q, parallel component of anisotropic ion pressure p 1, perpendicular component of anisotropic ion pressure p 2, obliqueness angle θ, and magnetic field strength Ω on the characteristics of SW structures is investigated. The present investigation could be useful in space and astrophysical plasma systems.

Recently, Holloway et al. (1997) compared simulations and measurements of the nonlinear evolution of internal tides on the Australian North West Shelf. The model they used for simulations was based on a Korteweg–de Vries (KdV) type equation, generalized to include dissipative and range varying effects (henceforth referred to as the gKdV equation). The purpose of this present note is to discuss certain issues concerning their inclusion of dissipation, of which there were two types: 1) turbulent horizontal eddy viscosity, represented by the coefficient n (m2 s21), and 2) quadratic bottom friction, represented by the dimensionless coefficient k. The authors pointed out that, for the value of n they used (n 5 2 3 1024 m2 s21), no apparent effect of eddy viscosity was observed. In considering the question as to what effects would have been produced had the value of n been significantly larger, and in investigating this issue further, we made several observations.

The current article devoted on the new method for finding the exact solutions of some time‐fractional Korteweg–de Vries (KdV) type equations appearing in shallow water waves. We employ the new method here for time‐fractional equations viz. time‐fractional KdV‐Burgers and KdV‐mKdV equations for finding the exact solutions. We use here the fractional complex transform accompanied by properties of local fractional calculus for reduction of fractional partial differential equations to ordinary differential equations. The obtained results are demonstrated by graphs for the new solutions. Copyright © 2016 John Wiley & Sons, Ltd.

The dissemination of positron-acoustic (PA) nonlinear structures, including the solitary waves (SWs) and cnoidal waves (CWs), is analyzed in an unmagnetized electron–positron–ion (e–p–i) plasma having inertial cold positrons and inertialess Cairns distributed electrons and Maxwellian positrons as well as immobile positive ions. The reductive perturbation method (RPM) is introduced to reduce the fluid equations to this model to the Korteweg–de Vries (KdV) type equation for studying small amplitude PA waves (PAWs). Moreover, the Kawahara (sometimes called the fifth-order KdV) equation is also obtained to investigate the characteristics of large amplitude PAWs. The effects of related parameters, such as nonthermal parameters, hot positron concentration, electron concentration, and temperature ratios, are numerically examined on the features of SWs and CWs.

Weakly nonlinear waves over the bottom disturbed topography: Korteweg–de Vries equation with variable coefficients

The study of heavy nucleus acoustic (HNA) periodic and solitary waves in degenerate relativistic magneto-rotating quantum plasma (DRMQP) composed of relativistic degenerate electrons, light nuclei and non-degenerate mobile heavy nuclei is presented. Reductive perturbation technique is adopted to derive the Korteweg–de Vries (KdV) type equation for studying the characteristics of HNA periodic waves. Further energy balance equation is derived by using Sagdeev potential approach and HNA cnoidal wave solution is determined. Only positive potential nonlinear periodic and solitary structures are observed. The impact of number density ratio, charge density, magnetic field strength and rotational effect is analyzed on the characteristics of HNA cnoidal and solitary waves. The results of this investigation may shed light on the salient features of different nonlinear HNA waves in different astrophysical environments, especially in white dwarfs.

Propagation and energy of the dressed solitons in the Thomas–Fermi magnetoplasma

On the theory and numerical simulation of acoustic and heat modes interaction in a liquid with bubbles: acoustic quasi-solitons