Solutions Of Korteweg-de Vries Research Articles

Numerical solutions of KDV and mKDV equations: Using sequence and multi-core parallelization implementation

In this study, we use the explicit difference method and also the combination of the explicit difference method with the implicit method to numerically solve the third order of Korteweg–de Vries (KDV) and modified Korteweg–de Vries (mKDV) equations. This method is applied to obtain soliton solutions on several problems and the results obtained are compared with each other and the exact solution of the problem. Also, due to that these problems are very ill-posed, when the time increases, the waveform of solitons and the input parameters of these methods are carefully examined. The L∞, L2 and mean-square errors of the solutions show that these methods can be applied to different problems and have very good accuracy. These methods are widely used in combination with other algorithms, but their very high execution time, especially in this category of problems, is always a big limitation. In this paper, a parallel approach for implementing these methods is presented and very good results have been obtained compared to sequential implementation. This is a reference to further study solutions of other models of KDV and mKDV problems as well as solving their inverse problems.

The characteristics of ion acoustic solitons in nonthermal regularized kappa distributed plasmas

Nonthermal equilibrium is an intrinsic characteristic of space and astrophysical plasmas, and in many space environments, the velocity distributions of charged particles with suprathermal tails may be well be fitted by the Kappa function, which becomes the Maxwellian distribution for κ→∞. Various studies of ion or dusty acoustic solitons, thus, have considered the Kappa distributed electrons in the model calculations. However, the Kappa velocity distribution (KVD) is theoretically not applicable for κ≤3/2. Alternatively, the recently proposed regularized Kappa distribution with two free parameters, κ and α, have been shown to be mathematically and physically smooth for all κ values, which may recover the standard KVD for α=0 and the Maxwellian distribution for κ→∞ and α=0. In this study, we examine the characteristics of ion acoustic solitons based on the linear, weakly nonlinear Korteweg–de Vries (KdV) and fully nonlinear theories with the regularized Kappa distributed electrons and warm ion fluids. These approaches may give rise to the dispersion relation with modified characteristic speed of acoustic waves, the analytical KdV solutions, and the Sagdeev's potential as well as the fully nonlinear solutions. It is shown that the model results are mathematically and physically valid for κ≤3/2 and the formulations with the charges being free parameters are applicable for general acoustic solitons.

Numerical Solution of the Korteweg-De Vries Equation Using Finite Difference Method

The Korteweg-de Vries (KdV) equation is a nonlinear partial differential equation that has a key role in wave physics and many other disciplines. In this article, we develop numerical solutions of the KdV equation using the finite difference method with the Crank-Nicolson scheme. We explain the basic theory behind the KdV equation and the finite difference method, and outline the implementation of the Crank-Nicolson scheme in this context. We also give an overview of the space and time discretization and initial conditions used in the simulation. The results of these simulations are presented through graphical visualizations, which allow us to understand how the KdV solution evolves over time. Through analysis of the results, we explore the behavior of the solutions and perform comparisons with exact solutions in certain cases. Our conclusion summarizes our findings and discusses the advantages and limitations of the method used. We also provide suggestions for future research in this area.

A Riemann–Hilbert method to algebro-geometric solutions of the Korteweg–de Vries equation

A proof without using Dubrovin’s equations is given for the Its–Matveev formula for algebro-geometric solutions of the Korteweg–de Vries (KdV) equation. It is shown that the Baker–Akhiezer (BA) function of the KdV equation can be described in terms of two types of holomorphic vector Riemann–Hilbert problems with σ1-symmetry condition in the complex k-plane. Further, a sufficient condition on determining whether the algebro-geometric KdV solution has mKdV-relevant property is presented. This provides an explanation for the classical Miura transformation in terms of the deformations of associated algebro-geometric datum. Our main tools include algebraic curve and Riemann surface, Riemann theta function, Riemann–Hilbert problem and associated deformation procedure.

Solution of Fractional Third-Order Dispersive Partial Differential Equations and Symmetric KdV via Sumudu–Generalized Laplace Transform Decomposition

This research work introduces a novel method called the Sumudu–generalized Laplace transform decomposition method (SGLDM) for solving linear and nonlinear non-homogeneous dispersive Korteweg–de Vries (KdV)-type equations. The SGLDM combines the Sumudu–generalized Laplace transform with the Adomian decomposition method, providing a powerful approach to tackle complex equations. To validate the efficacy of the method, several model problems of dispersive KdV-type equations are solved, and the resulting approximate solutions are expressed in series form. The findings demonstrate that the SGLDM is a reliable and robust method for addressing significant physical problems in various applications. Finally, we conclude that this transform is a symmetry to other symmetric transforms.

Solitary wave solutions in time-fractional Korteweg-de Vries equations with power law kernel

<abstract><p>The non-linear time-fractional Korteweg-de Vries and modified Korteweg-de Vries equations are studied with Caputo's fractional derivative. The general higher-order solitary wave solutions are derived using a novel technique called the Aboodh transform decomposition method. To validate the obtained results, two examples of Caputo's fractional derivative with appropriate subsidiary conditions are illustrated. The accuracy and efficiency are confirmed by using numerical simulations and error analysis, where good agreements are obtained. The numerical analysis shows that, in comparison to the time-fractional Korteweg-de Vries solution, the solitary wave solution for the time-fractional modified Korteweg-de Vries equation is less stable against the oscillations. The variations in the temporal variable $ t $ enhance the strength of the wave solutions. Moreover, the wave perturbations taper off as $ t $ attains large values. The parameter $ \alpha $ signifies the fractional derivative influence on the wave dispersion and nonlinearity effects. This affects the amplitude as well as the spatial extension of the solitary waves. With a relatively small value of $ t $, the obtained solutions admit pulse-shaped solitons. Moreover, the wave's solutions suffer from oscillations when the temporal variable attains large values. This effect cannot be noticed in the soliton solutions obtained in the integer order systems.</p></abstract>

Construction of new solutions of Korteweg-de Vries Caudrey-Dodd-Gibbon equation using two efficient integration methods

Korteweg-de Vries Caudrey-Dodd-Gibbon (KdV-CDG) equation describes many physical phenomena in plasma physics, optical fibers, dynamics of the ocean, quantum mechanics, acoustic waves and laser optical applications. In this paper, the KdV-CDG equation is analyzed via two reliable and efficient integrating approaches. The suggested techniques; the extended -expansion method and exponential (ψ(ξ))-expansion method successfully extract hyperbolic function solutions, trigonometric function solutions and rational function solutions. The existence criteria for all the obtained solutions are also discussed in this paper. At the end, various 3D and 2D contour plots have been constructed for better understanding of constructed solutions.

Electron acoustic singular solitons interaction in the Earth’s magnetotail region

Electron acoustic potential structures are generated in the presence of a large-amplitude electric field in Earth’s magnetotail region, Earth’s magnetosphere according to the numerous space observations by satellites. In this article, we study the electron-acoustic solitary waves (EASWs) in an unmagnetized electron–ion plasma consisting of cold electrons and isothermal ions with two different temperatures. Nonlinear evolution equations are derived from the hydrodynamic model of collisionless, unmagnetized electron–ion plasma. Exact analytic solutions are obtained from derived Korteweg-de Vries (KdV) equations and modified Korteweg-de Vries (MKdV) equations separately with the aid of the Hirota Bilinear method. Two singular soliton solutions of both KdV and MKdV equations are obtained separately. The efficiency and interactive approach of the heuristic Hirota Bilinear method leads to multiple singular soliton solutions for its beautiful algebraic technique which generates solitons solutions as well as singular solitons explicitly. Then the significant multi singular soliton interaction has been studied for KdV singular solitons and MKdV singular solitons in the critical parameter set.

Singular solitons interaction of dust ion acoustic waves in the framework of Korteweg de Vries and Modified Korteweg‐de Vries equations with (r,q) distributed electrons

AbstractTwo nonlinear evolution equations (NLEEs), namely, Korteweg‐de Vries (KdV) and modified Korteweg‐de Vries (MKdV) are derived from the dust hydrodynamic model of collisionless, unmagnetized dusty plasma with electrons following double spectral velocity distribution, that is, (r,q) distribution. Regular as well as singular analytic solutions of derived KdV equations and MKdV equations are obtained separately with the aid of the Hirota Bilinear method. Two singular soliton solutions of both KdV and MKdV equations are obtained separately. The efficiency and interactive approach of the heuristic Hirota Bilinear method leads to multiple singular soliton solutions for its beautiful algebraic technique which generates solitons solutions as well as singular solitons explicitly. Then the significance of multi singular soliton interaction has been studied for KdV singular solitons and for MKdV singular solitons in the critical parameter set.

Airy Kernel Determinant Solutions to the KdV Equation and Integro-Differential Painlevé Equations

We study a family of unbounded solutions to the Korteweg-de Vries equation which can be constructed as log-derivatives of deformed Airy kernel Fredholm determinants, and which are connected to an integro-differential version of the second Painlev\'e equation. The initial data of the Korteweg-de Vries solutions are well-defined for $x>0$, but not for $x<0$, where the solutions behave like $\frac{x}{2t}$ as $t\to 0$, and hence would be well-defined as solutions of the cylindrical Korteweg-de Vries equation. We provide uniform asymptotics in $x$ as $t\to 0$; for $x>0$ they involve an integro-differential analogue of the Painlev\'e V equation. A special case of our results yields improved estimates for the {tails} of the narrow wedge solution to the Kardar-Parisi-Zhang equation.

Modification of quintic B-spline differential quadrature method to nonlinear Korteweg-de Vries equation and numerical experiments

In this manuscript, a novel modification of quintic B-spline differential quadrature method is given. By this modification, the obtained algebraic system do not have any ghost points. Therefore, the equation system is solvable and doesn't need any additional equation. Another advantage of this modification is the fact that the obtained numerical results are better than classical methods. Finite difference method is used as another component of the algorithm to obtain numerical solution of Korteweg-de Vries (KdV) equation which is the important model of nonlinear phenomena. Nine effective experiments namely single soliton, double solitons, interaction of two solitons, splitting to the solitons, interaction of three-, four-, and five solitons, evolution of solitons via Maxwellian initial condition and train of solitons are given with comparison of error norms and invariants. All of the given results with earlier studies show that the present scheme has improved classical common methods. Rate of convergence for single soliton and double solitons problems are reported.

Numerical study of a Whitham equation exhibiting both breaking waves and continuous solutions

We consider a Whitham equation as an alternative for the Korteweg–de Vries (KdV) equation in which the third derivative is replaced by the integral of a kernel, i.e., ηxxx in the KdV equation is replaced by ∫−∞∞Kν(x−ξ)ηξ(ξ,t)dξ. The kernel Kν(x) satisfies the conditions limν→∞Kν(x) = δ″(x), where δ(x) is the Dirac delta function and limν→0Kν(x) = 0. The questions studied here, by means of numerical examples, are whether adjustment of the parameter ν produces both continuous solutions and shocks of the kernel equation and how well they represent KdV solutions and solutions of the underlying hyperbolic system. A typical example is for resonant forced oscillations in a closed shallow water tank governed by the kernel equation, which are compared with those governed by a partial differential equation. The continuous solutions of the kernel equation associated with frequency dispersion in the KdV equations limit to the shocks of the shallow water equations as ν → 0. Two experimental problems are solved in a single equation. As another example, suitable adjustment of ν in the kernel equation produces solutions reminiscent of a hydraulic and undular bore.

The effect of a positive bound state on the KdV solution: a case study * *The author is supported in part by the NSF Grant DMS 1716975.

We consider a slowly decaying oscillatory potential such that the corresponding 1D Schrödinger operator has a positive eigenvalue embedded into the absolutely continuous spectrum. This potential does not fall into a known class of initial data for which the Cauchy problem for the Korteweg–de Vries (KdV) equation can be solved by the inverse scattering transform. We nevertheless show that the KdV equation with our potential does admit a closed form classical solution in terms of Hankel operators. Comparing with rapidly decaying initial data our solution gains a new term responsible for the positive eigenvalue. To some extent this term resembles a positon (singular) solution but remains bounded. Our approach is based upon certain limiting arguments and techniques of Hankel operators.

Proliferation of soliton, explosive, shocklike, and periodic ion-acoustic waves in Titan's ionosphere

We report on many electrostatic nonlinear structures, namely, soliton, explosive, shocklike, and periodic waves, which may exist due to the dynamics of different ion species in Titan's ionosphere. Using the latest available observations of Titan at an altitude of ≈ 1300 km, the main ion components are HCNH+, C2H5+, and C3H5+. The dynamics of these waves are described by the Korteweg–de Vries (KdV) equation. Using the G′/G–expansion method, we obtain a class of solutions that elucidate these nonlinear structures. As the wave amplitude increases, both the width and velocity of the wave deviate from the prediction of the KdV equation. To describe the waves of larger amplitude, the higher-order dispersion has to be taken into account. The G′/G–expansion method is used, for the first time, to solve the KdV equation with higher-order dispersion. We compare between both KdV and higher-order dispersive KdV solutions, i.e., soliton, explosive, shocklike, and periodic waves.

On the propagation of regularity for solutions of the fractional Korteweg-de Vries equation

We consider the initial value problem (IVP) for the fractional Korteweg-de Vries equation (fKdV)(0.1){∂tu−Dxα∂xu+u∂xu=0,x,t∈R,0<α<1,u(x,0)=u0(x). It has been shown that the solutions to certain dispersive equations satisfy the propagation of regularity phenomena. More precisely, it deals in determine whether regularity of the initial data on the right hand side of the real line is propagated to the left hand side by the flow solution. This property was found originally in solutions of Korteweg-de Vries (KdV) equation and it has been also verified in other dispersive equations as the Benjamin-Ono (BO) equation.Recently, it has been shown that the solutions of the dispersive generalized Benjamin-Ono (DGBO) equation, this is α∈(2,3) in (0.1); also satisfy the propagation of regularity phenomena. This is achieved by introducing a commutator decomposition to handle the dispersive part in the equation. Following the approach used in the DGBO, we prove that the solutions of the fKdV also satisfies the propagation of regularity phenomena. Consequently, this type of regularity travels with infinite speed to its left as time evolves.

Electroacoustic Waves in a Collision-Free Magnetized Superthermal Bi-Ion Plasma

The electroacoustic waves, particularly ion-acoustic waves (IAWs), and their expansion in the medium of a magnetized collision-free plasma system has been investigated theoretically. The plasma system is assumed to be composed of both positively and negatively charged mobile ion species and kappa-distributed hot electron species. In the nonlinear perturbation regime, the magnetized Korteweg–de Vries (KdV) and magnetized modified KdV (mKdV) equations are derived by using reductive perturbation method. The prime features (i.e., amplitude, phase speed, width, etc.) of the IAWs are studied precisely by analyzing the stationary solitary wave solutions of the magnetized KdV and magnetized mKdV equations, respectively. It occurs that the basic properties of the IAWs are significantly modified in the presence of the excess superthermal hot electrons, obliqueness, the plasma particle number densities, etc. It is also observed that, in case of magnetized KdV solitary waves, both compressive and rarefactive structures are formed, whereas only compressive structures are found for the magnetized mKdV solitary waves. The implication of our results in some space and laboratory plasma situations is concisely discussed.

Remarks on Existence/Nonexistence of Analytic Solutions to Higher Order KdV Equations

In this note, we discuss the existence of analytic solutions to the nonlinear wave equations of the higher order than the ubiquitous Korteweg-de Vries (KdV) equation. First, we recall our recent results which show that the extended KdV equation (KdV2), that is, the equation obtained within second-order perturbation approach possesses three kinds of analytic solutions. These solutions have the same functional form as the corresponding KdV solutions. We show, however, that the most intriguing multi-soliton solutions, known for the KdV equation, do not exist for KdV2. Moreover, we show that for the equations obtained in the third order perturbation approach (and then in any higher order) analytic solutions in the forms known from KdV theory do not exist.

Iterative Solutions of Hirota Satsuma Coupled KDV and Modified Coupled KDV Systems

In this article the approximate solutions of nonlinear Hirota Satsuma coupled Korteweg De- Vries (KDV) and modified coupled KDV equations have been obtained by using reliable algorithm of New Iterative Method (NIM). The results obtained give higher accuracy than that of homotopy analysis method (HAM). The obtained solutions show that NIM is effective, simpler, easier, and explicit and gives a suitable way to control the convergence of the approximate solution.

Analytical solutions of conformable time, space, and time-space fractional KdV equations

In this study, we consider the Korteweg-de Vries (KdV) equation for solitary waves in the domain of conformable fractional calculus. By means of this fractional theory, we obtain exact solutions for time, space, and time-space fractional KdV equations and demonstrate our results graphically according to the fractional order of the related equations. Furthermore, we report that the fractional order in the solution of the time fractional KdV equation is associated with viscosity of the medium by comparing three-dimensional picture presentations of solutions of time fractional KdV and Burgers-KdV equations.

Simulations and observation of nonlinear internal waves on the continental shelf: Korteweg–de Vries and extended Korteweg–de Vries solutions

Abstract. Numerical solutions of the Korteweg–de Vries (KdV) and extended Korteweg–de Vries (eKdV) equations are used to model the transformation of a sinusoidal internal tide as it propagates across the continental shelf. The ocean is idealized as being a two-layer fluid, justified by the fact that most of the oceanic internal wave signal is contained in the gravest mode. The model accounts for nonlinear and dispersive effects but neglects friction, rotation and mean shear. The KdV model is run for a number of idealized stratifications and unique realistic topographies to study the role of the nonlinear and dispersive effects. In all model solutions the internal tide steepens forming a sharp front from which a packet of nonlinear solitary-like waves evolve. Comparisons between KdV and eKdV solutions are made. The model results for realistic topography and stratification are compared with observations made at moorings off Massachusetts in the Middle Atlantic Bight. Some features of the observations compare well with the model. The leading face of the internal tide steepens to form a shock-like front, while nonlinear high-frequency waves evolve shortly after the appearance of the jump. Although not rank ordered, the wave of maximum amplitude is always close to the jump. Some features of the observations are not found in the model. Nonlinear waves can be very widely spaced and persist over a tidal period.