Nonlinear Korteweg-de Vries Equation Research Articles

On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback

<p style='text-indent:20px;'>The aim of this work is to study the asymptotic stability of the nonlinear Korteweg-de Vries equation in the presence of a delayed term in the internal feedback. We first consider the case where the weight of the term with delay is smaller than the weight of the term without delay and we prove a semiglobal stability result for any lengths. Secondly we study the case where the support of the term without delay is not included in the support of the term with delay. In that case, we give a local exponential stability result if the weight of the delayed term is small enough. We illustrate these results by some numerical simulations.</p>

Constructing and Predicting Solutions for Different Families of Partial Differential Equations: A Reliable Algorithm

The construction of mathematical models for different phenomena, and developing their solutions, are critical issues in science and engineering. Among many, the Buckmaster and Korteweg‐de Vries (KdV) models are very important due to their ability of capturing different physical situations such as thin film flows and waves on shallow water surfaces. In this manuscript, a new approach based on the generalized Taylor series and residual function is proposed to predict and analyze Buckmaster and KdV type models. This algorithm estimates convergent series with an easy‐to‐use way of finding solution components through symbolic computation. The proposed algorithm is tested against the Buckmaster and KdV equations, and the results are compared with available solutions in the literature. At first, proposed algorithm is applied to Buckmaster‐type linear and nonlinear equations, and attained the closed‐form solutions. In the next phase, the proposed algorithm is applied to highly nonlinear KdV equations (namely, classical, modified, and generalized KdV) and approximate solutions are obtained. Simulations of the test problems clearly reassert the dominance and capability of the proposed methodology in terms of accuracy. Analysis reveals that the projected scheme is reliable, and hence, can be utilized for more complex problems in engineering and the sciences.

Comparison of modified ADM and classical finite difference method for some third-order and fifth-order KdV equations

Abstract The KdV equation, which appears as an asymptotic model in physical systems ranging from water waves to plasma physics, has been studied. In this paper, we are concerned with dispersive nonlinear KdV equations by using two reliable methods: Shehu Adomian decomposition method (STADM) and the classical finite difference method for solving three numerical experiments. STADM is constructed by combining Shehu’s transform and Adomian decomposition method, and the nonlinear terms can be easily handled using Adomian’s polynomials. The Shehu transform is used to accelerate the convergence of the solution series in most cases and to overcome the deficiency that is mainly caused by unsatisfied conditions in other analytical techniques. We compare the approximate and numerical results with the exact solution for the two numerical experiments. The third numerical experiment does not have an exact solution and we compare profiles from the two methods vs the space domain at some values of time. This study provides us with information about which of the two methods are effective based on the numerical experiment chosen. Knowledge acquired will enable us to construct methods for other related partial differential equations such as stochastic Korteweg-de Vries (KdV), KdV-Burgers, and fractional KdV equations.

Obliquely overtaking collisions of electrostatic N-soliton in the Thomas-Fermi dense magnetoplasma

Overtaking collision of unidirectional dust acoustic waves in the Thomas-Fermi dense magnetoplasma has been investigated. The system consists of classical negatively charged dust particles with degenerate electrons and ions. Nonlinear Korteweg-de Vries (KdV) equation that characterizing the dust acoustic solitons in such a plasma model is derived. The system admits only rarefactive solitary waves. Hirota's bilinear method is used to investigate the overtaking collision of two and three-soliton solutions. The exchange of energies due to the overtaking collisions is affected by physical parameters such as the electron equilibrium density, the dust temperature, and the obliquity angle. This causes the solitons behavior to be alternated. The amplitude, and width increase as the dust temperature and electron density decrease or when the dust density increases. The magnetic field affects only the soliton's width. The phase shifts are also affected by the system parameters. The current study is an attempt to illuminate the properties of N-soliton in Thomas-Fermi dense magnetoplasma that are applicable in compact astrophysical objects such as white dwarfs and high-intensity laser-solid matter interaction experiments.

New traveling solutions of the fractional nonlinear KdV and ZKBBM equations with 𝒜ℬℛ fractional operator

This research paper investigates novel explicit wave solutions of the fractional Korteweg–de Vries (KdV) equation and the fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation. These models are used as gravity models in water and an interaction model between the long waves. The Atangana–Baleanu ([Formula: see text]) fractional operator is utilized for the first time to convert the fractional form of both models into nonlinear partial differential equations with an integer order. The extended simplest equation method is employed to construct some distinct types of solitary wave solutions such as exponential, rational, hyperbolic and trigonometric functions. For more illustration of our obtained solutions, some figures for them are given. The power and practical properties of the used method are tested.

On Algorithm of Integrability Classification of the Nonlinear Dynamical Systems via Computer Algebra Methods

There is developed an algorithm to classify integrable nonlinear dynamical systems via Wolfram Mathematica. The hierarchy of conservation laws for the nonlinear dynamical system can be cal-culated by this algorithm. There are demonstrated some modifications of nonlinear Korteweg-de Vries equations integrated by inverse scatering method.

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.

Qualitative Analysis of the Dynamic for the Nonlinear Korteweg–de Vries Equation with a Boundary Memory

This paper addresses the impact of the presence of a boundary memory term in the third-order Korteweg–de Vries equation in a bounded interval $$[0,\ell ]$$ . First, an overall literature review is provided. Indeed, a comprehensive discussion on the literature constitutes a survey part of the current paper. Thereafter, it is shown that the system under consideration possesses a unique solution under a smallness assumption on the initial data and an appropriate condition on the parameters and the kernel involved in the memory term. Last but not least, we demonstrate that the zero solution is exponentially stable as long as the length $$\ell $$ is small enough by means of Lyapunov method, which permits to provide an estimate of the exponential decay rate. These findings improve and complement those of Zhang (in: Desch W, Kappel F, Kunisch K (eds) Proceedings of international conference on control and estimation of distributed parameter systems: nonlinear phenomena, International Series of Numerical Mathematics, vol 118, Birkhauser, Basel, pp 371–389, 1994) (resp. Baudouin et al. in IEEE Trans Autom Control 64:1403–1414, 2019), where no memory term is present (resp. a delay occurs instead of memory).

On the existence of the resolvent and separability of a class of the Korteweg-de Vriese type linear singular operators

Partial differential equations of the third order are the basis of mathematical models of many phenomena and processes, such as the phenomenon of energy transfer of hydrolysis of adenosine triphosphate molecules along protein molecules in the form of solitary waves, i.e. solitons, the process of transferring soil moisture in the aeration zone, taking into account its movement against the moisture potential. In particular, this class includes the nonlinear Korteweg-de Vries equation, which is the main equation of modern mathematical physics. It is known that various problems have been studied for the Korteweg-de Vries equation and many fundamental results obtained. In this paper, issues about the existence of a resolvent and separability (maximum smoothness of solutions) of a class of linear singular operators of the Korteweg-de Vries type in the case of an unbounded domain with strongly increasing coefficients are investigated.

Head-on Collisional Effects of Ion Acoustic Waves in Magnetized Plasma: Linear and Nonlinear Analyses

Head-on collisional phenomena of ion acoustic (IA) waves in electron–positron–ion magnetized plasmas are investigated through linear and nonlinear analyses. The plasma system is considered collisionless and comprising cold ions, kappa distributed cold electrons, and positrons. Dispersion relation is derived with the aid of the linear perturbation method. The two sided nonlinear Korteweg-de Vries (KdV) equations are derived employing extended Poincare-Lighthill-Kuo (ePLK) method. The effects of wave frequency on linear group velocity with species density ratio are studied. Further, the production of ion acoustic solitons in terms of ion gyrofrequency ( $${\omega }_{\mathrm{g}}$$ ), superthermality parameters for electrons ( $${\kappa }_{\mathrm{e}}$$ ), and positrons ( $${\kappa }_{\mathrm{p}}$$ ) is investigated. Additionally, the effects of obliqueness (angle between magnetic field and wave propagation vector) on nonlinearity coefficient and phase shifts with density ratios (positron to electron and ion to electron) are demonstrated.

Non-homogeneous boundary value problems for some KdV-type equations on a finite interval: A numerical approach

This paper addresses the approximation of solutions to some non-homogeneous boundary value problems associated with the nonlinear Korteweg-de Vries equation (KdV) and a system of two coupled KdV-type equations derived by Gear and Grimshaw posed on a bounded interval. An efficient Galerkin scheme that combines a finite element strategy for space discretization with a second-order implicit scheme for time-stepping is employed to approximate time dynamics of model equations studied. Several numerical experiments, including boundary controllability problems for nonlinear KdV and GG equations, are presented for different final states to show the performance of the numerical strategies proposed. The numerical results with nonlinear models agree with previous analytic theory and show the persistence of the behavior not uniform in time of the control functions computed already observed by Rosier [22] in the case of the linear KdV equation.

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.

An unconditionally stable space–time FE method for the Korteweg–de Vries equation

We introduce an unconditionally stable finite element (FE) method, the automatic variationally stable FE (AVS-FE) method for the numerical analysis of the Korteweg–de Vries (KdV) equation. The AVS-FE method is a Petrov–Galerkin method which employs the concept of optimal discontinuous test functions of the discontinuous Petrov–Galerkin (DPG) method. However, since AVS-FE method is a minimum residual method, we establish a global saddle point system instead of computing optimal test functions element-by-element. This system allows us to seek both the approximate solution of the KdV initial boundary value problem (IBVP) and a Riesz representer of the approximation error. The AVS-FE method distinguishes itself from other minimum residual methods by using globally continuous Hilbert spaces, such as H1, while at the same time using broken Hilbert spaces for the test. Consequently, the AVS-FE approximations are classical C0 continuous FE solutions. The unconditional stability of this method allows us to solve the KdV equation space and time without having to satisfy a CFL condition. We present several numerical verifications for both linear and nonlinear versions of the KdV equation leading to optimal convergence behavior. Finally, we present a numerical verification of adaptive mesh refinements in both space and time for the nonlinear KdV equation.

LHAM Approach to Fractional Order Rosenau-Hyman and Burgers' Equations

Fractional calculus has been found to be a great asset in finding fractional dimension in chaos theory, in viscoelasticity diffusion, in random optimal search etc. Various techniques have been proposed to solve differential equations of fractional order. In this paper, the Laplace-Homotopy Analysis Method (LHAM) is applied to obtain approximate analytic solutions of the nonlinear Rosenau-Hyman Korteweg-de Vries (KdV), K(2, 2), and Burgers' equations of fractional order with initial conditions. The solutions of these equations are calculated in the form of convergent series. The solutions obtained converge to the exact solution when α = 1, showing the reliability of LHAM.

Integration of the nonlinear Korteweg–de Vries equation with an additional term

We use the method of the inverse spectral problem to integrate the nonlinear Korteweg–de Vries equation with an additional term in the class of periodic functions.

Integration of the nonlinear Korteweg-de Vries equation with an additional term

Метод обратной спектральной задачи применяется к интегрированию нелинейного уравнения Кортевега-де Фриза с дополнительным членом в классе периодических функций.

Approximate Analytical Solutions of Nonlinear Korteweg-de Vries Equations Using Multistep Modified Reduced Differential Transform Method

This paper aims to propose and investigate the application of Multistep Modified Reduced Differential Transform Method (MMRDTM) for solving the nonlinear Korteweg-de Vries (KdV) equation. The proposed technique has the advantage of producing an analytical approximation in a fast converging sequence with a reduced number of calculated terms. MMRDTM is presented with some modification of the reduced differential transformation method (RDTM) which is the nonlinear term is replaced by related Adomian polynomials and then adopting a multistep approach. Consequently, the obtained approximation results do not only involve smaller number of calculated terms for the nonlinear KdV equation, but also converge rapidly in a broad time frame. We provided three examples to illustrates the advantages of the proposed method in obtaining the approximation solutions of the KdV equation. To depict the solution and show the validity and precision of the MMRDTM, graphical inputs are included.

A meshless method for the numerical solution of the seventh-order Korteweg-de Vries equation

This article describes a meshless method for the numerical solution of the seventh-order nonlinear one-dimensional non-stationary Korteweg-de Vries equation. The meshless scheme is based on the use of the collocation method and radial basis functions. In this approach, the solution is approximated by radial basis functions, and the collocation method is used to compute the unknown coefficients. The meshless method uses the following radial basis functions: Gaussian, inverse quadratic, multiquadric, inverse multiquadric and Wu’s compactly supported radial basis function. Time discretization of the nonlinear one-dimensional non-stationary Korteweg-de Vries equation is obtained using the θ-scheme. This meshless method has an advantage over traditional numerical methods, such as the finite difference method and the finite element method, because it doesn’t require constructing an interpolation grid inside the domain of the boundary-value problem. In this meshless scheme the domain of a boundary-value problem is a set of uniformly or arbitrarily distributed nodes to which the basic functions are “tied”. The paper presents the results of the numerical solutions of two benchmark problems which were obtained using this meshless approach. The graphs of the analytical and numerical solutions for benchmark problems were obtained. Accuracy of the method is assessed in terms of the average relative error, the average absolute error, and the maximum error. Numerical experiments demonstrate high accuracy and robustness of the method for solving the seventh-order nonlinear one-dimensional non-stationary Korteweg-de Vries equation.

BOUNDARY CONTROLLABILITY FOR THE TIME-FRACTIONAL NONLINEAR KORTEWEG-DE VRIES (KDV) EQUATION

In this paper, we study the time-fractional nonlinear Korteweg-de Vries (KdV) equation. By using the theory of semigroups, we prove the well-posedness of the time-fractional nonlinear KdV equation. Moreover, we present the boundary controllability result for the problem.

Internal null controllability of the generalized Hirota-Satsuma system

The generalized Hirota-Satsuma system consists of three coupled nonlinear Korteweg-de Vries (KdV) equations. By using two distributed controls it is proven in this paper that the local null controllability property holds when the system is posed on a bounded interval. First, the system is linearized around the origin obtaining two decoupled subsystems of third order dispersive equations. This linear system is controlled with two inputs, which is optimal. This is done with a duality approach and some appropriate Carleman estimates. Then, by means of an inverse function theorem, the local null controllability of the nonlinear system is proven.