Order Korteweg-de Vries Equation Research Articles

The Nonlinear Schrödinger Equation Derived from the Third Order Korteweg-de Vries Equation Using Multiple Scales Method

Nonlinear equations of evolution (NLEE) are mathematical models used in various branches of science. As a result, nonlinear equations of evolution have served as a language for formulating many engineering and scientific problems. For this reason, many different and effective techniques have been developed regarding nonlinear equations of evolution and solution methods. Although the origin of nonlinear equations of evolution dates back to ancient times, there have been significant developments regarding these equations from the past to the present. The main reason for this situation is that nonlinear equations of evolution involve the problem of nonlinear wave propagation. In recent years, equations of formation have become increasingly important in applied mathematics. This work focuses on the perturbation approach, often known as many scales, for nonlinear evolution equations. The article focuses on the analysis of the (1+1) dimensional third-order nonlinear Korteweg-de Vries (KdV) equation using the multiple scales method, which resulted in obtaining nonlinear Schrödinger (NLS) type equations.

A gradient reproducing kernel based stabilized collocation method for the 5th order Korteweg–de Vries equations

This paper presents a gradient reproducing kernel based stabilized collocation method (GRK-SCM) to solve the generalized nonlinear 5th order Korteweg–de Vries (KdV) equations. By introducing gradient reproducing kernel (GRK) approximations, one can circumvent the intricacy of high-order derivatives in RK approximations, while still satisfying high-order consistency requirements. This leads to the high accuracy and efficiency. GRKs are very good approximation candidates for addressing high order partial differential equations that require higher order derivatives formulated as strong form. The stabilized collocation method (SCM) effectively meets high-order integration constraints which can achieve accurate integration in the subdomains. This ensures both high precision and optimal convergence. Von Neumann analysis is utilized to establish the stability criteria for GRK-SCM when combined with forward difference temporal discretization. Numerical solutions for Sawada–Kotera (SK) equation and Kaup-Kupershmidt (KK) equation are studied where the solitary wave migration and collision, and periodic waves are represented. A fifth order forced KdV equation is also considered. The presented method's high accuracy and convergence are demonstrated through comparative studies with analytical solutions, and investigations of invariants and corresponding errors determine the good conservation properties of this algorithm.

The modulations of higher order solitonic pressure and energy of fluid filled elastic tubes

In this article, we investigate the weak higher order nonlinear solitonic pressure waves in elastic, incompressible, nonviscous fluid-filled tubes. The higher order Korteweg–de Vries equation has been developed from the perturbed nonlinear equation. The dependence of higher-order solitonic pressure, higher order fluid velocity, and higher order solitonic energy on the nonlinear coefficient and the perturbation parameter has been established. The presented solutions will be more beneficial in the several disciplines of blood flow and many other related fields of applied science.

Integration of the negative order Korteweg-de Vries equation by the inverse scattering method

In this paper, we consider the integration of the negative order Korteweg-de Vries equation by the inverse scattering method. The evolution of the spectral data of the Sturm-Liouville operator with a potential associated with the solution of the negative order Korteweg-de Vries equation is determined. The obtained results make it possible to apply the method of inverse scattering problem to solve the negative order Korteweg-de Vries equation in the class of rapidly decreasing functions.

Higher order corrections to beyond-all-order effects in a fifth order Korteweg–de Vries equation

Higher order corrections to beyond-all-order effects in a fifth order Korteweg–de Vries equation

Integration of the negative order Korteweg-de Vries equation with self-consistent source

Integration of the negative order Korteweg-de Vries equation with self-consistent source

Whitham shocks and resonant dispersive shock waves governed by the higher order Korteweg–de Vries equation

The addition of higher order asymptotic corrections to the Korteweg–de Vries equation results in the extended Korteweg–de Vries (eKdV) equation. These higher order terms destabilize the dispersive shock wave solution, also termed an undular bore in fluid dynamics, and result in the emission of resonant radiation. In broad terms, there are three possible dispersive shock wave regimes: radiating dispersive shock wave (RDSW), cross-over dispersive shock wave (CDSW) and travelling dispersive shock wave (TDSW). While there are existing solutions for the RDSW and TDSW regimes obtained using modulation theory, there is no existing solution for the CDSW regime. Modulation theory and the associated concept of a Whitham shock are used to obtain this CDSW solution. In addition, it is found that the resonant wavetrain emitted by the eKdV equation with water wave coefficients has a minimal amplitude. This minimal amplitude is explained based on the developed Whitham modulation theory.

Интегрирование уравнения Кортевега – де Фриза отрицательного порядка с источником специального вида

In this paper, we consider the negative order Korteweg-de Vries equation with a self-consistent source corresponding to the eigenvalues of the corresponding spectral problem. It is shown that the considered equation can be integrated by the method of the inverse spectral problem. The evolution of the spectral data of the Sturm-Liouville operator with a periodic potential associated with the solution of the considered equation is determined. The results obtained make it possible to apply the inverse problem method for solving the negative order Korteweg-de Vries equation with a self-consistent source corresponding to the eigenvalues of the corresponding spectral problem.

The lie symmetry analysis of third order Korteweg-de Vries equation

This study sought to analyse the Lie symmetry of third order Korteweg-de Vries equation. Solving nonlinear partial differential equations is of great importance in the world of dynamics. Korteweg-de Vries equations are partial differential equations arising from the theory of long waves, modelling of shallow water waves, fluid mechanics, plasma fluids and many other nonlinear physical systems, and their effects are relevant in real life. In this study, Lie symmetry analysis is demonstrated in finding the symmetry solutions of the third-order KdV equation of the form. The study systematically showed the formula to find the specific solution attained by developing prolongations, infinitesimal transformations and generators, adjoint symmetries, variation symmetries, invariant transformation and integrating factors to obtain all the lie groups presented by the equation. In conclusion, infinitesimal generators, group transformations and symmetry solutions of third-order KdV equation are acquired using a method of Lie symmetry analysis. This was achieved by generating infinitesimal generators which act on the KdV equation to form infinitesimal transformations. It can be seen from the solutions of this paper that the Lie symmetry analysis method is an effective and best mathematical technique for studying linear and nonlinear PDEs and ODEs.

Integration of the negative order Korteweg–de Vries equation with a self-consistent source in the class of periodic functions

In this paper, we consider the negative order Korteweg–de Vries equation with a self-consistent integral source. It is shown that the negative-order Korteweg–de Vries equation with a self-consistent integral source can be integrated by the method of the inverse spectral problem. The evolution of the spectral data of the Sturm–Liouville operator with a periodic potential associated with the solution of the negative order Korteweg–de Vries equation with a self-consistent integral source is determined. The obtained results make it possible to apply the inverse problem method to solve the negative order Korteweg–de Vries equation with a self-consistent source in the class of periodic functions.

New soliton solutions for the space-time fractional modified third order Korteweg–de Vries equation

In present research work, the soliton solutions of the space-time fractional modified third order Korteweg–de Vries (KdV) equation are explored by Sardar-subequation method (SSM). The (KdV) equation has an established version based on the shallow water waves in the oceans, oceanography theory and the ion-acoustic waves in plasma. The fractional order partial differential equation (PDE) is transform to a non-linear ordinary differential equation (ODE) by using traveling wave transformation. It is noted that in this work the fractional differential equation is solved in the sense of conformable derivative. Then soliton solutions of reduced equation are established by SSM. The present technique provides bright, dark, singular, periodic singular, combined bright-dark and combined dark-singular soliton solutions.

Double Elzaki Transform Decomposition Method for Solving Third Order Korteweg-De-Vries Equations

In this study, we used Double Elzaki Transform (DET) coupled with Adomian polynomial to produce a new method to solve Third Order Korteweg-De Vries Equations (KdV) equations. We will provide the necessary explanation for this method with addition some examples to demonstrate the effectiveness of this method.

New algorithm for the approximate solution of generalized seventh order Korteweg-Devries equation arising in shallow water waves

In the present article, we present for the first time optimal auxiliary function method (OAFM) for partial differential equation (PDEs). To find efficient and precision the proposed method, we take Lax’s seventh order korteweg-de Vries (KdV) and seventh order Sawada Kotera (SK) equations as test examples. The beauty of the planned method lies in auxiliary functions Ai and some parameters Ci which ensure a very rapid convergence of the solution. We compare the approximate solutions got by the proposed method with the homotopy perturbation method (HPM), the Optimal Homotopy asymptotic method. It should be emphasized that very good approximation is obtained at the first iteration. It has been shown, that OAFM is a simple and convergent method for the solution of nonlinear equations. The numerical results rendering that the applied method is explicit, efficacious and facile to utilize, for handling more general nonlinear equations.

Discontinuous shock solutions of the Whitham modulation equations as zero dispersion limits of traveling waves

Whitham modulation theory describes the zero dispersion limit of nonlinear disperesive partial differential equations (PDEs) by a system of conservation laws for the parameters of modulated periodic traveling waves (TWs). In this work, admissible, discontinuous, weak solutions of the Whitham modulation equations—termed Whitham shocks—are identified with zero dispersion limits of TW solutions to higher order dispersive PDEs. The far-field behavior of the TW solutions satisfies the Rankine–Hugoniot jump conditions for the Whitham modulation equations. Generally, the numerically computed traveling waves represent heteroclinic connections between two periodic orbits of an ordinary differential equation. The focus here is on the fifth order Korteweg–de Vries equation where three admissible one-parameter families of Whitham shocks are identified as solution components to the generalized Riemann problem for the Whitham modulation equations. Admissible KdV5–Whitham shocks are generally undercompressive, i.e., all characteristic families pass through the shock. The heteroclinic TWs that limit to admissible Whitham shocks are found to be ubiquitous in numerical simulations of smoothed step initial conditions for other higher order dispersive equations including the Kawahara equation (with third and fifth order dispersion) and a nonlocal model of weakly nonlinear gravity-capillary waves with full dispersion. Whitham shocks are linked to recent studies of nonlinear higher order dispersive waves in optics and ultracold atomic gases. The approach presented here provides a novel method for constructing new TW solutions to dispersive nonlinear wave equations and a framework to identify physically relevant, admissible shock solutions of the Whitham modulation equations.

Ion-acoustic waves at the night side of Titan’s ionosphere: higher-order approximation

Nonlinear solitary waves are investigated for a plasma system at the night side of Titan’s ionosphere. The plasma model consists of three positive ions, namely C2H, HCNH+, and C3H, as well as Maxwellian electrons. The basic set of fluid equations is reduced to a Korteweg de-Vries (KdV) equation and linear inhomogeneous higher order KdV (LIHO-KdV) equation. The solitary wave solutions of both equations are obtained using a renormalization method. The solitary waves’ existence region and the wave profile are investigated, and their dependences on the plasma parameters at the night side of Titan’s ionosphere are examined. The solitary waves’ phase velocities are subsonic or supersonic, and the propagating pulses are usually positive. The effect of higher-order corrections on the perturbation theory is investigated. It is found that the higher-order contribution makes the amplitude slightly taller, which is suitable for describing the solitary waves when the amplitude augments.

Comment on "Shallow-water soliton dynamics beyond the Korteweg-de Vries equation".

The authors of the paper "Shallow-water soliton dynamics beyond the Korteweg-de Vries equation" [1] write that they derived a new nonlinear equation describing shallow water gravity waves for an uneven bottom in the form of the higher (fifth)-order Korteweg-de Vries equation for surface elevation. The equation has been obtained by applying a perturbation method [2] for specific relations between the orders of the three small parameters of the problem α=O(β) and δ=O(β) up to the second order in β. In this comment, it is shown that the derivation presented in [1] is inconsistent because of an oversight concerning the orders of terms in equations of the Boussinesq system. Therefore the results, in particular, the new evolution equation and the dynamics that it describes, bear no relation to the problem under consideration. A consistent derivation is presented, and also results of applying the perturbation procedure with some other orderings between the small parameters are given to provide a broader view of the problem. Several new nonlinear evolution equations governing small amplitude shallow water waves for an uneven bottom have been derived.

On solutions of the combined KdV-nKdV equation

The aim of this work is to deal with a new integrable nonlinear equation of wave propagation, the combined of the Korteweg-de vries equation and the negative order Korteweg-de vries equation (combined KdV-nKdV) equation, which was more recently proposed by Wazwaz. Upon using wave reduction variable, it turns out that the reduced combined KdV-nKdV equation is alike the reduced (3+1)-dimensional Jimbo Miwa (JM) equation, the reduced (3+1)-dimensional Potential Yu-Toda-Sasa-Fukuyama (PYTSF) equation and the reduced (3 + 1)¬dimensional generalized shallow water (GSW) equation in the trav¬elling wave. In fact, the four transformed equations belong to the same class of ordinary differential equation. With the benefit of a well known general solutions for the reduced equation, we show that sub¬jects to some scaling and change of parameters, a variety of families of solutions are constructed for the combined KdV-nKdV equation which can be expressed in terms of rational functions, exponential functions and periodic solutions of trigonometric functions and hyperbolic func¬tions. In addition to that the equation admits solitary waves, and double periodic waves in terms of special functions such as Jacobian elliptic functions and Weierstrass elliptic functions.

Constructive Study of Modulational Instability in Higher Order Korteweg-de Vries Equations

Our present study is devoted to the constructive study of the modulational instability for the Korteweg-de Vries (KdV)-family of equations u t + s u p u x + u x x x (here s = ± 1 and p > 0 is an arbitrary integer). For deducing the conditions of the instability, we first computed the nonlinear corrections to the frequency of the Stokes wave and then explored the coefficients of the corresponding modified nonlinear Schrödinger equations, thus deducing explicit expressions for the instability growth rate, maximum of the increment and the boundaries of the instability interval. A brief discussion of the results, open questions and further research directions completes the paper.

New Cubic B-spline Approximations for Solving Non-linear Third-order Korteweg-de Vries Equation

Objectives: In this work, the approximate solution of non-linear third order Korteweg-de Vries equation has been studied. Methods: The proposed numerical technique engages finite difference formulation for temporal discretization, whereas, the discretization in space direction is achieved by means of a new cubic B-spline approximation. Findings: In order to corroborate this effort, three test problems have been considered and the computational outcomes are compared with the current methods. It is found that the proposed scheme involves straight forward computations and operates superior to the existing methods. Novelty/Improvements: The proposed numerical scheme is novel for Korteweg-de Vries equation and has never been employed for this purpose before. Keywords: Cubic B-spline Collocation Method, Cubic B-spline Functions, Finite Difference Formulation, Korteweg-de Vries Equation

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.