Coupled Korteweg-de Vries System Research Articles

Numerical simulations for fractional Hirota–Satsuma coupled Korteweg–de Vries systems

Abstract In this investigation, the fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) problem is solved using two modern semi-analytic techniques known as the Aboodh residual power series method (ARPSM) and Aboodh transform iteration method (ATIM). The two suggested approaches are briefly explained, along with how to use them to solve the fractional Hirota–Satsuma coupled KdV problem. Some analytical approximate solutions for the current problem are derived using the proposed techniques until the second-order approximation. To ensure high accuracy of the derived approximation, they are analyzed numerically and graphically and compared with the exact solutions of the integer cases. The offered techniques demonstrate more accuracy in their outcomes compared to other alternatives. The numerical results show that ARPSM and ATIM are highly accurate, practical, and beneficial for solving nonlinear equation systems. The current results are expected to help many physics researchers in modeling their different physical problems, especially those interested in plasma physics.

Generalized Darboux algorithms for Dunkl–Schrödinger equations with position-dependent mass at fixed energy

We apply generalized Darboux transformations to Schrödinger equations at fixed energy that contain Dunkl derivatives, and are equipped with a position-dependent mass. Our Darboux transformations, originating from coupled Korteweg–De Vries systems, consist of two algorithms that generate new solutions, potentials and position-dependent mass functions for the Dunkl–Schrödinger system.

A new analytical approximate solution of fractional coupled Korteweg-de Vries system

The main objective of this work is to present a modification of the Mittag- Leffler function to deduce a relatively new analytical approximate method (for short MMLFM) able to solve time-fractional nonlinear partial differential equations (PDEs). Moreover, we employ the MMLFM to solve the time-fractional coupled Korteweg?de Vries (KdV) model described by two nonlinear fractional partial differential equations (FPDEs) based upon Caputo fractional derivative (CFD). The simulation of projected results is presented in some figures and tables. Furthermore, we compare our solutions when ? = 1 with known exact solutions which indicate a good agreement, in addition, we compare our outcomes with the results obtained by other methods in the literature such as the Natural decomposing method (NDM) and homotopy decomposition method (HDM) in order to prove the reliability and efficiency of our used method. Also, we display solutions with different values of ? to present the effect of the fractional order on the proposed problem. The results of this article reveal the advantages of the MMLFM, which is simple, reliable, accurate, needs simple mathematical computations, is rapidly convergent to the exact solution, have a straightforward and easy algorithm compared to other analytical methods to study linear and nonlinear FPDEs, which makes this technique suited for real industrial or medical applications.

New Soliton Solutions of Time-Fractional Korteweg–de Vries Systems

Model construction for different physical situations, and developing their solutions, are the major characteristics of the scientific work in physics and engineering. Korteweg–de Vries (KdV) models are very important due to their ability to capture different physical situations such as thin film flows and waves on shallow water surfaces. In this work, a new approach for predicting and analyzing nonlinear time-fractional coupled KdV systems is proposed based on Laplace transform and homotopy perturbation along with Caputo fractional derivatives. This algorithm provides a convergent series solution by applying simple steps through symbolic computations. The efficiency of the proposed algorithm is tested against different nonlinear time-fractional KdV systems, including dispersive long wave and generalized Hirota–Satsuma KdV systems. For validity purposes, the obtained results are compared with the existing solutions from the literature. The convergence of the proposed algorithm over the entire fractional domain is confirmed by finding solutions and errors at various values of fractional parameters. Numerical simulations clearly reassert the supremacy and capability of the proposed technique in terms of accuracy and fewer computations as compared to other available schemes. Analysis reveals that the projected scheme is reliable and hence can be utilized with other kernels in more advanced systems in physics and engineering.

A nonhomogeneous boundary‐valued problem for the coupled Korteweg–de Vries system

In this paper, we study the initial‐boundary‐value problem (IBVP) for coupled Korteweg–de Vries equations posed on a finite interval with nonhomogeneous boundary conditions. We overcome the requirement for stronger smooth boundary conditions in the traditional method via the Laplace transform. Our approach uses the strong Kato smoothing property and the contraction mapping principle.

Closed-form solutions and conservation laws of a generalized Hirota–Satsuma coupled KdV system of fluid mechanics

Abstract In this article, a generalized Hirota–Satsuma coupled Korteweg–de Vries (KdV) system is investigated from the group standpoint. This system represents an interplay of long waves with distinct dispersion correlations. Using Lie’s theory several symmetry reductions are performed and the system is reduced to systems of non-linear ordinary differential equations (NLODEs). Subsequently, the simplest equation method is invoked to find exact solutions of the NLODE systems, which then provides the solitary wave solutions for the system under discussion. Finally, we construct conservation laws of generalized Hirota–Satsuma coupled KdV system with the aid of general multiplier approach.

A powerful approach to study the new modified coupled Korteweg–de Vries system

In this work, the q-homotopy analysis transform method (shortly q-HATM) which is a combined form of q-homotopy analysis method and Laplace transform method is employed to find numerical solution to the new modified coupled Korteweg–de Vries system. This method allows us to fine-tune the convergence region along with rate of convergence of the obtained series solution by allowing the auxiliary parameters n and ħ to vary. The obtained solution by the proposed method is presented in a refined convergent series form. The numerical results show that only few terms are sufficient to obtain an approximate solution which is accurate, efficient, and reliable. Furthermore, the graphical depictions of the obtained approximate solution of the system are presented.

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.

Bosonization and new interaction solutions for the coupled Korteweg-de Vries system

ABSTRACTThe novel coupled Korteweg-de Vries (KdV) system is derived under the condition named anyon-n algebra. The dark parameterization method is applied into the KdV system, thus, we can obtain new coupled integrable system represented by Lax pairs, and abundant new solution structures are found. It is proved that the dark parameterization procedure is very useful to find new integrable systems and solve nonlinear coupled systems.

Gauge symmetries for a coupled Korteweg-de Vries system

We introduce a gauge invariant Hamiltonian which generalizes one of the Hamiltonian structures of a parametric KdV system, which includes the complex KdV system. The associated canonical field equations have an infinite sequence of conserved quantities which are also gauge invariant. The gauge symmetry relates a wide class of integrable systems to the complex KdV system.

Duality relation among the Hamiltonian structures of a parametric coupled Korteweg-de Vries system

AbstractWe obtain the full Hamiltonian structure for a parametric coupled KdV system. The coupled system arises from four different real basic lagrangians. The associated Hamiltonian functionals and the corresponding Poisson structures follow from the geometry of a constrained phase space by using the Dirac approach for constrained systems. The overall algebraic structure for the system is given in terms of two pencils of Poisson structures with associated Hamiltonians depending on the parameter of the Poisson pencils. The algebraic construction we present admits the most general space of observables related to the coupled system. We then construct two master lagrangians for the coupled system whose field equations are theε-parametric Gardner equations obtained from the coupled KdV system through a Gardner transformation. In the weak limitε→ 0 the lagrangians reduce to the ones of the coupled KdV system while, after a suitable redefinition of the fields, in the strong limitε→ ∞ we obtain the lagrangians of the coupled modified KdV system. The Hamiltonian structures of the coupled KdV system follow from the Hamiltonian structures of the master system by taking the two limitsε→ 0 andε→ ∞.

A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equations

The time-fractional coupled Korteweg---de Vries (KdV) system is a generalization of the classical coupled KdV system and obtained by replacing the first order time derivatives by fractional derivatives of orders $$\nu _1$$?1 and $$\nu _2$$?2, $$(0<\nu _1,\nu _2\le 1).$$(0<?1,?2≤1). In this paper, an accurate and robust numerical technique is proposed for solving the time-fractional coupled KdV equations. The shifted Legendre polynomials are introduced as basis functions of the collocation spectral method together with the operational matrix of fractional derivatives (described in the Caputo sense) in order to reduce the time-fractional coupled KdV equations into a problem consisting of a system of algebraic equations that greatly simplifies the problem. In order to test the efficiency and validity of the proposed numerical technique, we apply it to solve two numerical examples.

Head-on collision of solitary waves in coupled Korteweg–de Vries systems modeling nonlinear transmission lines

The extended Poincaré–Lighthill–Kuo (PLK) method is applied to characterize head-on collisions of solitary waves in a coupled Korteweg–de Vries (KdV) system that has multiple modes supporting solitons. As a simple physically realizable system, we investigate two coupled electrical nonlinear transmission lines (NLTLs), and the proposed method successfully leads to the collision-induced phase shifts and the wave equation that governs the dynamics of the pulses generated by colliding solitary waves.

A Meshless Method of Lines for Numerical Solution of Some Coupled Nonlinear Evolution Equations

Abstract Method of lines (MOL) together with radial basis functions (RBFs) is proposed to find the numerical solution of coupled Korteweg-de Vries (KdV) equations, coupled modified Korteweg-de Vries (mKdV) equations and coupled KdV system. The initial-boudary-value problem is reduced to a system of ordinary differential equations. The system is then solved by the fourth order Runge-Kutta method. Numerical examples are given to illustrate practical usefulness of the present approach. The method is efficient and all the examples take very low CPU time. Accuracy of the method is assessed in terms of L∞ error norm, number of nodal points and time step size. Superiority of the suggested method is shown through comparison with the spectral collocation method.

A Method to Construct the Nonlocal Symmetries of Nonlinear Evolution Equations

A method is proposed to seek the nonlocal symmetries of nonlinear evolution equations. The validity and advantages of the proposed method are illustrated by the applications to the Boussinesq equation, the coupled Korteweg-de Vries system, the Kadomtsev—Petviashvili equation, the Ablowitz—Kaup—Newell—Segur equation and the potential Korteweg-de Vries equation. The facts show that this method can obtain not only the nonlocal symmetries but also the general Lie point symmetries of the given equations.

Exact Explicit Solutions and Conservation Laws for a Coupled Zakharov-Kuznetsov System

We study a coupled Zakharov-Kuznetsov system, which is an extension of a coupled Korteweg-de Vries system in the sense of the Zakharov-Kuznetsov equation. Firstly, we obtain some exact solutions of the coupled Zakharov-Kuznetsov system using the simplest equation method. Secondly, the conservation laws for the coupled Zakharov-Kuznetsov system will be constructed by using the multiplier approach.

Conservation laws for a generalized Ito-type coupled KdV system

Abstract In this paper, the conservation laws for a generalized Ito-type coupled Korteweg-de Vries (KdV) system are constructed by increasing the order of the partial differential equations. The generalized Ito-type coupled KdV system is a third-order system of two partial differential equations and does not have a Lagrangian. The transformation u = U x , v = V x converts the generalized Ito-type coupled KdV system into a system of fourth-order partial differential equations in U and V variables, which has a Lagrangian. Noether’s approach is then used to construct the conservation laws. Finally, the conservation laws are expressed in the original variables u and v. Some local and infinitely many nonlocal conserved quantities are found for the generalized Ito-typed coupled KdV system.

On completely integrable coupled Burgers and coupled Korteweg–de Vries systems

In this work, we study two completely integrable equations, namely, coupled Burgers and Korteweg–de Vries systems. The modified form of Hirota’s bilinear method, established by Hereman, is employed to formally derive multiple-soliton solutions and multiple-singular-soliton solutions for each system. Hirota’s bilinear method is reliable and effective and can also be applied to solve other types of higher-dimensional integrable and non-integrable systems.

Note on the Lax Pair of a Coupled Hybrid System

The hybrid lattice, known as a discrete Korteweg-de Vries (KdV) equation, is found to be a discrete modified Korteweg-de Vries (mKdV) equation. The coupled hybrid lattice, which is pointed out to be a discrete coupled KdV system, is also found to be a discrete form of a coupled mKdV system. New lax pairs for the single and coupled discrete hybrid systems are proposed as a different study from previous ones.

A Coupled Hybrid Lattice: Its Related Continuous Equation and Symmetries

The hybrid lattice, known as a discrete Korteweg-de Vries (KdV) equation, is found to be a discrete modified Korteweg-de Vries (mKdV) equation in this paper. The coupled hybrid lattice, which is pointed to be a discrete coupled KdV system, is also found to be discrete form of a coupled mKdV systems. Delayed differential reduction system and pure difference systems are derived from the coupled hybrid system by means of the symmetry reduction approach. Cnoidal wave, positon and negaton solutions for the coupled hybrid system are proposed.