Coupled Korteweg-de Vries Research Articles

On a coupled Kadomtsev–Petviashvili system associated with an elliptic curve

AbstractThe coupled Kadomtsev–Petviashvili system associated with an elliptic curve, proposed by Date, Jimbo, and Miwa [J. Phys. Soc. Jpn., 52:766–771, 1983], is reinvestigated within the direct linearization framework, which provides us with more insights into the integrability of this elliptic model from the perspective of a general linear integral equation. As a result, we successfully construct for the elliptic coupled Kadomtsev–Petviashvili system not only a Lax pair composed of differential operators in 2 × 2 matrix form but also multisoliton solutions with phases parametrized by points on the elliptic curve. Dimensional reductions based on the direct linearization, to the elliptic coupled Korteweg–de Vries and Boussinesq systems, are also discussed. In addition, a novel class of solutions is obtained for the ‐type Kadomtsev–Petviashvili equation with nonzero constant background as a by‐product.

Soliton solution of coupled Korteweg-de Vries equation by quintic UAH tension B-spline differential quadrature method

The concern of this paper is to acquire the numerical solution of Coupled Korteweg-de Vries (CKdV) equation describing the behaviour of shallow water surfaces in terms of solitons. A new regime quintic Uniform Algebraic Hyperbolic (QUAH) tension B-spline with a very well-known method called Differential Quadrature Method (DQM) is used. Over the past few years, the idea of consuming DQM to solve partial differential equations numerically has acknowledged much consideration all over the science community. Initially, the QUAH tension B-spline is used as a basis function in DQM to solve the weighting coefficients. After calculating weighting coefficients, the basis function converts the Initial boundary value problem into an ordinary differential equation, which is further solved by using a strong preserving Runge-Kutta 43 regime. Then, to check the accuracy and amendment of the used method, different test problems are solved numerically as well as error norms L2 and L∞ are calculated. Calculated results are shown graphically and in tabular form for quick and easy access. An obtained result gives the insight that the proposed technique is effective and efficient for elucidating the variety of complex partial differential equations.

Conserved Quantities of a Nonlinear Coupled System of Korteweg-De Vries Equations

The conserved vectors from a system of coupled Kortewegde Vries equations that have modelled the propagation of shallow water waves, ion-acoustic waves in plasmas, solitons, and nonlinear perturbations along internal surfaces between layers of different densities in stratified fluids, for example propagation of solitons of long internal waves in oceans. Notable applications have been to model shock wave formation, turbulence, boundary layer behavior, and mass transport. This paper illustrates the computation of conserved quantities using two approaches. First, by the multiplier method and by an application of new conservation theorem developed by Nail Ibragimov. 2010 Mathematics Subject Classification. 47B47; 47A30.

Collisional positron acoustic soliton and double layer in an unmagnetized plasma having multi-species

This paper explores the head-on collision between two-counter propagating positron acoustic solitons and double layers (DLs) in an unmagnetized collisionless plasma having mobile cold positrons fluid, immobile positive ions and (r,;q)-distributed hot positrons, and hot electrons. By employing the extended Poincaré–Lighthill–Kuo method, the coupled Korteweg–de Vries (KdV), modified KdV (mKdV) and Gardner equations are derived to archive this goal. The effect of dimensionless parameters on the propagation characteristics of interacting KdV solitons (KdVSs), mKdV solitons (mKdVSs), Gardner solitons (GSs) and DLs are examined in detail by considering the limiting cases of (r,;q)-distribution. It is noted that the interaction of GSs and DLs are reported for the first time. The outcomes might be comprehended and beneficial not only in space and astrophysical environments but also in laboratory studies.

Numerical Solutions of the Coupled Kortweg-de Vries Systems

A Homotopy analytical method (HAM) for solving nonlinear coupled Korteweg-de Vries (KdV) system is presented. The HAM contains an assistant parameter, which gives a helpful manner of controlling the merging locale of the arrangements. We will comparison exact solution with numerical results. This technique provides an efficient and effectives approximate analytical solution with high accuracy appeared graphically in figures demonstrate and affirm the tall precision and outlining the productivity and straightforwardness of the strategy.

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.

A novel narration for two long waves interaction through the elaboration scheme

The Jacobi elliptic function expansion scheme (JEFES) is broader than tan hyperbolic scheme, is derived to assemble new wave solutions for the modal of two long waves’ interactions. It’s proposing a new method to accumulate more general exact solutions (solitary and periodic) of nonlinear coupled Korteweg-de Vries (NCKDV) system and its craving to look up the works that are finished. It will be shown that limit on the solution may get more new solutions.

A Comparative Analysis of the Fractional‐Order Coupled Korteweg–De Vries Equations with the Mittag–Leffler Law

This article applies efficient methods, namely, modified decomposition method and new iterative transformation method, to analyze a nonlinear system of Korteweg–de Vries equations with the Atangana–Baleanu fractional derivative. The nonlinear fractional coupled systems investigated in this current analysis are the system of Korteweg–de Vries and the modified system of Korteweg–de Vries equations applied as a model in nonlinear physical phenomena arising in chemistry, biology, physics, and applied sciences. Approximate analytical results are represented in the form of a series with straightforward components, and some aspects showed an appropriate dependence on the values of the fractional‐order derivatives. The convergence and uniqueness analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time‐fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. The series result achieved applying this technique is proved to be accurate and reliable with minimal calculations. The numerical simulations for obtained solutions are discussed for different values of the fractional order.

Physics-informed neural network for solving coupled Korteweg-de Vries equations

The studies of coupled partial differential equations are focus of engineering and applied mathematics. Although traditional numerical methods have been widely used, researchers are still looking for new methods to solve coupled partial differential equations. In this paper, physical information neural network (PINN) is introduced to solve one-dimensional coupled Korteweg-de Vries (cKdV) equations. Compared with the traditional neural network, the innovation of PINN is to embed the physical constraints of the equations into the network loss function. Moreover, within the acceptable relative error range, the solution can take a longer single time step than the presently available. The results revealed that PINN can solve the cKdV equations with reasonable errors only by training a small amount of data.

The Analysis of Fractional-Order Kersten–Krasil Shchik Coupled KdV System, via a New Integral Transform

In this article, we use the homotopy perturbation transform method to find the fractional Kersten–Krasil’shchik coupled Korteweg–de Vries (KdV) non-linear system. This coupled non-linear system is typically used to describe electric circuits, traffic flow, shallow water waves, elastic media, electrodynamics, etc. The homotopy perturbation method is modified with the help of the ρ-Laplace transformation to investigate the solution of the given examples to show the accuracy of the current technique. The solution of the given technique and the actual results are shown and analyzed with figures.

Characterization of Darboux transformations for quantum systems with quadratically energy-dependent potentials

We construct three classes of higher-order Darboux transformations for Schrödinger equations with quadratically energy-dependent potentials by means of generalized Wronskian determinants. Particular even-order cases reduce to the Darboux transformation for conventional (energy-independent) potentials. Our construction is based on an adaptation of the results for coupled Korteweg–de Vries equations [N. V. Ustinov and S. B. Leble, J. Math. Phys. 34, 1421 (1993)].

A Novel Analytical View of Time-Fractional Korteweg-De Vries Equations via a New Integral Transform

We put into practice relatively new analytical techniques, the Shehu decomposition method and the Shehu iterative transform method, for solving the nonlinear fractional coupled Korteweg-de Vries (KdV) equation. The KdV equation has been developed to represent a broad spectrum of physics behaviors of the evolution and association of nonlinear waves. Approximate-analytical solutions are presented in the form of a series with simple and straightforward components, and some aspects show an appropriate dependence on the values of the fractional-order derivatives that are, in a certain sense, symmetric. The fractional derivative is proposed in the Caputo sense. The uniqueness and convergence analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. It is worth mentioning that the proposed methods are powerful and are some of the best procedures to tackle nonlinear fractional PDEs.

Crank–Nicolson finite difference method for time-fractional coupled KdV equation

A Crank–Nicolson finite difference method is presented to solve a system of time fractional coupled Korteweg–de Vries (KdV) equation, the Grünwald–Letnikov definition is used for the time-fractional derivative. The approximation solutions for some numerical examples by the proposed method clarify that the method is applicable, simple and efficient for such problems.

Fractional System of Korteweg-De Vries Equations via Elzaki Transform

In this article, a hybrid technique, called the Iteration transform method, has been implemented to solve the fractional-order coupled Korteweg-de Vries (KdV) equation. In this method, the Elzaki transform and New Iteration method are combined. The iteration transform method solutions are obtained in series form to analyze the analytical results of fractional-order coupled Korteweg-de Vries equations. To understand the analytical procedure of Iteration transform method, some numerical problems are presented for the analytical result of fractional-order coupled Korteweg-de Vries equations. It is also demonstrated that the current technique’s solutions are in good agreement with the exact results. The numerical solutions show that only a few terms are sufficient for obtaining an approximate result, which is efficient, accurate, and reliable.

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.

Numerical solution of time-fractional coupled Korteweg–de Vries and Klein–Gordon equations by local meshless method

This article provides numerical simulations of the time-fractional coupled Korteweg–de Vries and Klein–Gordon equations via the local meshless collocation method (LMCM) utilising the radial basis functions. The recommended local meshless technique is utilised for the space derivatives of the models whereas Caputo fractional definition is used for time-fractional derivative. Numerical experiments are performed for one-dimensional coupled Korteweg–de Vries and two-dimensional Klein–Gordon equations. In order to verify the efficiency and accuracy of the proposed meshless method, numerical results are compared with exact and numerical techniques reported in recent literature which reveals that the method is computationally attractive and produces better results.

Numerical calculation of N-periodic wave solutions to coupled KdV-Toda-type equations.

In this paper, we study the N-periodic wave solutions of coupled Korteweg-de Vries (KdV)-Toda-type equations. We present a numerical process to calculate the N-periodic waves based on the direct method of calculating periodic wave solutions proposed by Akira Nakamura. Particularly, in the case of N = 3, we give some detailed examples to show the N-periodic wave solutions to the coupled Ramani equation, the Hirota-Satsuma coupled KdV equation, the coupled Ito equation, the Blaszak-Marciniak lattice, the semi-discrete KdV equation, the Leznov lattice and a relativistic Toda lattice.

Alternative solitons in the Hirota–Satsuma system via the direct method

Multisolitons and multi-hump solitons are constructed for the coupled Korteweg–de Vries (cKdV) system introduced by Hirota and Satsuma (1981) in the context of the propagation of two waves interacting with different dispersion relations. The direct method analytically exhibit a new class of complementary or alternative soliton solutions functionally different to the traditional or classical multisolitons known (Hirota and Satsuma, 1981; Ramani et al., 1983; Parra Prado and Cisneros-Ake, 2020). As in the classical case, our alternative multisolitons are found to obey elastic interactions of the N-KdV and M-cKdV solitons. To show this, we inductively construct tables to describe in detail the mechanism of the different wave interactions and the coefficients involved in them. As a particular scenario, it is found that each of the 1-cKdV and 1-KdV-1cKdV interaction cases may produce two independent criteria for the appearance of one-soliton solutions consisting of two humps in shape. A proper combination of these two conditions gives place to multi-hump solitons. We explicitly show the two-one, two-two, three-one and three-two humps for the two-component wave solutions. • The direct method due to Hirota allows to find complementary two-component multisolitons in a coupled KdV system. • Symmetric one-soliton solutions with two humps in shape are obtained from two independent criteria. • Multi-hump alternative solitons are constructed from two different two-hump conditions.

Resonant coupling of mode-1 and mode-2 internal waves by topography

Abstract

New multi-hump exact solitons of a coupled Korteweg-de-Vries system with conformable derivative describing shallow water waves via RCAM

In this article, a modification of the rapidly convergent approximation method is proposed to solve a coupled Korteweg–de Vries equations with conformable derivative that govern shallow-water waves. Based on the Leibniz and chain rule of conformable derivative, these equations reduced into ODEs with integer-order using traveling wave transformation. Adopting the modified scheme a new novel exact solution of the reduced coupled ordinary differential equations is obtained in terms of exponential functions. Finally, by putting them back into traveling wave transformation the solutions of the considered partial differential equations with conformable derivative are derived. To ensure the boundedness of the derived solutions few theorems have been proposed and proved. The derived results of the theorems are utilized to plot the solutions. Graphics exhibit that solutions have variant multi-hump soliton peculiarities and their tails decay to zero exponentially in a monotonic manner. These results not only show the efficiency of the modified scheme but also establish that the solution is enriched with new multi-hump features.