Coupled mKdV Equations Research Articles

Riemann–Hilbert Method Equipped with Mixed Spectrum for N-Soliton Solutions of New Three-Component Coupled Time-Varying Coefficient Complex mKdV Equations

Riemann–Hilbert Method Equipped with Mixed Spectrum for N-Soliton Solutions of New Three-Component Coupled Time-Varying Coefficient Complex mKdV Equations

Multiple and Singular Soliton Solutions for Space–Time Fractional Coupled Modified Korteweg–De Vries Equations

The focus of this paper is on the nonlinear coupled evolution equations, specifically within the context of the fractional coupled modified Korteweg–de Vries (mKdV) equation, employing the conformable fractional derivative (CFD) approach. The primary objective of this paper is to thoroughly investigate the applicability of the Hirota bilinear method for deriving analytical solutions to the fractional mKdV equations. A range of exact analytical solutions for the fractional coupled mKdV equations is obtained. The findings in general indicate that the Hirota bilinear method is an effective approach for resolving the complexities associated with the fractional coupled mKdV equations.

Diverse soliton solutions and dynamical analysis of the discrete coupled mKdV equation with 4×4 Lax pair

Under consideration in this study is the discrete coupled modified Korteweg–de Vries (mKdV) equation with 4 × 4 Lax pair. Firstly, through using continuous limit technique, this discrete equation can be mapped to the coupled KdV and mKdV equations, which may depict the development of shallow water waves, the optical soliton propagation in cubic nonlinear media and the Alfven wave in a cold collision-free plasma. Secondly, the discrete generalized (r, N–r)-fold Darboux transformation is constructed and extended to solve this discrete coupled equation with the fourth-order linear spectral problem, from which diverse exact solutions including usual multi-soliton and semi-rational soliton solutions on the vanishing background, higher-order rational soliton and mixed hyperbolic-rational soliton solutions on the non-vanishing background are derived, and the limit states of some soliton and rational soliton solutions are analyzed by the asymptotic analysis technique. Finally, the numerical simulations are used to explore the dynamical behaviors of some exact soliton solutions. These results may be helpful for understanding some physical phenomena in fields of shallow water wave, optics, and plasma physics.

Higher-order regulatable rogue wave and hybrid interaction patterns for a new discrete complex coupled mKdV equation associated with the fourth-order linear spectral problem

Higher-order regulatable rogue wave and hybrid interaction patterns for a new discrete complex coupled mKdV equation associated with the fourth-order linear spectral problem

The investigation of travelling wave solutions to some coupled space-time fractional differential equations in fluids

Solutions of nonlinear coupled fractional differential equations (FDEs), namely, the space-time fractional coupled Burgers’ equations, coupled MKdV equations, and coupled KdV equations are obtained by the improved fractional sub-equation method in the sense of Jumarie’s modified Riemann-Liouville derivative. The form of these solutions suggests ansatz suitable for finding solutions of much more general FDEs of coupled type.

Closed form soliton solutions to the space-time fractional foam drainage equation and coupled mKdV evolution equations

Abstract Fractional nonlinear evolution equations concerning conformable fractional derivative are effective models to interpret intricate physical phenomena in the real world. The space-time fractional foam drainage equation and the coupled mKdV equations with conformable fractional derivative are important model equations for shallow water waves, the waves of flow of liquid between bubbles, the capillary waves, the waves of foam density, the electro-hydro-dynamical model, the ion acoustic plasma waves etc. In this study, we extract the functional and further general exact wave solutions comprising the rational, trigonometric, exponential and hyperbolic functions to the above stated models taking the advantage of the auxiliary equation method with the assistance of the fractional complex transformation. The method is convenient, reliable and delivers fresh and useful solutions to fractional differential equations ascend in physical and engineering sciences. We depict figures of the obtained solutions in order to illustrate the inner structure associated to the phenomena.

The N-soliton solutions for the matrix modified Korteweg–de Vries equation via the Riemann–Hilbert approach

This paper focus on seeking the N -soliton solutions for the matrix mKdV equation, which generalizes to the multi-component mKdV equations and the coupled mKdV equations. By utilizing the technique of Riemann–Hilbert and investigating the spectral problem of the Lax pair, a class of Riemann–Hilbert problem will be discussed and established. Finally, in the case of irregularity and reflectionless, we derive the N -soliton solutions for a system of the matrix mKdV equation.

A Lumped Galerkin finite element method for the generalized Hirota-Satsuma coupled KdV and coupled MKdV equations

In the present study, a Lumped Galerkin finite element method using quadratic B-splines has been applied to the generalized Hirota-Satsuma coupled Korteweg de Vries (KdV) and coupled modified Korteweg-de Vries (mKdV) equations. The numerical solutions of discretized equations using Lumped Galerkin finite element method have been obtained using the fourth order Runge-Kutta method which is widely used for the solution of ordinary differential equation system. The numerical solutions obtained for various space and time values have been compared with exact ones using the error norms $L_{2}$ and $L_{\infty}$. Lumped Galerkin finite element method is an effective one which can be applied to a wide range of nonlinear evolution equations.

Riemann-Hilbert Problems of a Six-Component mKdV System and its Soliton Solutions

Based on a 4 × 4 matrix spectral problem, an AKNS soliton hierarchy with six potentials is generated. Associated with this spectral problem, a kind of Riemann-Hilbert problems is formulated for a six-component system of mKdV equations in the resulting AKNS hierarchy. Soliton solutions to the considered system of coupled mKdV equations are computed, through a reduced Riemann-Hilbert problem where an identity jump matrix is taken.

Analytical and Numerical Solutions for Time-Fractional New Coupled mKdV Equation Arising in Interaction of Two Long Waves

Analytical and Numerical Solutions for Time-Fractional New Coupled mKdV Equation Arising in Interaction of Two Long Waves

Riemann–Hilbert problems and N-soliton solutions for a coupled mKdV system

A 3 × 3 matrix spectral problem is introduced and its associated AKNS integrable hierarchy with four components is generated. From this spectral problem, a kind of Riemann–Hilbert problems is formulated for a system of coupled mKdV equations in the resulting AKNS integrable hierarchy. N -soliton solutions to the coupled mKdV system are presented through a specific Riemann–Hilbert problem with an identity jump matrix.

Fractional sub-equation method for Hirota–Satsuma-coupled KdV equation and coupled mKdV equation using the Atangana’s conformable derivative

ABSTRACTIn this paper, we present the exact solutions obtained for the space–time conformable generalized Hirota–Satsuma-coupled KdV equation and coupled mKdV equation using the Atangana’s conformable derivative. The conformable sub-equation method is applied to obtain the solutions; the solutions obtained are compared with the extended tanh-function method for the special case when the fractional order takes the integer order. The analytical solutions show that the conformable sub-equation method is very effective for the conformable-coupled KdV and mKdV equations.

Explicit Solutions of the Coupled mKdV Equation by the Dressing Method via Local Riemann-Hilbert Problem

We study the coupled mKdV equation by the dressing method via local Riemann-Hilbert problem. With the help of the Lax pairs, we obtain the matrix Riemann-Hilbert problem with zeros. The explicit solutions for the coupled mKdV equation are derived with the aid of the regularization of the Riemann-Hilbert problem.

Series Solution for the Time-Fractional Coupled mKdV Equation Using the Homotopy Analysis Method

We present new analytical approximated solutions for the space-time fractional nonlinear partial differential coupled mKdV equation. A homotopy analysis method is considered to obtain an infinite series solution. The effectiveness of this method is demonstrated by finding exact solutions of the fractional equation proposed, for the special case when the limit of the integral order of the time derivative is considered. The comparison shows a precise agreement between these solutions.

Solitons collision and freak waves in a plasma with Cairns-Tsallis particle distributions

The solitons collision (head-on collision) and rogue waves in an unmagnetized plasma comprising nonthermal-nonextensive distributed (Cairns-Tsallis) electrons and cold ions are investigated. For solitons collision, the extended Poincaré–Lighthill–Kuo (PLK) method is employed to derive the coupled Korteweg-de Vries (KdV) equations and their corresponding phase shifts. It is found that solitons having two polarities can propagate in the present model. The coefficients of the nonlinear terms of the coupled KdV equations vanish at a critical value of nonthermality. Therefore, another set of coupled modified KdV (mKdV) equations with cubic nonlinearity is derived and the corresponding phase shifts are calculated. It is found analytically and numerically that the solutions of the coupled KdV equations allow solitons collision only when the solitons have the same polarity, whereas the coupled mKdV equations allow the collisions between the two solitons of the same and opposite polarities. The influence of the nonthermal-nonextensive parameters on the phase shifts of the solitons collision is examined. Furthermore, the rogue waves are studied in the framework of the mKdV equation. The behavior of the rogue waves is analyzed using the nonlinear Schrödinger equation (NLSE), derived from the mKdV equation. It is found that the rogue wave amplitude shrinks with the increase of the nonextensive parameter. The NLSE derived from the KdV equation cannot support the presence of rogue waves.

Numerical simulation of coupled MKdV equation by reduced differential transform method

The main goal of this article is to demonstrate the use of the Reduced Differential Transform Method (RDTM). This method has been applied directly without using bilinear forms, Wronskian, or inverse scattering method. Also it is worth pointing out that if the terms of the series increase, the RDTM provides better convergence to the analytical solution. In this paper, this method is used for solving nonlinear coupled MKdV equation with given initial conditions having arbitrary constants. The numerical solutions obtained by RDTM are compared with the known exact solutions by fixing the arbitrary constants and with Adomian Decomposition Method (ADM) in tables as well as in figures.

The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics

In this article, the modified extended tanh-function method is employed to solve fractional partial differential equations in the sense of the modified Riemann–Liouville derivative. Based on a nonlinear fractional complex transformation, certain fractional partial differential equations can be turned into nonlinear ordinary differential equations of integer orders. For illustrating the validity of this method, we apply it to four nonlinear equations namely, the space–time fractional generalized nonlinear Hirota–Satsuma coupled KdV equations, the space–time fractional nonlinear Whitham–Broer–Kaup equations, the space–time fractional nonlinear coupled Burgers equations and the space–time fractional nonlinear coupled mKdV equations.

Fractional Sub-equation Method and Analytical Solutions to the Hirota-satsuma Coupled KdV Equation and Coupled mKdV Equation

Fractional Sub-equation Method and Analytical Solutions to the Hirota-satsuma Coupled KdV Equation and Coupled mKdV Equation

THE RATIONAL SOLUTIONS TO A GENERALIZED RICCATI EQUATION AND THEIR APPLICATION

Based on the rational solutions to a generalized Riccati equation, a new method which is called as rational function method is proposed. We apply this method to solve the coupled mKdV equations and derive a set of rational solutions. This method is also available for seeking their solutions to other NLPDEs. It shows that the rational function method is more universal and powerful than the auxiliary Riccati equation method.

Approximate Analytic Solutions of Time-Fractional Hirota-Satsuma Coupled KdV Equation and Coupled MKdV Equation

By introducing the fractional derivative in the sense of Caputo and combining the pretreatment technique to deal with long nonlinear items, the generalized two-dimensional differential transform method is proposed for solving the time-fractional Hirota-Satsuma coupled KdV equation and coupled MKdV equation. The presented method is a numerical method based on the generalized Taylor series expansion which constructs an analytical solution in the form of a polynomial. The numerical results show that the generalized two-dimensional differential transform method is very effective for the fractional coupled equations.