Bifurcation, Traveling Wave Solutions, and Stability Analysis of the Fractional Generalized Hirota–Satsuma Coupled KdV Equations

In this paper, the bifurcation, phase portraits, traveling wave solutions, and stability analysis of the fractional generalized Hirota–Satsuma coupled KdV equations are investigated by utilizing the bifurcation theory. Firstly, the fractional generalized Hirota–Satsuma coupled KdV equations are transformed into two-dimensional Hamiltonian system by traveling wave transformation and the bifurcation theory. Then, the traveling wave solutions of the fractional generalized Hirota–Satsuma coupled KdV equations corresponding to phase orbits are easily obtained by applying the method of planar dynamical systems; these solutions include not only the bell solitary wave solutions, kink solitary wave solutions, anti-kink solitary wave solutions, and periodic wave solutions but also Jacobian elliptic function solutions. Finally, the stability criteria of the generalized Hirota–Satsuma coupled KdV equations are given.

SOLITARY WAVE SOLUTIONS OF SOME CONFORMABLE TIME-FRACTIONAL COUPLED SYSTEMS VIA AN ANALYTIC APPROACH

The main purpose of this research is to inquire the new solitary wave solution of the coupled time-fractional models to validate the influence and proficiency of the planned variational iteration method (VIM) using conformable derivative definition. Applications to four demanding nonlinear problems like Hirota-Satsuma coupled KdV equations, modified Boussinesq (MB) equation, approximate long wave (ALW) equation and Drinfeld-Sokolov-Wilson (DSW) equation demonstrate the efficiency and the robustness of the method. An analysis of the consequences with effects of relevant parameters and comparison with the exact solution presented with the help of graphs tables and gives the further understanding of numerical results by others. The convergence of the method is illustrated numerical and their physical significance is discussed

Lie symmetry reductions and conservation laws for fractional order coupled KdV system

Lie symmetry analysis is achieved on a new system of coupled KdV equations with fractional order, which arise in the analysis of several problems in theoretical physics and numerous scientific phenomena. We determine the reduced fractional ODE system corresponding to the governing factional PDE system.In addition, we develop the conservation laws for the system of fractional order coupled KdV equations.

Nonlinear Stability of the Periodic Traveling Wave Solution for a Class of Coupled KdV Equations

In this paper, by applying the Jacobian ellipse function method, we obtain a group of periodic traveling wave solution of coupled KdV equations. Furthermore, by the implicit function theorem, the relation between some wave velocity and the solution’s period is researched. Lastly, we show that this type of solution is orbitally stable by periodic perturbations of the same wavelength as the underlying wave.

Fundamental Solutions for the Coupled KdV System and Its Stability

In this paper, we establish exact solutions for the non-linear coupled KdV equations. The exp-function method is used to construct the solitary travelling wave solutions for these equations. The numerical adaptive moving mesh PDEs (MMPDEs) method is also implemented in order to solve the proposed coupled KdV equations. The achieved results may be applicable to some plasma environments, such as ionosphere plasma. Some numerical simulations compared with the exact solutions are provided to illustrate the validity of the proposed methods. Furthermore, the modulational instability is analyzed based on the standard linear-stability analysis. The depiction of the techniques are straight, powerful, robust and can be applied to other nonlinear systems of partial differential equations.

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.