New exact and numerical solutions for the KdV system arising in physical applications

Abstract The Kortweg–de Vries (KdV) equation is more appropriate to simulate some natural phenomena and gives more accurate results for some physical systems such as the movement of water waves. In this work, novel analytical traveling wave solutions for a nonlinear KdV system are explored using the sech method. The exact solutions are presented in the form of hyperbolic functions. These solutions show the propagation of water waves on the surface. We also implement the numerical adaptive moving technique (MMPDEs) to construct the relevant system’s numerical solutions. A detailed comparison between the numerical and analytical solutions is also presented to confirm the accuracy of the numerical technique. We present some new 2D and 3D figures to illustrate the behaviour of the exact and numerical solutions. The obtained numerical solutions verify the accuracy of the considered methods qualitatively and quantitatively. The stability of the obtained solutions is investigated using the Hamiltonian system. The achieved results can be applied for some new observations in the ocean, coastal water, macroscopic phenomena, and processes. The proposed techniques are potent tools to solve many other non-linear partial differential equations in applied mathematics.


Introduction
The Kortweg-de Vries (KdV) equation is generally recognized as a model for representing weakly nonlinear long waves in various branches of fluid mechanics, physics and applied sciences. This equation depicts the development of waves under the competing influences of weak nonlinearity and weak dispersion. Moreover, this model is encountered in many unrelated phenomena such as aerodynamics, continuum mechanics, and fluids as a model for solitons, shock wave formation, turbulence, and mass transport. These applications are widely formed from some common physical models which are described by the KdV system by considering a characteristic limit of the physical model. Ultimately, integrable equations have a significant and considerable effect in hydrodynamics and nonlinear optics since they emerge in several vital approximations to the underlying problems. The nonlinear Schr€ odinger equation is used to investigate the rogue waves in deep water, while the KdV model depicts the behaviour in shallow water. Furthermore, the KdV system deals with the ion-acoustic solutions in plasma physics (Gao & Tian, 2001).
A third-order KdV equation (Dubard, Gaillard, Klein, & Matveev, 2010) reads as u t þ 6uu x þ u xxx ¼ 0: (1) Here, u t represents the time evolution, uu x is nonlinear and u xxx is a dispersion. Equation (1) is specifically considered as an extremely exquisite model of a nonlinear solvable equation whose solutions can be exactly and accurately determined. Furthermore, the analysis of the KdV equation has mainly attracted a massive number of active scientists since it is a tremendously interesting topic.
The principal objective of this work is to apply the sech technique (Seadawy, Lu, & Yue, 2017) and the MMPDEs method on the considered problem to extract its exact and numerical solutions, respectively. The stability of the obtained solutions is examined using the Hamiltonian system (Russell, Huang, & Ren, 1994). The obtained numerical solutions are successfully compared with the analytical solution. We also present some 2D and 3D figures to show the behaviour of the obtained solutions. The obtained new solutions may be applicable to verify and confirm the wave observations and wave progress in plasma physics. Indeed, the proposed two methods can be also applied for further models arising in applied sciences.
This paper is organized as follows. In Section 2, we discuss the exact solution of system (2). Section 3 presents the numerical solution of system (2) and shows some relevant figures. Section 4 discusses the most important results that we achieve in this work. In Section 5, we conclude this article.

Exact solution
This part is devoted to discover the exact solutions of system (2). Applying the transformation where w is the wave speed, system (2) is converted into the following ODEs: We now integrate each equation in system (5) once with respect to n: Then, we equate the constant of integrals to zero to end up with Balancing u nn with u 2 appearing in the first equation of system (6) gives N ¼ 2: Similarly, we balance v nnn with u v n in the second equation of system (6) to have M ¼ 2: As a result, the solutions take the following forms: vðnÞ Substituting Eqs. (7) and (8) into system (6) and equating the whole coefficients of sech j n to zero give nonlinear algebraic equations. Obtaining the exact solutions of these equations leads to different cases explained as follows.
Case 1 Hence, the exact traveling wave solutions are given by Case 2 Therefore, the exact solutions are shown as follows: It is worth illustrating that the boundary conditions are originally initiated from the long behaviour of the exact solutions. It can be observed from Figure 2 (left) that the travelling wave solution u(x, t) approaches À0.5 at the boundaries of the domain while Figure 3 (left) demonstrates that G(x, t) tends to zero at the boundaries. The parameters are fixed by a ¼ 1:2, b ¼ 0:2, w ¼ 0:5 and x 0 ¼ À 12: Thus, it can be concluded that Uðx, tÞ ! À 0:5, and Gðx, tÞ ! 0, as x ! 61: In order to establish a numerical solution for a given PDE, we discretize the physical domain into subintervals. In other words, the proposed PDE is discretized on a uniform or non-uniform mesh. In a uniform mesh, the used domain is equally partitioned into subintervals so that Dx is fixed for the entire domain. The resultant error is commonly reduced by using small enough Dx: Consequently, this strategy is intensive. However, when we distribute the mesh so that the region where the solution has rapidly changed takes more points and fewer points where the solution is not changed, the resultant error is diminished. Hence, the employed approach plays a principal role in decreasing the value of error. For instance, the r-adaptive moving mesh methods randomly distribute the nodes and send more points to the regions with high error to reduce the error occurred in the solution. These methods produce reliable, appropriate, and acceptable results. Therefore, we apply the MMPDE to find the numerical results for the considered PDEs.

Numerical results
In this article, we utilize a vital method called the MMPDE (Huang & Russell, 1998) to find the numerical solution of the considered problem. This approach was developed in Huang and Russell (1998) to obtain the numerical solutions of onedimensional PDE while it was clearly described in Budd and Williams (2009) to solve multi-dimensional PDEs. The solution of the proposed PDE is determined by using novel meshes called gradient flow equations. The vital purpose of using such technique is to initiate these meshes and to reduce the resultant error in the results by sending more points to the regions with high errors. Another method named the Parabolic Monge-Ampere (PMA) (Alharbi & Naire, 2017) is employed to some equations to approximate the general solutions of multi-dimensional PDEs. The first approach is utilized here to discover the numerical solutions of system (2) which is rewritten as subject to Uðx L , tÞ ¼ À 0:5, Uðx R , tÞ ¼ À 0:5, and the initial conditions where x L ¼ a, x R ¼ b and T e is an appropriate time.
For the non-uniform mesh, we utilize the coordinate transform x ¼ xðg, tÞ : ½0, 1 ! ½x L , x R , where x 2 ½x L , x R and g 2 ½0, 1 present the space and computational coordinates, respectively. The discretisation of the spatial and temporal derivatives, using a chain rule, is given by Equation (20) can be also written as In order to establish xðg, tÞ, we utilize the following equation ( with the boundary conditions and the initial condition i ¼ 1, 2, 3, :::, N þ 1: Here, FðU, G, xÞ describes the mesh density function, s indicates a relaxation parameter and s ðI À k @ gg Þ is called the smoothing operator developed by Ceniceros and Hou (2001). The movement of the mesh is mainly controlled by the function FðU, G, xÞ: According to Alharbi (Ceniceros & Hou, 2001), the proposed technique gives appropriate results when a suitable function F is selected. Cook (2016) chose a modified density function given by where a 0 presents a non-negative constant related to the average quantity of ð x g Þ 2 and ðU 2 þ G 2 Þ, respectively. It is worth mentioning that the spatial derivative is semi-discretised while the temporal one is kept continuous. Achieving this, the KdV problem becomes a system of ðN þ 1Þ ODEs which can be simply solved by using line methods. The MATLAB ODE solver (ode15i) is utilized to integrate the resultant system numerically. We begin by separating the physical domain as follows: x i <x iþ1 , i ¼ 0, 1, 2, :::, N: The computational coordinates can be written as Moreover, the boundary mesh is fixed by x 0 ¼ x L and x Nþ1 ¼ x R , while the interior points are approximated by solving MMPDE7 (Eq. 22) subject to the ODE boundary conditions x t, 0 ¼ x t, N ¼ 0: Therefore, the discretisation of Eq. (21) is established by employing finite differences as follows: where

Results and discussion
Here, we discuss the acquired results presented in this study. The sech technique and the MMPDEs method have been worthily applied to construct vital solutions. It has been noted that new structures for the KdV system have been clearly reported and all of these solutions may be serviceable in fluid dynamics. However, the physical situations in which the KdV system emerge tend to be highly idealized because the assumption of constant coefficients. Figures 1-7 illustrate the profile pictures of the presented solutions for the KdV system. These solutions provide an excellent performance to distinguish the kinds of solitary solutions according to the physical parameters. Namely, the behaviour of the solutions being solitons, super-solitons, dissipative or periodic and so on, is an indication for the values of the physical parameters. The adaptive moving method is an appropriate technique for dealing with most partial differential equations. Specifically, this scheme is an effective tool for solving KdV system. This technique approximates the numerical results and reduces the error perfectly. The comparison between the exact and numerical solutions shows that the MMPDEs scheme works efficiently. Furthermore, the presented 2D graphs compare the achieved numerical results to the exact ones. As can be seen in the above figures, both solutions are nearly coincided and almost have the same behaviours. Therefore, the resultant error is very small and can be neglected. Finally, the proposed two methods can be applied to many other nonlinear models arising in new physics.

Conclusions
In this article, the sech method and the MMPDEs technique have been applied to extract new soliton solutions of the KdV system. Some novel hyperbolic solutions have been successfully presented. In particular, the exact solutions are presented in the form of the sech function. The derived solutions are stable inside the interval [0; 20]. The sech method is an effective and successful mathematical means providing excellent analytical solutions for the KdV equation. The MMPDEs method gives reliable approximations when it is applied to these analytical solutions. This can be easily seen in the presented figures where the numerical solutions are nearly the same as the exact solutions. The presented new solutions may be applicable to verify and confirm the wave observations and wave progress in plasma fluids. The graphical comparison shows that the error is very small. These two methods are not only powerful but also have the merit of being extensively applicable. We conclude that the proposed methods can be employed to deal with other PDEs in physics, applied science and engineering fields.