The time-fractional mZK equation for gravity solitary waves and solutions using sech-tanh and radial basis function method

Abstract. In recent years, we know that gravity solitary waves have gradually become the research spots and aroused extensive attention; on the other hand, the fractional calculus have been applied to the biology, optics and other fields, and it also has attracted more and more attention. In the paper, by employing multi-scale analysis and perturbation methods, we derive a new modified Zakharov–Kuznetsov (mZK) equation to describe the propagation features of gravity solitary waves. Furthermore, based on semi-inverse and Agrawal methods, the integer-order mZK equation is converted into the time-fractional mZK equation. In the past, fractional calculus was rarely used in ocean and atmosphere studies. Now, the study on nonlinear fluctuations of the gravity solitary waves is a hot area of research by using fractional calculus. It has potential value for deep understanding of the real ocean–atmosphere. Furthermore, by virtue of the sech-tanh method, the analytical solution of the time-fractional mZK equation is obtained. Next, using the above analytical solution, a numerical solution of the time-fractional mZK equation is given by using radial basis function method. Finally, the effect of time-fractional order on the wave propagation is explained.


Introduction
Gravity solitary waves are important part of solitary waves and can cause the suddenly change of all kinds of medium and small scale local circulation.The emergence and development of solitary waves theory is a major event on the study of nonlinear partial integer-order equation, we seek the time-fractional mZK equation by using the semiinverse method [17] and Agrawal method [2,3].Further, we get the solution of timefractional mZK equation.So, the structure of the article is as follows.In Section 2, according to two layers fluid shallow water wave equation, we get a new (2+1)-dimensional integer-order mZK equation by using the multi-scale analysis and perturbation method.
Based on the new model, we acquire the time-fractional mZK equation in Section 3. In the next section, using the sech-tanh method, we get the analytical solution of timefractional mZK equation.According to the analytical solution, the numerical solution of time-fractional mZK equation is obtained by using radial basis function method, and the effect of time-fractional order on the wave propagation is explained in Section 5.In the end, a summary of the article is presented.

The derivation of integer-order mZK equation
The two layers fluid shallow water wave equation is described as follows: where φ = φ 0 + φ , φ = g * H = g * H 0 + g * H , g * = g(ρ − ρ )/ρ, ρ is the upper fluid density, and ρ is the lower fluid density.Importing the following characteristic quantities into Eq.( 1) we can gain the two layers of dimensionless fluid diving waves equation ( The boundary condition is in the following form: In previous studies, researchers have obtained new models on the scale of 1/2 or 1.However, as the disturbance increases, the nonlinearity of gravity solitary waves is also increasing.With the development of the theory and the nonlinearity increasing, the KdV equation is replaced by the mKdV equation to describe the propagation of gravity solitary waves.Furthermore, in the plane, the (1 + 1)-dimensional model cannot accurately describe the actual phenomenon.Thus, the high-order model -the ZK equation -was subsequently deduced.But, with the nonlinearity increasing continuously, what is the development trend of the new model?That is a question that cause our great attention.In this paper, we will solve this problem.In order to refine the gravity solitary waves model, we first generalize the scale of predecessors.And we research the new model on a scale of 1/4.
Introducing some conversion in the following form Eq. ( 3) can be rewritten as So, we can extend u, v, φ as follows: Substituting Eqs. ( 4) and (5) into Eq.( 2), we can get all level of approximation equations about .
Firstly, we use the first-order approximation equation Assume that the solution of Eq. ( 6) has the following form: Substituting Eq. ( 7) into Eq.( 6), we cannot get m(X, Y, T ), where m(X, Y, T ) is the amplitude of gravity solitary waves.Next, we describe the second-order approximation equation as follows: and we assume that the solution of Eq. ( 8) has the following form: Substituting Eq. ( 9) into Eq.( 8), we cannot gain m(X, Y, T ).Keep on using the thirdorder approximation equation and assume that it has the solution as follows: Similarly, m(X, Y, T ) can be not gained.Thus, we must continue to solve the highorder problem, and the fourth-order approximation equation is similar to the third-order approximation equation.So, we use the fifth-order approximation equation According to Eq. ( 10), assuming U − c = 0 and eliminating u 4 , φ 4 , we can gain the equation about v 4 as follows: Multiplying Eq. ( 11) by v 0 , integrating Eq. ( 11) with respect to y from 0 to H and using the identical equation we can have the following equation: where Remark.Based on the above derivation, we get Eq.( 12).Equation ( 12) is a new model.When a 4 = 0, it can be reduced to mKdV equation.When m 2 m X → mm X , we call it ZK equation.So, Eq. ( 12) is mZK equation.Generally speaking, the ZK equation governs the behaviour of nonlinearity gravity solitary waves.But the nonlinearity of ZK equation is rather weak, and it has more external disturbance.While mZK equation has the strong nonlinearity, it is a (2 + 1)-dimensional model, which is more practical.Thus, the mZK equation is more suitable than ZK equation to describe the propagation of gravity solitary waves.

The derivation of time-fractional mZK equation
In this section, we seek for the time-fractional mZK equation.We start by introducing the notion of fractional derivative.Let g is a continuous function on the interval [a, b], and β is a positive real number.
Definition 1. (See [30,48,51].)The left Riemann-Liouville fractional derivative a D β t of a function g(t) is defined as Definition 2. (See [30,48,51].)The right Riemann-Liouville fractional derivative t D β b of a function g(t) is defined as [30,48,51].)The well-known Riesz fractional derivative R a D β t of a function g(t) is defined as According to Section 2, we get the following mZK equation: where [X, Y ] ⊆ R 2 is the space coordinate, T × [0, T * ] is the time coordinate.Letting m = n X , here n(X, Y, T ) is a potential function.Then the above equation can be written as Next, we use the semi-inverse method to seek the Lagrangian of mZK equation.The functional of Eq. ( 13) has the following form: ) are unknown constants, we can get it in the later calculation.Making use of integration by parts and assuming Taking advantage of variation method, integration by parts and the formula the Euler equation can be obtained as follows: Making a comparison between Eqs. ( 14) and ( 13), we can know that Thus, we can get the integer-order Lagrangian form of mZK equation as follows: At this time, we can gain the following Lagrangian form of time-fractional mZK equation by using Definition 1: So, the functional of mZK equation is as follows: https://www.mii.vu.lt/NAAccording to Agrawal method [2,3], the variation of Eq. ( 16) is in the following form: Using Definition 2 and the transformation Eq. ( 17) can be indicated as follows:

Relying on the above assumption n
When δJ = 0, we can gain the optimization of function J (n) by using variation principle.So, the Euler-Lagrange equation for the time-fractional mZK equation is as follows: Thus, the following equation can be obtained: Letting n X (X, Y, T ) = u(X, Y, T ), we obtain According to Definition 3, Eq. ( 21) can be rewritten as 22) is new model, which is obtained by the timefractional calculus.So, we call it time-fractional mZK equation.

The analytical solution of time-fractional mZK equation
In Section 3, we get the time-fractional mZK equation.In this section, based on sech-tanh method, we seek the explicit solution of time-fractional mZK equation.By applying the fractional complex transformations , where a, l are unknown constants, which are ensured in the later calculation, Eq. ( 22) can be written in the following form: Assume that the solution of Eq. ( 22) can be expressed by a polynomial in λ and will take the following form: We know that n = 1 by balancing the highest-order derivative term and nonlinear term of Eq. ( 22).Then we have Thus, we can get According to the above equations, we obtain Gathering all of coefficients of same index about λ in the above equation and making them to 0, respectively, we can have Solving the above equation, we can gain So, m(X, Y, T ) can be written as Expression ( 23) is an analytical solution of time-fractional mZK equation.

The numerical solution of time-fractional mZK equation
In the previous section, we get an analytical solution of time-fractional mZK equation by using sech-tanh method.However, analytical solution sometimes cannot be used to study the property of equation.So, according to the analytical solution, we seek the numerical solution of time-fractional mZK equation by using radial basis function method.According to Eq. ( 23), we can get the initial condition of Eq. ( 22) as follows: The boundary conditions of Eq. ( 22) are as follows: and Assuming that and using we can get a finite difference approximation where the first-order derivative is defined by So, according to Eq. ( 26), we obtain That means that Assuming Eq. ( 27) can be rewritten as https://www.mii.vu.lt/NASubstituting Eq. ( 28) into Eq.( 22), we get two cases in the following form: So, Eq. ( 22) is converted into the following equation: According to radial basis function approximation, the function m(X, Y, T ) can be written as a linear combination of N radial functions as follows: where τ j , j = 1, 2, . . ., N , are unknown coefficients, which can be calculated, N is the numbers of data points, ρ(X, X j ) = ρ(r j ), r j = X − X j , j = 1, 2, . . ., N , are the Euclidean norm.Based on N equations, which are caused by Eq. ( 30) at N points, and the regularization conditions, we can have additional two equations When Eqs. (30) and (31) are combined together, we can get the matrix equation T , and P = (a ij ) is a (N + 3) × (N + 3) matrix, which is defined by  According to Eqs. ( 24), ( 25) and ( 31) and substituting Eq. ( 30) into Eq.( 29), we can obtain the discretization equation where Q is a (N + 3) × (N + 3) matrix, which is defined by Here J represents an operate, which is defined by Based on Eq. ( 32), we can calculate the undetermined coefficient (τ ) n+1 .Substituting the (τ ) n+1 into Eq.( 30), we can obtain a numerical solution of time-fractional mZK equation.
Next, we discuss the influence of fractional order (β) and time (T ) on the solitary waves solution of mZK equation.According to the numerical solution of time-fractional mZK equation and changing β and T , we can study the propagation property of gravity solitary waves, and waves forms are represented in different figures.The effect of timefractional order β on the soliton shapes has been studied in Figs.1(a)-1(d).These figures show that the fractional order of differentiation has a small effect only on the position of the turning points.And with the march of time, the distances among the turning points become more and more wide.In other words, the time-fractional order do not change the shape of gravity solitary waves and only has a small effect on the position of these waves.It is judged that gravity solitary waves maintain its shape during the propagation process.And the main evolving features of gravity solitary waves can be not changed.

Conclusion
In this paper, we obtain a new (2 + 1)-dimensional mZK equation by using the multiscale analysis and perturbation method.Based on the semi-inverse method and Agrawal method, we gain the time-fractional mZK equation.In the end, in order to study the properties of gravity solitary waves, we seek the solution of time-fractional mZK equation.The focus of the article are as follows.
1. We obtain a new (2 + 1)-dimensional integer-order mZK equation.It can accurately describe the gravity solitary waves.According to semi-inverse and Agrawal methods, we get a time-fractional mZK equation.It is the generalization of integerorder mZK equations.2. According to the time-fractional mZK equation, we get an analytical solution by using sech-tanh method.According to the analytical solution, a numerical solution can be obtained by using radial basis function method.Then, based on the solution of time-fractional mZK equation, we discuss the effect of fractional order on wave propagation.

Figure 1 .
Figure 1.The (1 + 1)-dimensional plot of numerical solution for gravity solitary waves for different values of β.

Figure 2 .
Figure 2. The (2 + 1)-dimensional plot of numerical solution for gravity solitary waves for different values of β ant T .