Multiple-soliton and periodic solutions to space–time fractional Whitham–Broer–Kaup equations

In this paper, the space–time fractional Whitham–Broer–Kaup equations are investigated. By means of new fractional scaling transformations, the fractional nonlinear system of different time and space orders is transformed to the integer one. The multiple solitary solutions and periodic solutions are obtained, respectively. All those solutions are given exactly by introducing new scaling transformations, which makes our study unique and different from most existing studies. It is expected that exact solutions for nonlinear wave system of fractional order can be handled in the similar way.


Introduction
Recently, fractional-order differential equations [1] have been received much attention because they have been argued to be more appropriate than traditional integerorder ones to describe nonlinear phenomena in real worlds [2][3][4][5]. Many different analytical and numerical methods have been employed to capture solutions of fractional differential equations, such as the homotopy analysis method [6], the reproducing kernel Hilbert space method [7], the iterative Adomian decomposition method [8], the Hermite wavelet method [9], the implicit collocation method [10], the finite element method [11], the B-spline wavelet operational method [12], the generalized fractional-order Legendre function method [13], the boundary element method [14].
On the other hand, many efforts have been done towards finding solutions of shallow water equations either analytically or numerically. For example, Garner et al. [15] presented multiple solitary solutions to the well-known Korteweg-de Vries equation. Date et al. [16] gave quasi-periodic solutions to the orthogonal Kadomtsev-Petviashvili equations. Wang [17] obtained solitary solutions for the variant Boussinesq equations. Fan and Zhang [18] exhibited exact solutions of the Whitham-Broer-Kaup equations in shallow water. Naranmandula and Chen [19] further showed multia e-mail: cjf@imut.edu.cn (corresponding author) ple soliton solutions of the Whitham-Broer-Kaup equations. Wang et al. [20] captured a class of novel exact solutions of periodic, solitary, and kink types of the Whitham-Broer-Kaup equations.
In most aforementioned studies, researchers aimed to obtain solutions by solving the fractional differential equations directly. However, Guo and Zhou [24] introduced a fractional scaling transformation to convert the fractional differential equations into the integral ones. As a result, solution techniques for regular wave equations can be used to solve fractional ones straightforwardly. In this paper, the space-time fractional Whitham-Broer-Kaup equations are to be examined. The multiple soliton solutions and travelling solutions will be captured. By means of the fractional scaling transformations, the fractional differential equations are transformed to integer ones. Then, the homogeneous balance method [19] and the first time integration method [24] are employed to give exact solutions to the fractional nonlinear systems, respectively.

Fractional calculus and governing equations
The modified Riemann-Liouville fractional calculus [31] is defined as where μ is a decimal, and Γ denotes the gamma function.
According to the definitions of the fractional calculus (1), the following properties are held: where F (x) and G(x) are arbitrary functions, γ > −1 is a constant. The governing equations describe the space-time fractional Whitham-Broer-Kaup equations are written by where u and h are, respectively, the horizontal velocity and the height deviating from the equilibrium position of the liquid, c 1 and c 2 are the coefficients associated with the different diffusion capabilities of the medium. α and β are the fractional numbers that satisfy 0 < α < 1 and 0 < 3β < 1.

Multiple solitary solutions
By means of the following fractional scaling transformations: the governing Eqs. (3) and (4) are converted to The homogeneous balance method [19] is used to give multiple solitary solutions to Eqs. (6) and (7). With this technique, we define where f (Ω) and s(Ω) are intermediate transition functions, the prime denotes the derivation to Ω, and Ω is a function of X, T . Substituting the transformations (8) into Eqs. (6) and (7), we obtain bf To seek solutions of Eqs. (9) and (10), it is required that the coefficients of Ω and its derivatives for certain orders have to be equal to zero. Particularly, we are able to obtain the following particular solutions, by enforcing the coefficients of Ω 3 X for Eq. (9) and Ω 4 X for Eq. (10) to be equal to zero, as where c 0 = c 2 + c 1 + c 2 2 . From Eq. (11), we are readily to obtain the following correlations: Substituting the correlations (12) into Eqs. (9) and (10), and then enforcing the coefficients of f , f and f to be equal to zero, we are able to obtain the following core equation: whose solution is in which N denotes the quantities of solitary waves, k j is the wave number of the kth solitary wave. Here, b = 0 is taken so that the consistence of the coefficients is satisfied. We, therefore, obtain the solutions of Eqs. (3) and (4), by taking account of the scaling transformations denoted in (5), as where

Periodic solutions
To obtain the travelling wave solution of fractional order, we introduce the following scaling transformations: Substituting the scaling transformations (17) into Eqs. (3) and (4), we obtain Integrating Eqs. (18) and (19) with η once time, we obtain where P 1 and P 2 are integral constants. The travelling solutions of Eqs. (18) and (19) will be determined using the first integration method, which is briefly depicted below and can be found in Refs. [24] for more details.
For the caseω 2 + 2P 1 < 0, its solution is given as Here, h(x, t) can be obtained using Eq. (20) and travelling transformation (17), which is not shown here.
For other values of m, solution is obtained via similar procedures. For example, when m = 2, we finally obtain the Riccati-type equation of the form X = ± 1 2 c 1 + c 2 2 −X 2 + 2ω 0X + 2P 1 , (36) whose solution is the exactly same as that of the case m = 1.

Conclusion
The space-time fractional Whitham-Broer-Kaup equations have been investigated. By means of fractional scaling transformations, the fractional nonlinear system is transformed to the integer one. The multiple solitary solutions and periodic solutions are obtained respectively. In summary, the novel aspects of the paper are: • The Whitham-Broer-Kaup equations of different space and time fractional orders are studied. • The novel scaling transforms are introduced.
• The exact multiple solitons solutions are obtained.
• The exact periodic solutions are presented.