Abstract
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.
Highlights
By means of new fractional scaling transformations, the fractional nonlinear system of different time and space orders is transformed to the integer one
It is expected that exact solutions for nonlinear wave system of fractional order can be handled in the similar way
Fractional-order differential equations[1] have been received much attention because they have been argued to be more appropriate than traditional integer-order ones to describe nonlinear phenomena in real worlds [2–5]
Summary
Fractional-order differential equations[1] have been received much attention because they have been argued to be more appropriate than traditional integer-order ones to describe nonlinear phenomena in real worlds [2–5]. Wang et al.[20] captured a class of novel exact solutions of periodic, solitary, and kink types of the Whitham-Broer-Kaup equations. Substituting the correlations (12) into Eqs.(9) and (10), and enforcing the coefficients of f ′′′, f ′′ and f ′ to be equal to zero, we are able to obtain the following core equation. We obtain the solutions of Eqs.(3) and (4), by taking account of the scaling transformations denoted in (5), as. To obtain the travelling wave solution of fractional order, we introduce the following scaling transformations η xβ Γ(1 + β ). The solution of Eq(31) is obtained in different cases, after taking account of Eq(32) and travelling variables (17), as shown below.
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