Abstract
We obtain exact solutions to the fractional generalized higher order Korteweg-de Vries (KdV) equation using the complex method. It has showed that the applied method is very useful and is practically well suited for the nonlinear differential equations, those arising in mathematical physics.
1. Introduction
Nonlinear fractional differential equations (NFDEs) are universally applied in signal processing, electrical networks, acoustics, fluid dynamics, biology, chemistry, physics, etc. For example, the singular behaviours [1–9] and impulsive phenomena [10–19] often exhibit some blow-up properties [20–25] which occur in a lot of complex physical processes. NFDEs have been attracted extensive attention and have been widely investigated [26–38]. Exact solutions of NFDEs play an important role in the study of mathematical physics phenomena. Therefore, seeking exact solutions of NFDEs is an interesting and significant subject.
The fractional generalized higher order KdV equation is a useful model. Applying the generalized exp(-)-expansion method, Lu et al. [39] obtained exact solutions of this equation. In this article, we would like to utilize the complex method [40–43] to seek exact solutions to the fractional generalized higher order KdV equation.
2. Preliminaries
Let , be a continuous function and denote a constant discrete span. Define the operator as follows:then the fractional difference of of order can be expressed aswhere , and its fractional derivative of order can be expressed asThe above is expressed asFurther, Jumarie’s modified Riemann-Liouville derivative [44, 45] is given bythen its related NFDE is given byLet , , , , andthen the degree of is denoted by . We define the differential polynomial asin which is a finite index set and are constants. The degree of can be denoted by .
The ordinary differential equation (ODE) is given bywhere are constants, .
Suppose that the meromorphic solutions of (9) have at least one pole. Let and insert the Laurent seriesinto (9); if it is determined different Laurent singular parts:then (9) is said to satisfy the weak condition.
Give two complex numbers such that , let be the discrete subset , and L is isomorphic to . Let the discriminant and
A meromorphic function with double periods , which satisfies the equationin which , , and , is called the Weierstrass elliptic function.
If a meromorphic function is a rational function of , an elliptic function, or a rational function of , then we say that belongs to the class .
In 2009, Eremenko et al. [46] considered the following -order Briot-Bouquet equation (BBEq):where and are constant coefficient polynomials. For the -order BBEq, we have the lemma as follows.
Lemma 1 (see [41, 47, 48]). Let , and , and the -order BBEqsatisfies weak condition, then the meromorphic solutions belong to the class . Suppose that for some values of parameters such solutions exist, then other meromorphic solutions should form one-parametric family , . Then each elliptic solution with a pole at can be expressed asin which are determined by (10), , and .
Each rational function solution can be expressed asand it has distinct poles of multiplicity .
Each simply periodic solution is a rational function of and can be expressed asand it has distinct poles of multiplicity .
Lemma 2 (see [48, 49]). Weierstrass elliptic functions have the following addition formula:When , it can be degenerated to rational functions according toWhen , it can also be degenerated to simple periodic functions according to
3. Main Results
The fractional generalized higher order KdV equation [39] is given as
Substituting traveling wave transforminto (22), we getIntegrating (24) yieldswhere and are constants and is the integral constant.
Theorem 3. If , then the meromorphic solutions of (25) have the following forms.
The rational function solutionswhere , , and .
The simply periodic solutionswhere , , and .
The elliptic function solutionswhere , , and .
Proof. Substituting (10) into (25) we have , , , , , , , and is an arbitrary constant.
Therefore, (25) satisfies the weak condition. In the following, we will show the meromorphic solutions of (25).
By (17), we infer that the indeterminant rational solutions of (25) arewith pole at .
Inserting into (25), we havethen we get , , and .
So we can determine thatwhere and .
Therefore the rational solutions of (25) arewhere , , and .
Let . To derive simply periodic solutions, we substitute into (25) to yieldSubstitutinginto (33), we obtain thatwhere and .
Substituting into (35), we achieve simply periodic solutions to (25) with pole at where and .
Thurs, simply periodic solutions of (25) arewhere , , and .
From (16) of Lemma 1, the elliptic solutions of (25) are expressed aswith pole at .
Putting into (25), we getwhere and .
So, the elliptic solutions of (25) arewhere .
We can apply the addition formula to rewrite it aswhere , , and .
Substitute traveling wave transform into the meromorphic solutions of (25) to get traveling wave exact solutions to the fractional generalized higher order KdV equation. So we obtain Theorem 4 as follows.
Theorem 4. If , then traveling wave solutions of (25) have the following forms.
The rational function solutionswhere , , and .
The simply periodic solutionswhere , , and .
The elliptic function solutionswhere , , and .
4. Conclusions
In this note, we have used the complex method to construct exact solutions to the mentioned NFDE. Although we do not show that the meromorphic solutions of the fractional generalized higher order KdV equation belong to the class , we can still obtain the meromorphic solutions to this NFDE and then get its traveling wave exact solutions. The results demonstrate that the applied method is direct and efficient method, which allows us to do tedious and complicated algebraic calculation. We can utilize these ideas to other NFDEs.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the NSF of China (11701111 and 11271090); the NSF of Guangdong Province (2016A030310257); Guangdong Universities (Basic and Applied Research) Major Project (2017KZDXM038); Guangzhou City Social Science Federation “Yangcheng Young Scholars” Project (18QNXR35).