Two Different Classes of Wronskian Conditions to a (3 + 1)-Dimensional Generalized Shallow Water Equation
Based on the Hirota bilinear method and Wronskian technique, two different classes of sufficient conditions consisting of linear partial differential equations system are presented, which guarantee that the Wronskian determinant is a solution to the corresponding Hirota bilinear equation of a ()-dimensional generalized shallow water equation. Our results show that the nonlinear equation possesses rich and diverse exact solutions such as rational solutions, solitons, negatons, and positons.
Wronskian formulations are a common feature for soliton equations, and it is a powerful tool to construct exact solutions to the soliton equations [1–4]. The resulting technique has been applied to many soliton equations such as the MKdV, NLS, derivative NLS, sine-Gordon, and other equations [5–10]. Within Wronskian formulations, soliton solutions, rational solutions, positons, and negatons are usually expressed as some kind of logarithmic derivatives of Wronskian-type determinants [11–16].
The following (3+1)-dimensional generalized shallow water equation was investigated in different ways (see, e.g., [17, 18]). In , soliton-typed solutions for (1.1) were constructed by a generalized tanh algorithm with symbolic computation. In , the traveling wave solutions of (1.1) expressed by hyperbolic, trigonometric, and rational functions were established by the -expansion method, where satisfies a second order linear ordinary differential equation.
As we know, in the process of employing Wronskian technique, the main difficulty lies in looking for the linear differential conditions, which the functions in the Wronskian determinant should satisfy. Moreover, the differential conditions for the Wronskian determinant solutions of many soliton equations are not unique [5, 7, 10, 12]. In this paper, we will give two different classes of linear differential conditions for the Nth order Wronskian determinant solutions (simply, Wronskian conditions) of (1.4) based on the special structure of the Hirota bilinear form (1.4). Our results will show that (1.4) has diverse Wronskian determinant solutions under different linear differential conditions and further (1.2) will have diverse exact solutions such as rational solutions, solitons, negatons, and positons.
2. The First Class of Wronskian Conditions
In this section, we present the first class of linear differential conditions for the Wronskian determinant solutions of (1.4).
Theorem 2.1. Let a set of functions , satisfy the following linear differential conditions: where the coefficient matrix is an arbitrary real constant matrix (see [10, 12]), is an arbitrary nonzero constant, is an arbitrary positive integer, and denotes the th order derivative of with respect to . Then the Wronskian determinant defined by (2.1) solves (1.4).
The proof of Theorem 2.1 needs the following two useful known Lemmas.
Lemma 2.2. Set to be an N-dimensional column vector, and to be a real nonzero constant. Then one has where .
Proof of Theorem 2.1. Under the properties of the Wronskian determinant and the conditions (2.3) and (2.4), we can compute various derivatives of the Wronskian determinant with respect to the variables as follows: Therefore, we can now compute that Using Lemma 2.3, we can further obtain that This last equality is just the Plücker relation for determinants: where denotes an matrix, and are four N-dimensional column vectors. Therefore, we have shown that solve (1.4) under the conditions (2.3) and (2.4). Further, the corresponding solution to (1.2) is
Remarks 1. The condition (2.4) is a generalized linear differential condition which includes many different special cases.
For example, when , the condition (2.4) is reduced to When , the condition (2.4) is reduced to When , the condition (2.4) is reduced to
Using the linear differential conditions (2.3) and (2.4) as well as the transformation (1.3), we can compute many exact solutions of (1.2) such as rational solutions, solitons, negatons, and positons.
As an example, in the special case of and , the conditions (2.3) and (2.4) read
If we let the coefficient matrix of condition (2.15) has the following form (see [10–12, 16] for details), using the same method as that in , we can obtain N-soliton solutions of (1.2).
For example, when , we can compute two exact 1-soliton solutions for (1.2), with being a constant.
3. The Second Class of Wronskian Condition
In this section, we show another linear differential condition to the Wronskian determinant solutions of (1.4).
Proof. Under the properties of the Wronskian determinant and the condition (3.1), various derivatives of the Wronskian determinant with respect to the variables are obtained as follows:
Therefore, we can now compute that
then, substituting the above results into (1.4), we can further obtain that
From (3.5), it is easy to see that the above expression (3.4) is nothing but zero because they both satisfy the Plücker relation (2.10). Therefore, we have shown that also solve (1.4) under the condition (3.1).
The condition (3.1) has an exponential-type function solution: where and are free parameters and is an arbitrary natural number.
In particular, we can have the following Wronskian solutions of (1.2): where with and being free parameters.
4. Conclusions and Remarks
In summary, we have established two different kinds of linear differential conditions for the Wronskian determinant solutions of the (3+1)-dimensional generalized shallow water equation (1.1) or equivalently (1.2). Especially, the first Wronskian conditions are generalized linear differential conditions which include many different special cases. Our results show that the nonlinear equation (1.1) carry rich and diverse Wronskian determinant solutions.
This work was supported in part by the National Science Foundation of China (under Grant nos. 11172233, 11102156, and 11002110) and Northwestern Polytechnical University Foundation for Fundamental Research (no. GBKY1034).
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