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 3 1 -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.


Introduction
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 gives the corresponding Hirota bilinear equation of 1.2 where D x , D y , D z , and D t are the Hirota operators 19 .
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.

The First Class of Wronskian Conditions
The Nth order Wronskian determinant was introduced firstly by Freeman and Nimmo 1, 20 : where Solutions determined by f | N − 1| to 1.4 are called Wronskian determinant solutions.
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 φ i φ i x, y, z, t , 1 ≤ i ≤ N satisfy the following linear differential conditions:

2.4
where the coefficient matrix A λ ij 1≤i,j≤N is an arbitrary real constant matrix (see [10,12]), n is an arbitrary nonzero constant, m is an arbitrary positive integer, and φ i, mx denotes the mth order derivative of φ i with respect to x.Then the Wronskian determinant f | N − 1| defined by 2.1 solves 1.4 .
The proof of Theorem 2.1 needs the following two useful known Lemmas.Lemma 2.2.Set a j , j 1, 2, . . ., N to be an N-dimensional column vector, and b j , j 1, 2, . . ., N to be a real nonzero constant.Then one has

4 ISRN Mathematical Analysis
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 f | N − 1| with respect to the variables x, y, z, t as follows:

2.7
Therefore, we can now compute that

2.8
Using Lemma 2.3, we can further obtain that

2.9
This last equality is just the Pl ücker relation for determinants:

2.14
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 m 2 and n 1, the conditions 2.3 and 2.4 read

2.16
If we let the coefficient matrix A λ ij 1≤i,j≤N of condition 2.15 has the following form see 10-12, 16 for details , λ ii / λ jj , i / j , 2.17 using the same method as that in 16 , we can obtain N-soliton solutions of 1.2 .For example, when λ 11 > 0, λ ii 0, i 2, . . ., N, we can compute two exact 1-soliton solutions for 1.2 ,

2.19
with γ 1 being a constant.

The Second Class of Wronskian Condition
In this section, we show another linear differential condition to the Wronskian determinant solutions of 1.4 .
Theorem 3.1.Let a group of functions φ i φ i x, y, z, t , 1 ≤ i ≤ N satisfy the following linear differential condition: where k is an arbitrary nonzero constant.Then the Wronskian determinant f | N − 1| defined by 2.1 solves 1.4 .
Proof.Under the properties of the Wronskian determinant and the condition 3.1 , various derivatives of the Wronskian determinant f | N − 1| with respect to the variables x, y, z, t are obtained as follows:

3.2
Therefore, we can now compute that then, substituting the above results into 1.4 , we can further obtain that where

3.5
From 3.5 , it is easy to see that the above expression 3.

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.
u xxxy − 3u xx u y − 3u x u xy u yt − u xz 0 1.1 was investigated in different ways see, e.g., 17, 18 .In 17 , soliton-typed solutions for 1.1 were constructed by a generalized tanh algorithm with symbolic computation.In 18 , ISRN Mathematical Analysis the traveling wave solutions of 1.1 expressed by hyperbolic, trigonometric, and rational functions were established by the G /G-expansion method, where G G ξ satisfies a second order linear ordinary differential equation.Under a scale transformation x → −x, 1.1 is reduced equivalently to u xxxy 3u xx u y 3u x u xy − u yt − u xz 0 4 is nothing but zero because they both satisfy the Pl ücker relation 2.10 .Therefore, we have shown that f | N − 1| also solve 1.4 under the condition 3.1 .wheredij and l ij are free parameters and p is an arbitrary natural number.In particular, we can have the following Wronskian solutions of 1.2 : i t , i 1, . . ., N, 3.8 with l i and w i being free parameters.