Abstract

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.

1. 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 𝑢𝑥𝑥𝑥𝑦−3𝑢𝑥𝑥𝑢𝑦−3𝑢𝑥𝑢𝑥𝑦+𝑢𝑦𝑡−𝑢𝑥𝑧=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], the traveling wave solutions of (1.1) expressed by hyperbolic, trigonometric, and rational functions were established by the ğºî…ž/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 𝑢𝑥𝑥𝑥𝑦+3𝑢𝑥𝑥𝑢𝑦+3𝑢𝑥𝑢𝑥𝑦−𝑢𝑦𝑡−𝑢𝑥𝑧=0,(1.2) and the Cole-Hopf transformation 𝑢=2(ln𝑓)𝑥(1.3) gives the corresponding Hirota bilinear equation of (1.2) 𝐷3𝑥𝐷𝑦−𝐷𝑦𝐷𝑡−𝐷𝑥𝐷𝑧=𝑓𝑓⋅𝑓𝑥𝑥𝑥𝑦−𝑓𝑦𝑡−𝑓𝑥𝑧𝑓−𝑓𝑥𝑥𝑥𝑓𝑦−3𝑓𝑥𝑥𝑦𝑓𝑥+𝑓𝑦𝑓𝑡+𝑓𝑥𝑓𝑧+3𝑓𝑥𝑥𝑓𝑥𝑦=0,(1.4) where 𝐷𝑥,𝐷𝑦,𝐷𝑧, and 𝐷𝑡 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.

2. The First Class of Wronskian Conditions

The Nth order Wronskian determinant was introduced firstly by Freeman and Nimmo [1, 20]: 𝑊𝜙1,𝜙2,…,𝜙𝑁=||||||=||||||||||||𝜙𝑁−11(0)𝜙1(1)…𝜙1(𝑁−1)𝜙2(0)𝜙2(1)…𝜙2(𝑁−1)𝜙⋮⋮⋱⋮𝑁(0)𝜙𝑁(1)…𝜙𝑁(𝑁−1)||||||||||||,𝑁≥1,(2.1) where 𝜙𝑖(𝑗)=𝜕𝑗𝜕𝑥𝑗𝜙𝑖,0≤𝑗≤𝑁−1,1≤𝑖≤𝑁.(2.2) Solutions determined by î„žğ‘“=|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 𝜙𝑖=𝜙𝑖(𝑥,𝑦,𝑧,𝑡), 1≤𝑖≤𝑁 satisfy the following linear differential conditions: 𝜙𝑖,𝑥𝑥=𝑁𝑗=1𝜆𝑖𝑗𝜙𝑗,𝜙(2.3)𝑖,𝑦=𝑛𝜙𝑖,𝑥,𝜙𝑖,𝑡=−𝜙𝑖,(𝑚𝑥),𝜙𝑖,𝑧=4𝑛𝜙𝑖,𝑥𝑥𝑥+𝑛𝜙𝑖,(𝑚𝑥),(2.4) where the coefficient matrix 𝐴=(𝜆𝑖𝑗)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 𝜙𝑖,(𝑚𝑥) denotes the 𝑚th order derivative of 𝜙𝑖 with respect to 𝑥. Then the Wronskian determinant î„žğ‘“=|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 ğ‘Žğ‘—,𝑗=1,2,…,𝑁 to be an N-dimensional column vector, and 𝑏𝑗,𝑗=1,2,…,𝑁 to be a real nonzero constant. Then one has 𝑁𝑖=1𝑏𝑖||ğ‘Ž1,ğ‘Ž2,…,ğ‘Žğ‘||=𝑁𝑗=1||ğ‘Ž1,ğ‘Ž2,…,ğ‘ğ‘Žğ‘—,…,ğ‘Žğ‘||,(2.5) where ğ‘ğ‘Žğ‘—=(𝑏1ğ‘Ž1𝑗,𝑏2ğ‘Ž2𝑗,…,ğ‘ğ‘ğ‘Žğ‘ğ‘—)𝑇.

Lemma 2.3 (see [11]). Under the condition (2.3) and Lemma 2.2, the following equalities hold: ||||||𝑁−1𝑁𝑖=1𝜆𝑖𝑖𝑁𝑖=1𝜆𝑖𝑖||||||=𝑁−1𝑁𝑖=1𝜆𝑖𝑖||||||𝑁−12=−||||||+||||||𝑁−3,𝑁−1,𝑁𝑁−2,𝑁+12=||||||||||||−||||||||||||−||||||+||||||.𝑁−1𝑁−5,𝑁−3,𝑁−2,𝑁−1,𝑁𝑁−4,𝑁−2,𝑁−1,𝑁+1+2𝑁−3,𝑁,𝑁+1𝑁−3,𝑁−1,𝑁+2𝑁−2,𝑁+3(2.6)

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 î„žğ‘“=|𝑁−1| with respect to the variables 𝑥,𝑦,𝑧,𝑡 as follows: 𝑓𝑥=||||||𝑁−2,𝑁,𝑓𝑥𝑥=||||||+||||||,𝑓𝑁−3,𝑁−1,𝑁𝑁−2,𝑁+1𝑥𝑥𝑥=||||||||||||+||||||,𝑓𝑁−4,𝑁−2,𝑁−1,𝑁+2𝑁−3,𝑁−1,𝑁+1𝑁−2,𝑁+2𝑦||||||=𝑛𝑁−2,𝑁,𝑓𝑥𝑦||||||+||||||,𝑓=𝑛𝑁−3,𝑁−1,𝑁𝑁−2,𝑁+1𝑥𝑥𝑦||||||||||||+||||||,𝑓=𝑛𝑁−4,𝑁−2,𝑁−1,𝑁+2𝑁−3,𝑁−1,𝑁+1𝑁−2,𝑁+2𝑥𝑥𝑥𝑦||||||||||||||||||||||||+||||||,𝑓=𝑛𝑁−5,𝑁−3,𝑁−2,𝑁−1,𝑁+3𝑁−4,𝑁−2,𝑁−1,𝑁+1+2𝑁−3,𝑁,𝑁+1+3𝑁−3,𝑁−1,𝑁+2𝑁−2,𝑁+3𝑡𝜕=−𝑚𝑓𝜕𝑥𝑚𝜕=−𝑚||||||𝑁−1𝜕𝑥𝑚,𝑓𝑦𝑡𝜕=−𝜕𝜕𝑦𝑚||||||𝑁−1𝜕𝑥𝑚𝜕=−𝑛𝜕𝜕𝑥𝑚||||||𝑁−1𝜕𝑥𝑚,𝑓𝑧||||||−||||||+||||||𝜕=4𝑛𝑁−4,𝑁−2,𝑁−1,𝑁𝑁−3,𝑁−1,𝑁+1𝑁−2,𝑁+2+𝑛𝑚||||||𝑁−1𝜕𝑥𝑚,𝑓𝑥𝑧||||||−||||||+||||||𝜕=4𝑛𝑁−5,𝑁−3,𝑁−2,𝑁−1,𝑁𝑁−3,𝑁,𝑁+1𝑁−2,𝑁+3+𝑛𝜕𝜕𝑥𝑚||||||𝑁−1𝜕𝑥𝑚.(2.7) Therefore, we can now compute that 𝑓𝑥𝑥𝑥𝑦−𝑓𝑦𝑡−𝑓𝑥𝑧𝑓−||||||+||||||||||||+||||||−||||||||||||,=3𝑛𝑁−5,𝑁−3,𝑁−2,𝑁−1,𝑁𝑁−4,𝑁−2,𝑁−1,𝑁+1+2𝑁−3,𝑁,𝑁+1𝑁−3,𝑁−1,𝑁+2𝑁−2,𝑁+3𝑁−1−𝑓𝑥𝑥𝑥𝑓𝑦−3𝑓𝑥𝑥𝑦𝑓𝑥+𝑓𝑦𝑓𝑡+𝑓𝑥𝑓𝑧||||||||||||,=−12𝑛𝑁−3,𝑁−1,𝑁+1𝑁−2,𝑁3𝑓𝑥𝑥𝑓𝑥𝑦||||||+||||||=3𝑛𝑁−3,𝑁−1,𝑁𝑁−2,𝑁+12−||||||+||||||=3𝑛𝑁−3,𝑁−1,𝑁𝑁−2,𝑁+12||||||||||||.+12𝑛𝑁−3,𝑁−1,𝑁𝑁−2,𝑁+1(2.8) Using Lemma 2.3, we can further obtain that 𝐷3𝑥𝐷𝑦−𝐷𝑦𝐷𝑡−𝐷𝑥𝐷𝑧||||||||||||−||||||||||||+||||||||||||𝑓⋅𝑓=12𝑛𝑁−3,𝑁,𝑁+1𝑁−1𝑁−3,𝑁−1,𝑁+1𝑁−2,𝑁𝑁−3,𝑁−1,𝑁𝑁−2,𝑁+1=0.(2.9) This last equality is just the Plücker relation for determinants: ||𝐵,𝐴1,𝐴2||||𝐵,𝐴3,𝐴4||−||𝐵,𝐴1,𝐴3||||𝐵,𝐴2,𝐴4||+||𝐵,𝐴1,𝐴4||||𝐵,𝐴2,𝐴3||=0,(2.10) where 𝐵 denotes an 𝑁×(𝑁−2) matrix, and 𝐴𝑖,1≤𝑖≤4 are four N-dimensional column vectors. Therefore, we have shown that î„žğ‘“=|𝑁−1| solve (1.4) under the conditions (2.3) and (2.4). Further, the corresponding solution to (1.2) is 𝑢=2(ln𝑓)𝑥||||||,𝑓=𝑁−1.(2.11)

Remarks 1. The condition (2.4) is a generalized linear differential condition which includes many different special cases.
For example, when 𝑚=1, the condition (2.4) is reduced to 𝜙𝑖,𝑦=𝑛𝜙𝑖,𝑥,𝜙𝑖,𝑡=−𝜙𝑖,𝑥,𝜙𝑖,𝑧=4𝑛𝜙𝑖,𝑥𝑥𝑥+𝑛𝜙𝑖,𝑥.(2.12) When 𝑚=2, the condition (2.4) is reduced to 𝜙𝑖,𝑦=𝑛𝜙𝑖,𝑥,𝜙𝑖,𝑡=−𝜙𝑖,𝑥𝑥,𝜙𝑖,𝑧=4𝑛𝜙𝑖,𝑥𝑥𝑥+𝑛𝜙𝑖,𝑥𝑥.(2.13) When 𝑚=3, the condition (2.4) is reduced to 𝜙𝑖,𝑦=𝑛𝜙𝑖,𝑥,𝜙𝑖,𝑡=−𝜙𝑖,𝑥𝑥𝑥,𝜙𝑖,𝑧=5𝑛𝜙𝑖,𝑥𝑥𝑥.(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 𝜙𝑖,𝑥𝑥=𝑁𝑗=1𝜆𝑖𝑗𝜙𝑗𝜙,𝑖=1,…,𝑁,(2.15)𝑖,𝑦=𝜙𝑖,𝑥,𝜙𝑖,𝑡=−𝜙𝑖,𝑥𝑥,𝜙𝑖,𝑧=4𝜙𝑖,𝑥𝑥𝑥+𝜙𝑖,𝑥𝑥.(2.16)
If we let the coefficient matrix 𝐴=(𝜆𝑖𝑗)1≤𝑖,𝑗≤𝑁 of condition (2.15) has the following form (see [10–12, 16] for details), âŽ›âŽœâŽœâŽœâŽœâŽœâŽœâŽğœ†ğ´=110𝜆22⋱0ğœ†ğ‘ğ‘âŽžâŽŸâŽŸâŽŸâŽŸâŽŸâŽŸâŽ ğ‘Ã—ğ‘,𝜆𝑖𝑖≠𝜆𝑗𝑗,,𝑖≠𝑗(2.17) using the same method as that in [16], we can obtain N-soliton solutions of (1.2).
For example, when 𝜆11>0,𝜆𝑖𝑖=0,𝑖=2,…,𝑁, we can compute two exact 1-soliton solutions for (1.2), 𝑢1=2𝜕𝑥lncosh4𝜆3/211𝑧+𝜆11√𝑧+𝜆11√𝑦+𝜆11𝑥−𝜆11𝑡+𝛾1√=2𝜆11tanh4𝜆3/211𝑧+𝜆11√𝑧+𝜆11√𝑦+𝜆11𝑥−𝜆11𝑡+𝛾1,𝑢(2.18)2=2𝜕𝑥lnsinh4𝜆3/211𝑧+𝜆11√𝑧+𝜆11√𝑦+𝜆11𝑥−𝜆11𝑡+𝛾1√=2𝜆11coth4𝜆3/211𝑧+𝜆11√𝑧+𝜆11√𝑦+𝜆11𝑥−𝜆11𝑡+𝛾1,(2.19) with 𝛾1 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).

Theorem 3.1. Let a group of functions 𝜙𝑖=𝜙𝑖(𝑥,𝑦,𝑧,𝑡),1≤𝑖≤𝑁 satisfy the following linear differential condition: 𝜙𝑖,𝑦=𝑘𝜙𝑖,𝑥𝑥,𝜙𝑖,𝑡=−2𝜙𝑖,𝑥𝑥𝑥,𝜙𝑖,𝑧=3𝑘𝜙𝑖,𝑥𝑥𝑥𝑥,(3.1) where 𝑘 is an arbitrary nonzero constant. Then the Wronskian determinant î„žğ‘“=|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 î„žğ‘“=|𝑁−1| with respect to the variables 𝑥,𝑦,𝑧,𝑡 are obtained as follows: 𝑓𝑥=||||||𝑁−2,𝑁,𝑓𝑥𝑥=||||||+||||||,𝑓𝑁−3,𝑁−1,𝑁𝑁−2,𝑁+1𝑥𝑥𝑥=||||||||||||+||||||,𝑓𝑁−4,𝑁−2,𝑁−1,𝑁+2𝑁−3,𝑁−1,𝑁+1𝑁−2,𝑁+2𝑦−||||||+||||||,𝑓=𝑘𝑁−3,𝑁−1,𝑁𝑁−2,𝑁+1𝑥𝑦−||||||+||||||,𝑓=𝑘𝑁−4,𝑁−2,𝑁−1,𝑁𝑁−2,𝑁+2𝑥𝑥𝑦−||||||−||||||+||||||+||||||,𝑓=𝑘𝑁−5,𝑁−3,𝑁−2,𝑁−1,𝑁𝑁−4,𝑁−2,𝑁−1,𝑁+1𝑁−3,𝑁−1,𝑁+2𝑁−2,𝑁+3𝑥𝑥𝑥𝑦−||||||−||||||||||||+||||||||||||+||||||,𝑓=𝑘𝑁−6,𝑁−4,𝑁−3,𝑁−2,𝑁−1,𝑁𝑁−4,𝑁−2,𝑁,𝑁+1−2𝑁−5,𝑁−3,𝑁−2,𝑁−1,𝑁+1𝑁−3,𝑁,𝑁+2+2𝑁−3,𝑁−1,𝑁+3𝑁−2,𝑁+4𝑡||||||−||||||+||||||,𝑓=−2𝑁−4,𝑁−2,𝑁−1,𝑁𝑁−3,𝑁−1,𝑁+1𝑁−2,𝑁+2𝑦𝑡−||||||−||||||+||||||+||||||−||||||+||||||,𝑓=−2𝑘𝑁−6,𝑁−4,𝑁−3,𝑁−2,𝑁−1,𝑁𝑁−4,𝑁−2,𝑁,𝑁+1𝑁−5,𝑁−3,𝑁−2,𝑁−1,𝑁+1𝑁−3,𝑁,𝑁+2𝑁−3,𝑁−1,𝑁+3𝑁−2,𝑁+4𝑧−||||||+||||||−||||||+||||||,𝑓=3𝑘𝑁−5,𝑁−3,𝑁−2,𝑁−1,𝑁𝑁−4,𝑁−2,𝑁−1,𝑁+1𝑁−3,𝑁−1,𝑁+2𝑁−2,𝑁+3𝑥𝑧−||||||+||||||−||||||+||||||.=3𝑘𝑁−6,𝑁−4,𝑁−3,𝑁−2,𝑁−1,𝑁𝑁−4,𝑁−2,𝑁,𝑁+1𝑁−3,𝑁,𝑁+2𝑁−2,𝑁+4(3.2) Therefore, we can now compute that 𝑓𝑥𝑥𝑥𝑦−𝑓𝑦𝑡−𝑓𝑥𝑧−||||||+||||||||||||,𝑓=6𝑘𝑁−4,𝑁−2,𝑁,𝑁+1𝑁−3,𝑁,𝑁+2𝑁−1−3𝑓𝑥𝑥𝑦𝑓𝑥+𝑓𝑥𝑓𝑧||||||−||||||||||||,=6𝑘𝑁−4,𝑁−2,𝑁−1,𝑁+1𝑁−3,𝑁−1,𝑁+2𝑁−2,𝑁−𝑓𝑥𝑥𝑥𝑓𝑦+𝑓𝑦𝑓𝑡+3𝑓𝑥𝑥𝑓𝑥𝑦−||||||||||||+||||||||||||,=6𝑘𝑁−4,𝑁−2,𝑁−1,𝑁𝑁−2,𝑁+1𝑁−3,𝑁−1,𝑁𝑁−2,𝑁+2(3.3) then, substituting the above results into (1.4), we can further obtain that 𝐷3𝑥𝐷𝑦−𝐷𝑦𝐷𝑡−𝐷𝑥𝐷𝑧𝐴𝑓⋅𝑓=6𝑘1+𝐴2,(3.4) where 𝐴1||||||||||||+||||||||||||−||||||||||||=||||||||||||−||||||||||||+||||||||||||,𝐴=−𝑁−4,𝑁−2,𝑁,𝑁+1𝑁−1𝑁−4,𝑁−2,𝑁−1,𝑁+1𝑁−2,𝑁𝑁−4,𝑁−2,𝑁−1,𝑁𝑁−2,𝑁+1𝑁−4,𝑁−2,𝑁−3,𝑁−1𝑁−4,𝑁−2,𝑁,𝑁+1𝑁−4,𝑁−2,𝑁−3,𝑁𝑁−4,𝑁−2,𝑁−1,𝑁+1𝑁−4,𝑁−2,𝑁−3,𝑁+1𝑁−4,𝑁−2,𝑁−1,𝑁2=||||||||||||−||||||||||||+||||||||||||=||||||||||||−||||||||||||+||||||||||||.𝑁−3,𝑁,𝑁+2𝑁−1𝑁−3,𝑁−1,𝑁+2𝑁−2,𝑁𝑁−3,𝑁−1,𝑁𝑁−2,𝑁+2𝑁−3,𝑁,𝑁+2𝑁−3,𝑁−2,𝑁−1𝑁−3,𝑁−2,𝑁𝑁−3,𝑁−1,𝑁+2𝑁−3,𝑁−2,𝑁+2𝑁−3,𝑁−1,𝑁(3.5) 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 î„žğ‘“=|𝑁−1| also solve (1.4) under the condition (3.1).
The condition (3.1) has an exponential-type function solution: 𝜙𝑖=𝑝𝑗=1𝑑𝑖𝑗𝑒𝜂𝑖𝑗,𝜂𝑖𝑗=𝑙𝑖𝑗𝑥+𝑘𝑙2𝑖𝑗𝑦+3𝑘𝑙4𝑖𝑗𝑧−2𝑙3𝑖𝑗𝑡,𝑖=1,2,…,𝑁,(3.6) where 𝑑𝑖𝑗 and 𝑙𝑖𝑗 are free parameters and 𝑝 is an arbitrary natural number.
In particular, we can have the following Wronskian solutions of (1.2): 𝑢=2(ln𝑓)𝑥𝜙,𝑓=𝑊1,𝜙2,…,𝜙𝑁,(3.7) where 𝜙𝑖=𝑒𝑙𝑖𝑥+𝑘𝑙2𝑖𝑦+3𝑘𝑙4𝑖𝑧−2𝑙3𝑖𝑡+𝑒𝑤𝑖𝑥+𝑘𝑤2𝑖𝑦+3𝑘𝑤4𝑖𝑧−2𝑤3𝑖𝑡,𝑖=1,…,𝑁,(3.8) with 𝑙𝑖 and 𝑤𝑖 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.

Acknowledgments

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).