Abstract
Under investigation in this paper is a generalized (2+1)-dimensional dispersive long-wave system, describing the nonlinear and dispersive long gravity waves in two horizontal directions in the shallow water of a wide channel of finite depth or an open sea. Via symbolic computation, we derive the same bilinear forms as those reported, but through a different method. Four sets of the similarity reductions are obtained, each of which leads to a known ordinary differential equation. The results rely on the coefficients in the original system, with respect to the horizontal velocity and wave elevation above the undisturbed water surface.
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1 Introduction
Studies on fluids have been reported [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. For investigating the nonlinear and dispersive long gravity waves in two horizontal directions, especially those in the shallow water of a wide channel or an open sea with finite depth, Ref. [15] has proposed the following generalized (2+1)-dimensional dispersive long-wave system:
with u(x, y, t) as the horizontal velocity, v(x, y, t) as the wave elevation above the undisturbed water surface, u(x, y, t) and v(x, y, t) as the real differentiable functions in respect of the variables x, y and t, the subscripts as the partial derivatives, \(\alpha \ne 0\), \(\beta \) and \(\delta \ne 0\) implying the real constants, while t and (x, y) denoting the time and propagation plane, separately. Also in Ref. [15], some special cases which can report the applications of System (1) have been listed.
Ref. [15] has derived two sets of the bilinear forms of System (1), i.e.,
in which \(\theta _4\) indicates a real constant, f(x, y, t) and g(x, y, t) imply the \(C^{\infty }\) functions of x, y and t, while \(D_{x}\), \(D_{y}\) and \(D_{t}\) represent the Hirota operators defined as [16]
with \(x'\), \(y'\) and \(t'\) denoting the formal variables, while m, r and n meaning three non-negative integers. Besides, Ref. [15] has also obtained certain scaling transformations, hetero-Bäcklund transformations and N-soliton solutions for System (1), where N is a positive integer. For System (1), Ref. [17] has constructed certain hetero- and auto-Bäcklund transformations with some soliton solutions, while Ref. [18] has given out some similarity reductions.Footnote 1
To System (1), contributions of this paper could be introduced in the following aspects:
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Background: Nowadays, many nonlinear evolution equations/systems have been put into use in some physical studies, e.g., optical fibers, fluids and plasmas [17,18,19,20,21,22,23,24,25,26,27,28].
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Motivations: On the one hand, we plan to construct the same bilinear forms as Bilinear Forms (2) with a different method, to confirm the correctness of Bilinear Forms (2). On the other hand, we would like to find out more similarity reductions, which link System (1) to some ordinary differential equations (ODEs), to complement the existing results.
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Novelty and outlines: Bäcklund transformations and solutions of System (1) could be derived via the bilinear forms [15]. In comparison with the Bell polynomials in Ref. [15], the Hirota method may give rise to more potential bilinear forms [29]. Besides, similarity reductions in this paper, which are different from those in Ref. [18], might fit some other situations.
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Originality: To date, for System (1), similarity reductions different from those in Ref. [18] have not been investigated. In Sect. 2, we will derive two sets of the bilinear forms, which are the same as those in Ref. [15], but through a different method, i.e., the Hirota method [16, 30,31,32,33]. In Sect. 3, with symbolic computationFootnote 2 [34,35,36,37,38], we will obtain four sets of the similarity reductions for System (1), which are different from those in Ref. [18]. Conclusions will be given in Sect. 4.
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Significance and potential applications: This paper could be of some use for the future studies on the nonlinear and dispersive long gravity waves in two horizontal directions, especially those in the shallow water of a wide channel or an open sea with finite depth.
2 Two Sets of the Bilinear Forms for System (1) through the Hirota Method
Since our goal is to construct some bilinear forms for System (1) in respect of f(x, y, t) and g(x, y, t), the Hirota method brings about the assumptions
where \(\zeta _2\) and \(\zeta _4\) are two real constants, while \(\zeta _1\) and \(\zeta _3\) imply two real non-zero constants.
Integrating Eq. (1a) once in respect of x and y, respectively, with the integration function vanishing, we get
To bring in the Hirota operators, based on the following formulae [16]:
with the assumption that
we convert Eq. (1a) into
Similarly, we integrate Eq. (1b) once in respect of x with the integration function vanishing, to find
According to Formulae (5) and the following formulae [16]:
Based on the above derivation, we are able to come up with the theorem:
Theorem 2.1
In brief, via Assumptions (3), we construct the following bilinear forms for System (1) via the Hirota method:
which are the same as Bilinear Forms (2) when \(\zeta _{4}=\theta _{4}\).
3 Four Sets of the Similarity Reductions for System (1)
For obtaining some similarity reductions, we give rise to the assumptionsFootnote 3
where \(\theta (x,y,t)\), \(\omega (x,y,t) \ne 0\), \(\gamma (x,y,t)\), \(\kappa (x,y,t) \ne 0\) and \(z(x,y,t) \ne 0\) imply some real differentiable functions to be determined, while p[z(x, y, t)] and q[z(x, y, t)] are two real differentiable functions of z.
Making use of symbolic computation and inserting Assumptions (12) into System (1), we obtain that
in which
\(\chi _{i}\)’s (\(i=0,...,9\)) and \(\tau _{j}\)’s (\(j=0,...,8\)) are some real differentiable functions with respect to x, y and t, while the prime sign means d/dz. Because p(z) and q(z) are the functions of z only, we are able to convert Eq. (13) into a set of the ODEs in respect of p(z) and q(z). Each set of \(\theta (x,y,t)\), \(\omega (x,y,t)\), \(\gamma (x,y,t)\), \(\kappa (x,y,t)\) and z(x, y, t) could lead to, at least, a similarity reduction of System (1). In this paper, we consider the case of \(z_{x} z_{y} \ne 0\), so that \(\chi _{0} \ne 0\) and \(\tau _{0} \ne 0\), to obtain that
with \(\Omega _{i}(z)\)’s and \(\Gamma _{j}(z)\)’s as some real to-be-determined functions of z only.
For the sake of simplicity, we give out the assumption thatFootnote 4
with \(\lambda _1\), \(\lambda _2\) and \(\lambda _3\) as the real non-zero constants, while \(\lambda _4\) as a real constant. Substituting Eqs. (14q), (14k) and (16) into Eqs. (15) turns to
According to the second freedom of Remark 3 in Ref. [62], Eq. (14g) results in
With the first freedom of Remark 3 in Ref. [62], Eq. (14b) leads to
and Eq. (14m) helps us derive
Based on the first and the second freedom of Remark 3 in Ref. [62], respectively, we will obtain two branches of the results.
Branch 1: \(\gamma (x,y,t)=-\beta \;, \;\; \Gamma _{2}(z)=0\)
Inserting \(\gamma (x,y,t)=-\beta \) into Eqs. (14) brings about
Eqs. (13) can turn into
Then we integrate ODE (22a) twice about z, to obtain
with \(\phi _1\) and \(\phi _2\) being two real constants of integration. Integrating ODE (22b) once in respect of z and considering ODE (23), we can transfer ODEs (22) to a simple ODE, written as
where \(\phi _3\) denotes a real constant of integration.
Thus, we derive two sets of the similarity reductions for System (1), i.e.,
ODE (25d) is a known ODE, reported in Ref. [63].
Branch 2: \(\gamma (x,y,t)=\delta ^2 \lambda _1 \lambda _2 -\beta \;, \;\; \Gamma _{2}(z)=1\)
When \(\gamma (x,y,t)=\delta ^2 \lambda _1 \lambda _2 -\beta \), we propose to derive
Eqs. (13) are converted into
Similarly, we integrate ODE (27a) twice about z to find
with \(\phi _4\) and \(\phi _5\) as two real constants of integration. Integrating ODE (27b) once about z and considering ODE (28) could develop into
with \(\phi _6\) as a real constants of integration.
Thus, we require into another two sets of the similarity reductions for System (1), i.e.,
ODE (30d) is a known ODE, reported in Ref. [63].
With respect to the horizontal velocity and the wave elevation above the undisturbed water surface, we derive the following theorem about System (1), describing the nonlinear and dispersive long gravity waves in two horizontal directions in the shallow water of a wide channel of finite depth or an open sea.
Theorem 3.1
Similarity Reductions (25) and Similarity Reductions (30), both of which are different from those in Ref. [18], depend on all the constant coefficients in System (1), i.e., \(\alpha \), \(\beta \) and \(\delta \). The reason why there are two sets of Similarity Reductions (25)/Similarity Reductions (30) is the existence of “±" sign.
4 Discussions
We have noticed that both Similarity Reductions (25) and Similarity Reductions (30) are different from those in Ref. [18], while both ODE (25d) and ODE (30d) are the known ODEs. Our results have been shown to depend on \(\alpha \), \(\beta \) and \(\delta \), all the constant coefficients in System (1), and might be of some use in the studies on the nonlinear and dispersive long gravity waves in two horizontal directions in the shallow water of a wide channel of finite depth or an open sea.
5 Conclusions
As for a generalized (2+1)-dimensional dispersive long-wave system in respect of the horizontal velocity and the wave elevation above the undisturbed water surface, i.e., System (1), we have obtained the following:
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Two sets of the bilinear forms, i.e., Bilinear Forms (11), which are the same as Bilinear Forms (2), but through a different method, i.e., the Hirota method. Thus, the correctness of Bilinear Forms (2) can be confirmed.
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Four sets of the similarity reductions for System (1), i.e., Similarity Reductions (25), from System (1) to ODE (25d), and Similarity Reductions (30), from System (1) to ODE (30d).
Notes
Note that ODE (14) and ODE (15d) in Ref. [18] are wrong, and we need to correct them to \(p''-\frac{3}{2}p^2-\frac{1}{2}p^3+(\phi _1 z+\phi _2-1)p+(\phi _1 z+\phi _2-\phi _3)=0\).
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Acknowledgements
We express our sincere thanks to the Editors and Reviewers for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11772017, 11272023 and 11471050, by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China (IPOC: 2017ZZ05) and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02YS. Y. Shen also thanks the BUPT Excellent Ph.D. Students Foundation under Grant No. CX2022156.
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Gao, XT., Tian, B., Shen, Y. et al. Considering the Shallow Water of a Wide Channel or an Open Sea Through a Generalized (2+1)-dimensional Dispersive Long-wave System. Qual. Theory Dyn. Syst. 21, 104 (2022). https://doi.org/10.1007/s12346-022-00617-7
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DOI: https://doi.org/10.1007/s12346-022-00617-7