Abstract
We propose a generalized long-water wave system that reduces to the standard water wave system. We also obtain the Lax pair and symmetries of the generalized shallow-water wave system and single out some their similarity reductions, group-invariant solutions, and series solutions. We further investigate the corresponding self-adjointness and the conservation laws of the generalized system.
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1 Introduction
The classical dispersiveless long wave equations
have a number of dispersive generalizations [1]. Kupershmidt [2] considered the following extension of (1):
where α, β are arbitrary constants. The invertible change of variables \(u=\bar{u}, h=\bar{h}+\gamma \bar{u}x\), turns (2) into
Broer [1] derived system (2) for \(\alpha =\frac{1}{3}, \beta =0\), for which it is the proper Boussinesq equation. In terms of the potential \(\varphi :u=\varphi _{x}\), system (2) was derived by Kaup [3]. Later, Matveev and Yavor [4] found algebrogeometrically a large class of almost periodic solutions. Li, Ma, and Zhang [5] used a scaling transformation to transfer a nonlinear long wave equation of Boussinesq class to the Broer–Kaup (BK) system, a type of long water wave equations:
Furthermore, some exact solutions and Darboux transformations of (1) were obtained by applying the Lax-pair method. In terms of [5], we can study the similarity reduction, exact solutions, and conservation laws of the Boussinesq system through the scalar transformation
that is, we can transform the BK system to the Boussinesq system
where ξ is the elevation of the water wave, u is the surface velocity of water along the x-direction. Hence, the results of the paper have certain physical sense.
In the paper, we construct a generalized BK system as follows:
where α, β are constants, so that some symmetries of (2) are produced by the symmetry group method [6]. It follows that some similarity solutions, group-invariant solutions, and series solutions are produced. In addition, Ibragimov and Avdonina [7] showed how to apply the symmetries of differential equations to study the self-adjointness and conservation laws. Thus we would like to follow the approach to investigate the quasiself-adjointness and conservation laws of the generalized BK system (2).
2 Integrability of (2)
Set
where
Then the compatibility condition of (3)
admits the generalized BK system, which can be directly verified. Hence the generalized BK system (2) is Lax integrable. Using (3), we can get some Darboux transformations for deducing solutions of the system. Here we omit them.
3 Similarity solutions and group-invariant solutions
Applying the Lie symmetry analysis, we can get the symmetry of system (2):
The vector field \(X_{3}\) has the following characteristic equation:
which gives rise to
One integration factor of (6) is given by
which transfers (6) to the complete integration equation
from which we have the invariant variable \(\xi =2\beta t^{\frac{1}{2}}-t ^{-\frac{1}{2}}x\). In terms of Eq. (5), we have the formal invariants
where \(f(\xi )\) and \(g(\xi )\) are arbitrary smooth functions of ξ. Substituting (7) into system (2) yields the ordinary differential system
Let \(\beta =0\), Then system (8) reduces to
where \(\tau =-t^{-\frac{1}{2}}x\), which is a reduction of ξ. In facr, system (9) is an ordinary differential system corresponding to the BK system (1).
The group-invariant transformations of the generalized BK system are as follows:
In what follows, we consider solutions to the BK system. Set \(v=V(\rho )\), \(w=W(\rho )\), \(\rho =x+lt\). Then (2) becomes
from which we have
A special solution to (11) is given by
in the case of \(l=-\beta \), \(c_{1}=c_{2}=0\). Hence we get a set of solutions to the generalized BK system (2):
Applying the group-invariant transformation (10), we can deduce some other new solutions to system (2):
Taking \(\beta =0\), we can obtain group-invariant solutions to the BK system (1). In particular, we can get the series solutions to the BK system. Indeed, let
and substituting into (9), we have that
from which we infer that
where \(c_{0}\), \(d_{0}\), \(c_{1}\), \(d_{1}\) are arbitrary parameters. Inserting these expressions into (14), we get the series solutions of the BK system. The second equation of system (9) can be reduced to
under the condition
As long as the solution of (15) is obtained, we can get the solution \(f(\tau )\) from (16). If \(g_{1}(\tau )\) is the known solution of (15), then we assume that \(g(\tau )=u(\tau )g_{1}(\tau )\). If \(u(\tau )\) is known, then the solution \(g(\tau )\) to Eq. (15) can be presented. It is easy to see that
Substituting (17) into Eq. (15) yields
Since
we have
Assume that \(u'(\tau )=z(\tau )\). Then Eq. (18) becomes
which has the solution
where c is a constant. Thus we have
Substituting (19) into Eq. (16), we can get \(f(\tau )\). Thus a type of special solutions to system (9) can be obtained.
4 The self-adjointness of system (2)
Ibragimov [8] introduced a few related notations of the strict self-adjointness, the nonlinear self-adjointness, and the quasiself-adjointness. Let us recall them.
Let H be a Hilbert space with the scalar product \((u,v)\) defined by
where \(F^{*}\) is the adjoint operator to a linear operator F. A special Hilbert space is given by
along with an inner product
Let F be a linear differential operator in H whose action on the function u is expressed by \(F[u]\). Then Eq. (20) becomes
which means that
where \(D_{i}=\frac{\partial }{\partial x^{i}}+u_{i}^{\alpha } \partial _{u^{\alpha }}+u_{ij}^{\alpha }\partial _{u_{j}^{\alpha }}+ \cdots \) .
For the differential equations
where \(u=(u^{1},\ldots ,u^{m})\). The adjoint equations to (22) are as follows:
with \(F_{\alpha }^{*}=\frac{\delta \mathcal{\varphi }}{\delta u^{ \alpha }}\). The Lagrangian φ for (22) is defined by
Definition 1
The differential Eqs. (22) are said to be strictly self-adjoint if their adjoint Eqs. (23) are equivalent to (23) upon the substitution \(v=u\). That is, the equation
holds with a coefficient λ.
Definition 2
([8])
Upon a substitution
if (23) becomes (22), then we call (22) is quasiself-adjoint.
Definition 3
Upon a substitution
if (26) solves the adjoint Eqs. (23) for all the solutions of (22), then we call system (22) nonlinearly self-adjoint, that is, we have the following equations:
It is easy to find that the strictly self-adjoint and quasiself-adjoint equations both are particular cases of the nonlinear self-adjoint equations.
For the generalized BK system (2), denoted by
the formal Lagrangian \(\mathcal{L}\) can be written as \(\mathcal{L}=pF+qG\), and the adjoint system of (2) is as follows:
Setting \(p=\varphi (v,w)\) and \(q=\psi (v,w)\) and substituting into (27), along with (28), we have
where \(\lambda _{1}\), \(\lambda _{2}\), \(\mu _{1}\), \(\mu _{2}\) are undetermined functions. It is easy to get
Inserting all these results into (29) yields that
Therefore, for all solutions of system (2), (28) holds. Thus system (2) is nonlinearly self-adjoint.
5 Another expression of system (2) and some properties
Set
Then system (2) becomes
which has the infinitesimal symmetries
where \(c_{1}\), \(c_{2}\), \(c_{3}\), \(c_{4}\) are constants. Obviously, when \(c_{1}=c_{2}=c_{3}=0\) and \(c_{4}=1\), we get \(X_{1}=\partial _{x}\). When \(c_{1}=c_{3}=c_{4}=0\) and \(c_{2}=1\), we have \(X_{2}=\partial _{t}\). When \(c_{2}=c_{3}=c_{4}=0\) and \(c_{1}=1\), we find \(X_{3}=2t\partial _{t}+x \partial _{x}+\partial _{V}+2\partial _{W}\); \(X_{i}\) \((i=1,2,3)\) all are particular cases of X.
Next, we consider the characteristic equation of X so that we can obtain the similarity reductions of system (30). The characteristic equation of X reads as
Case 1: \(c_{1}=1\).
System (30) reduces to
Case 2: \(c_{1}=c_{2}=0\). Equation (31) becomes
We take
Then system (30) reduces to
The two equations are in fact the same.
Case 3: \(c_{1}=0\), \(c_{2}\neq0\). Equation (31) reduces to
We choose
Thus system (30) turns to
System (35) has the particular solutions
where ξ satisfies the constraint
Thus from (32) we get a set of new solutions of system (2):
In what follows, we consider the series solutions of (33).
Setting
and substituting into system (33), we infer that
from which we get
where \(a_{0}\), \(b_{0}\), \(a_{1}\), \(b_{1}\) are arbitrary parameters. Thus we obtain the following formal series solutions of system (33):
where \(a_{i},b_{i}\) (\(i=4,5,\ldots \)) satisfy (36) and (37). Substituting (38)and (39) into (32), we can get the series solutions of the generalized BK system.
Next, we consider the solutions to system (34). It is easy to see that
where ĉ is an integration constant.
System (35) is solvable similarly to system (33), and we omit the computations.
6 Conservation laws
In this section, we consider the conservation laws of the generalized BK system by using the method in [7, 8]. From the identity
we find that
where
and \(\mathcal{L}\) is the Euler–Lagrange function, which satisfies
Since system (28) holds, we can investigate the conservation laws by using (41), where the components of the conservation laws are the following:
which satisfy the conservation equations
For \(X_{1}=\frac{\partial }{\partial x}\), we find that
Substituting (44) into (42) yields
For \(X_{2}=\frac{\partial }{\partial t}\), we get
For \(X_{3}=(\frac{1}{2}x+\beta t)\partial _{x}+t\partial _{t}- \frac{1}{2}v\partial _{v}-w\partial _{w}\), we infer
where \(v_{t}\), \(w_{t}\) are given by system (2).
Remark
Anco and Bluman [9] proposed a method for constructing conservation laws of differential equations, which uses a formula directly generating the conservation laws and independent of the system having a Lagrangian formulation, in contrast to Noether’s theorem, which requires a Lagrangian. They adopted the linear equations and the adjoint equations of the original differential equations to study conservation laws. Essentially, the algorithm presented by Ibragimov et al. is the same as that of Anco and Bluman. Besides, Anco [10] also gave some comments on the work of Ibragimov.
7 Conclusions
In the paper, we have investigated various similarity reductions and exact solutions of the generalized BK system and various its conservation laws by the Lie group analysis. We have pointed out that the standard BK system is only a paticular case of the generalized BK system (2) when \(\alpha =-1\) and \(\beta =0\). In addition, Lou [11, 12] applied the symmetry group method to study some coherent solutions of nonlocal KdV systems and primary branch solutions of a first-order autonomous system. We hope to extend the methods to the systems presented in the paper in the forthcoming days. In addition, Ma [13] obtained some new conservation laws of some discrete evolution equation by symmetries and adjoint symmetries. Zhang, et al. [14, 15] considered symmetry properties of some fractional equations. Therefore there is an open problem how we can look for the fractional systems that correspond to the systems presented in the paper and how we can to solve them. Besides, Liu, Zhang, and Zhou [16] constructed the fractional Volterra hierarchy, gave a definition of the hierarchy in terms of Lax pair and Hamiltonian formalisms, and constructed its tau functions and multisoliton solutions. Bridgman, Hereman, Quispel, and Kamp [17] and El-Nabulsi [18] studied the peakon and Toda lattice. The approaches adopted in [16,17,18] can lead us to investigate some related properties of the generalized BK system presented in the paper. These questions will be discussed in the future.
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The authors wish to thank the anonymous referees for their valuable suggestions.
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This work is supported by the Fundamental Research Funds for the Central University (No. 2017XKZD11).
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Zhang, X., Zhang, Y. Some invariant solutions and conservation laws of a type of long-water wave system. Adv Differ Equ 2019, 496 (2019). https://doi.org/10.1186/s13662-019-2422-8
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DOI: https://doi.org/10.1186/s13662-019-2422-8