In this paper, Lie symmetry analysis is performed for the equation derived from $(2+1)$-dimensional higher order Broer-Kaup equation. Meanwhile, the optimal system and similarity reductions based on the Lie group method are obtained. Furthermore, the conservation law is studied via the Ibragimov’s method.
Nonlinear partial differential equations (PDEs) arising in many physical fields like the condense matter physics, plasma physics, fluid mechanics and optics and so on. In order to investigate the exact solution of PDEs, a fruitful techniques have been developed, such as traveling wave transformations, inverse scattering method , Darboux and Bäcklund transformations , Lie symmetry analysis [3, 4, 5]. Lie symmetry analysis is a very useful method to find the new solutions of PDEs, which was distribution by Sophus Lie (1842-1899). In addition on the base of symmetries, the integrability of the nonlinear PDEs, such as group classification, optimal system and conservation laws, can be considered. Lie groups, as a type of trandformation groups, can tranfer one solution to another one of a given PDE. In other words, if we get one solution of a PDE, we can obtain the other ones via the symmetry of the PDE. Based on this, we will investigate the Lie symmetry analysis of the given PDE.
Noether’s theorem  establishes a connection between symmetries of differential equations and conservation laws. However, there are other methods to study the conservation laws, such as partial Noether’s approach, multiplier approach and Ibragimov’s method. As stated in , the former three methods are not applicable to the nonlinear PDEs that do not admit a Lagrangian. In order to overcome these difficulties, Ibragimov’s method was proposed . Especially state, on the contribution of Lie symmetry method, significant researches have been done on the integrability of the nonlinear PDEs, group classification, optimal system, reduced solutions and conservation laws, such as [9, 10, 11, 12, 13, 14] and [15, 16, 17, 18, 19, 20, 21] published this year and last year.
The (2+1)-dimensional higher order Broer-Kaup equation was considered in  and , whose expression is as follows:
Li et al. and Mei et al. took the Bäcklund transformation of system (1.1) and obtained the relationship:
Such that (1.1) becomes a single differential equation:
For (1.2), we consider its special case. That is, U = U(x, t) is regarded as (1 + 1)-dimensional and replaced by u, then (1.2) becomes
For convenience to cite later, we call (1.3) to be Li-Mei system, which is equivalent to
The exact traveling wave solutions have been investigated in . However, to the best of our knowledge, the Lie symmetry, optional system and conservation law of Li-Mei equation have not been researched, which is the original intention of this work.
This paper is organized as follows. In section 2, we perform Lie symmetry analysis of Li-Mei system. In section 3, the optimal system and similarity reductions are studied. section 4 distributes to studying the conservation law in the method of Ibragimov’s and construction the conserved vectors.
2 Lie symmetries of Li-Mei equation (1.3)
Lie symmetries analysis will be performed of Eq. (1.3) in this section. Consider a one-parameter Lie group of transformations:
With a small parameter ε ≪ 1. The vector field associated with the above transformation group can assumed as:
Thus the third prolongation pr(3)V is:
where only the terms involved in (1.3) appear in (2.3). In (2.3), ϕx, ϕt, ϕxx and ϕxxx are all undetermined functions, which are given by the following formulae.
where Dx, Dt are denoted the total derivatives with respect to x and t, respectively.
The determining equation of Eq. (1.3) arises from the following invariance condition:
By (2.8), we have the following symmetry condition:
which ξ(x, t, u), τ(x, t, u) and ϕ(x, t, u) must satisfy.
Substituting (2.4)-(2.7) into (2.10), replacing ut by −(4uxxx + 12u2 ux + + 12uuxx) whenever it appears, and comparing the coefficients of the various monomials in the first, second and third order partial derivatives, and solving the system, we obtain the expression of ξ(x, t, u), τ(x, t, u) and ϕ(x, t, u).
where c1, c2, c3 are arbitrary constants.
Hence the infinitesimal generators of Eq. (1.3) can be listed as follows
By solving the following ordinary differential equations with initial condition:
We therefore obtain the group transformation which is generated by the infinitesimal generators V1, V2, V3, respectively:
Here G1, G2, G3 are all one-dimensional Lie groups generated by their own generators gi,ε, whose operation is manifested by (2.14), (2.15), (2.16), respectively.
It is trivial that V1, V2, V3 form a 3-dimensional Lie algebra L with the following Lie bracket:
In (2.14)-(2.16), an arbitrary element in Gi(i = 1, 2, 3) can transfer one solution of Eq. (1.3) to another one, so do the products of the elements from G1, G2 and G3.
The Lie group G1 × G2 is a normal Lie subgroup of G1G2G3. The Lie algebra generated by V1 and V2 is an ideal of L.
The vector fields V1, V2 and V3 supply a representation of the Lie algebra
where the Lie bracket is
The definition of representations of Lie algebras see .
It is suffice if we take the representation space to be the set of all the analytic functions and the linear mapping ρ : xi ↦ Vi for i = 1, 2, 3. □
The vector fields V1 and V2 have trivial prolongation. However, the prolongation of V3 can be computed:
It is easy to check pr(3)V3(Δ) = −4⋅Δ, which is called the symmetry invariance of differential equation (1.3).
We are now to take an example to illustrate the applications of Lie symmetry analysis. We take ut = uxx as an example rather than Eq. (1.3) since it is difficult to find the analytical solution. The vector fields of this equation is V1 = ∂x, V2 = ∂t, V3 = ∂u, V4 = x/2∂x + t∂t + u∂u. It is not difficult to find a special solution u(x, t) = et(ex + e−x). Under the operation of Lie group generated by V1 − V4, we can check that
are all the solutions of ut = uxx.
3 Optimal system of one-dimensional subalgebras
The more technical matters arose in order to classify the subalgebra of Lie algebra generated by Lie point symmetries, for instance  and . A concise method to get the optimal system was presented by Ibragimov in 2010 . In this section we shall construct an optimal system of one-dimensional subalgebra.
The following operators provide two optimal systems of one-dimensional subalgebras of the Lie algebra spanned by V1, V2, V3 of Eq. (1.3):
where both ν and μ are arbitrary constants.
Suppose W and V are two vector field and
By solving this ODE we have
by summing the Lie series
In view of (3.4), we obtain
For an arbitrary nonzero vector
our task is to simplify as many of the coefficients ai as possible through the applications of adjoint maps to V.
a3 ≠ 0. Scaling V if necessary, we can assume that a3 = 1. By making use of (3.5) and acting on such a V by we can make the coefficient of V2 vanish:
Next we act on V′ by Ad(exp(ε a1V1)), to cancel the coefficient of V1. Hence V is equivalent to V3 under the adjoint representation.
a3 = 0.
a2 ≠ 0, a1 ≠ 0. Without losing generality, we can assume that a2 = 1. One can easily figure out that the adjoint representation induced by any combinations of V1, V2, V3 shall make a1V1 + V2 invariant. In other words, any one-dimensional subalgebra generated by V is equivalent to the subalgebra generated by a1V1 + V2.
a2 = 0, a1 ≠ 0. Similarly to the discussion of Subcase 1, we can conclude that V is equivalent to V1 under the adjoint representation.
The other optimal system can be obtained similarly. □
4 Similarity reductions and exact solutions for Eq. (1.3)
In the preceding section, we got the optimal system of Eq. (1.3). We are now in the position to deal with the symmetry reduction and exact solutions via constructing similarity variables.
For the generator V1, we assume ζ = t, u = f(ζ) and the we obtain the trivial solution f = c, where c is an arbitrary constant.
For the linear combination V1 + μ V2, we have(4.7)
where ζ = μ x − t, which follows that this ODE(4.8)
where and μ ≠ 0.
In the above, we sketch the graphs of f(ζ) in Eqs. (4.2), (4.4), (4.6), (4.8) and 3D-plot of u(x, t) in Eqs. (4.3), (4.4) under the initial conditions f(0) = , f(1) = 1, f′(0) = 0.
5 Nonlinear self-adjointness and conservation law
First of all we show that Li-Mei equation is nonlinearly self-adjoint.
For a given PDEs
define the Euler-Lagrange operator
and the formal Lagrangian
The adjoint equations
The system (5.1) is said to be nonlinearly self-adjoint if the adjoint system (5.4) is satisfied for all solutions u of system (5.1) upon a substitution v = φ(x, u) such that φ(x, u) ≠ 0, which is equivalent to the following identity holding for the undetermined functions
In this paper α = 1 and Rα(x, u, u(1), ⋯, u(k)) = Δ(x, t, ut, ux, uxx, uxxx) = Eq. (2.9). The formal Lagrangian is
Substituting it into (5.2) = 0, we have the adjoint equation to Eq. (1.3)
By means of
it leads us to
Firstly we obtain λ = −φu by comparing the terms with the third-order derivative of u. And then
We therefore get φu = 0 and 4φxxx − 12uφxx + 12u2φx + φt = 0. In view of φu = 0, one has φx = φt = 0, so φ = c ≠ 0, which proves that Eq. (1.3) is self-adjoint.
We are now in the position to construct the conservation law of Eq. (1.3).
(Ibragimov’s method). Let the system of differential Eq. (5.1) be nonlinearly self-adjoint. Then every Lie point, Lie-Bäcklund, nonlocal symmetry
admitted by the system of Eq. (5.1) gives rise to a conservation law, where the components 𝓒i of the conserved vector 𝓒 = (𝓒1, ⋯, 𝓒n) are determined by
For the generator we have W = ϕ − ξ ux − τ ut, we therefore obtain the following components of conserved vector
Taking the formal Lagrangian 𝓛 given by (5.6) into (5.13) and (5.14), we can simplify the expressions of 𝓒x and 𝓒t as follows
For the generator V1 = , it has the Lie characteristic function W = −ux. By using of the formulae (5.15) and (5.16), it can give rise to the following components of the conserved vector
For the generator V2 = , we have W = −ut, the formulae (5.15) and (5.16) yield the following components of the conserved vector
For the generator we have W = − u − xux − 3tut, the formulae (5.15) and (5.16) imply the following components of the conserved vector
These vectors involve an arbitrary solution v of the adjoint equation (5.7) and hence provide an infinite number of conservation laws.
In this paper, we have obtained the symmetries and the corresponding Lie algebras of Li-Mei system by using Lie symmetry analysis method. Meanwhile, the optimal system and its similarity reductions are investigated. Furthermore, we proved that it is nonlinearly self-adjoint. Finally, the conserved vectors were constructed via the Ibragimov’s method.
The vector fields generate the equation under consideration supply a representation of a Lie algebra. However, for a given finitely dimensional Lie algebra, such as nine types of simply Lie algebras, how to get its representation via vector fields? If we have already obtained the vector fields, can we get the differential equation which generates the vector field? If the differential equation is obtained, is it unique? All of them are the aims that we will study in the near future.
Authors’ contributions Hengtai Wang denotes to studying of the whole system and writing the main body of the article. Huiwen Chen contributes the graphs of the paper. Zigen Ouyang checks all the errors of this paper. Fubin Li denotes some calculations of this article.
The authors are grateful to the editor for his/her valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant No. 11801264, 11601222) and Hunan Provincial Natural Science Foundation of China (Grant Nos. 2019JJ40240, 2019JJ50487).
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