# Approximate Symmetry Analysis and Approximate Conservation Laws of Perturbed KdV Equation

Authors: Yu-Shan Bai; Qi Zhang
Journal: Advances in Mathematical Physics
 (2018)

---

## Abstract

Approximate symmetries, which are admitted by the perturbed KdV equation, are obtained. The optimal system of one-dimensional subalgebra of symmetry algebra is obtained. The approximate invariants of the presented approximate symmetries and some new approximately invariant solutions to the equation are constructed. Moreover, the conservation laws have been constructed by using partial Lagrangian method.


---

## Body

## 1. Introduction

The partial differential equations (PDEs) with small parameter have been arisen in mathematics, physics, mechanics, etc. For these perturbed equations, the finding of analytic solutions, conservation laws, symmetries, etc. is essential in these related fields. Traditionally, various perturbation methods are used to solve such equations. In addition, emerging innovative methods such as the inverse scattering transformation method, homotopy perturbation method, Adomian decomposition method, and approximate symmetry method have been developed fast in the past few decades [1–4]. The approximate symmetry method, which is developed by Baikov, Gazizov, and Ibragimov [3, 4] in the 1980s, shows an effectiveness to obtain approximate solutions to a perturbed PDEs. The idea behind this development was the extension of Lie’s theory in which a small perturbation of the original equation is encountered. The new method maintains the essential features of the standard Lie symmetry method and provides us with the most widely applicable technique to find the approximate solutions to a perturbed differential equations [5, 6].The construction of conservation laws is one of the key applications of symmetries to physical problems [7, 8]. But for perturbed systems, one hindrance is how to construct conservation laws. Combined with the Noether theory and Lagrangian principle, the approximate symmetry method yields the approximate conservation laws for a perturbed PDEs. There are mainly two methods. One approach, based on generalizations of the Noether’s theorem to perturbed equations, is given to get approximate conservation laws via the approximate Noether symmetries associated with a Lagrangian of the perturbed equations [5, 9]. This method depends on the existence of the Lagrangian functional for underlying differential equations. The other approach (also called partial Lagrangian method), which is presented by Johnpillai, Kara, and Mahomed [10, 11], demonstrates how one can construct approximate conservation laws of Eular-Type PDEs via approximate Noether-type symmetry operators associated with partial Lagrangian. Partial Lagrangian method can construct approximate conservation laws of perturbed equations via approximate operators that are not necessarily approximate symmetry operators of the underlying system of equations. This method is used for more general equations as even the equations do not admit essential Lagrangian.In this paper, we consider perturbed KdV equation(1)ut-6uux+uxxx+εαu+2xux-εβ2u+xux=0,where ε is a small parameter, while α and β are arbitrary constants. Obviously, the equation has no Lagrangian because of its odd order. In [12], E. S. Benilov and B. A. Malomed discussed the equation based on the inverse scattering transformation and showed its integrability. When β=2α (1) has important applications in the description of nonlinear ion acoustic waves in an inhomogeneous plasma. For many other applications of (1), please refer to [2, 12, 13] and the references therein.In this article, with the application of approximate symmetry method and partial Lagrangian method, we will investigate (1) and show its all first-order approximate symmetries. Furthermore, we will also construct several approximate solutions and approximate conservation laws of the equation. The rest of the paper is organized as follows. Section 2 gives some basic concepts and notations. Section 3 analyzes approximate symmetries of (1) by applying the approximate symmetry method, developed by Baikov, Gazizov, and Ibragimov. We compute the optimal system of the presented approximate symmetries. In Section 4 we generate the approximate invariants of presented approximate symmetries and construct corresponding approximately invariant solutions. Section 5, the final part, presents approximate conservation laws via partial Lagrangian method.


## 2. Notations and Definitions

We will use the following notations and definitions. LetG be a one-parameter approximate transformation group:(2)z~i≈fx,a,ε≡f0iz,a+εf1iz,a,i=1,…,N.An approximate equation(3)Fz,ε≡F0z+εF1z≈0is said to beapproximately invariant with respect to G or admits G if(4)Fz~,ε≈Ffz,a,ε,ε=oεwhenever z=(z1,…,zN) satisfies (3).Ifz=(x,u,u(1),…,u(k)), where independent variables x=(x1,…,xn), dependent variables u and u(k) denote the collections of all kth-order partial derivatives, then (3) becomes an approximate differential equation of k, and G is an approximate symmetry group of the differential equation.Theorem 1 (see [5]).
Equation (3) is approximately invariant under the approximate transformation group (2) with the generator(5)X=X0+εX1≡ξ0iz∂∂zi+εξ1iz∂∂zi,if and only if(6)XkFz,εF≈0=oε,or(7)X0kF0z+εX1kF0z+X0kF1z3=oεin which k is order of equation and X(k) is k-th order prolongation of X. The operator (5) satisfying (7) is called an infinitesimal approximate symmetry or an approximate operator admitted by (3). Accordingly, (7) is termed the determining equation for approximate symmetries.Remark 2.
The determining equation (7) can be written as follows:(8)X0kF0z=λzF0z,(9)X1kF0z+X0kF1z=λzF1z.The factor λ(z) is determined by (8) and then substituted in (9). The latter equation must hold for all solutions of F0(z)=0. Comparing (8) with the determining equation of exact symmetries, we obtain the following statement.Theorem 3 (see [5]).
If (3) admits an approximate transformation group with the generator X=X0+εX1, where X0≠0, then the operator(10)X0=ξ0i∂∂ziis an exact symmetry of (11)F0z=0.Remark 4.
It is manifested from (8) and (9) that if X0 is an exact symmetry of (11), then X=εX0 is an approximate symmetry of (3).Definition 5 (see [5]).
Equations (11) and (3) are termed an unperturbed equation and a perturbed equation, respectively. Under the conditions of Theorem 3, the operator X0 is called a stable symmetry of the unperturbed equation (11). The corresponding approximate symmetry generator X=X0+εX1 for the perturbed equation (10) is called a deformation of the infinitesimal symmetry X0 of (11) caused by the perturbation εF1(x). In particular, if the most general symmetry Lie algebra of Eq.(11) is stable, we say that the perturbed equation (3) inherits the symmetries of the unperturbed equation.


## 3. Approximate Symmetry Analysis

### 3.1. Exact Symmetries

Let us write the approximate group generator in the form(12)X=X0+εX1=ξ0+εξ1∂∂x+τ0+ετ1∂∂y+η0+εη1∂∂u,where ξi, τi, and ηi(i=0,1) are unknown functions of x, t, and u.Solving the determining equation(13)X0kF0zF0z=0=0,for the exact symmetries of the unperturbed equation ut-6uux+uxxx=0, we can get(14)η0t=0,η0x=0,τ0x=0,τ0u=0,η0u+2ξ0x=0,ξ0xx=0,ξ0u=0,6η0+ξ0t+12uξ0x=0,3ξ0x-τ0t=0.Then, we obtain ξ0=c1x-6c2t+c3,τ0=3c1t+c2,η0=-2c1u+c2, where c1,c2,c3,c4 are arbitrary constants. Hence,(15)X0=c1x-6c2t+c3∂∂x+3c1t+c4∂∂t+-2c1u+c2∂∂u.In other words, ut-6uux+uxxx=0 admits the four-dimensional Lie algebra with the basis (16)X01=x∂∂x+3t∂∂t-2u∂∂u,X02=-6t∂∂x+∂∂u,X03=∂∂x,X04=∂∂u.


### 3.2. Approximate Symmetries and Optimal System

#### 3.2.1. Approximate Symmetries: The Casesα and β Are Arbitrary

First we need to determine the auxiliary functionH by virtue of (15), by (17)H=1εX0kF0z+εF1zF0z+εF1z=0.Substituting the expression (15) of the generator X0 into above equation we obtain the auxiliary function:(18)H=c2α-2c2β+3c1α-6c1βu+2c3α-12c2αt+6c1αx-c3β+6c2βt-3c1βxux.Now, calculate the operators X1 by solving the inhomogeneous determining equation for deformations: (19)X1kF0zF0z=0+H=0.Above determining equation yields(20)c2α-2c2β-6c1αu-6c1βu+η1t=0,2c1α-2η1x=0,τ1x=0,η1u=0,η1u+2ξ1x=0,ξ1xx=0,τ1u=0,3ξ1x-τ1t=0,2c3α-c3β+6c1αx-3c1βx-12c2αt+6c2βt-6η1-ξ1t-12uξ1x=0.Solving this system, we obtain (21)X1=-3c1α+c1βtx+2c3α-c3β-6d1t-3c2α+c2βt2+d2∂∂x+-9c1α+c1βt22+d3∂∂t+3c1αx2+2c2β+6c1αu-c2α+6c1βut+d1∂∂u.Then, we obtain the following approximate symmetries of (1)(22)X=c1x-6c2t+c3+ε-3c1α+c1βtx+2c3α-c3β-6d1t-3c2α+c2βt2+d2∂∂x+3c1t+c2+ε-9c1α+c1βt22+d3∂∂t+-2c1u+c2+ε3c1αx2+2c2β+6c1αu-c2α+6c1βut+d1∂∂u,and we have(23)v1=x∂∂x+3t∂∂t-2u∂∂u+ε-3α+βtx∂∂x-92α+βt2∂∂t+3αx2+6αut+6βut∂∂u,v2=-6t∂∂x+∂∂u+ε-3α+βt2∂∂x+2β-αt∂∂u,v3=∂∂x+ε2α-βt∂∂x,v4=∂∂u,v5=εv2,v6=εv3,v7=εv4.Tables1, 2, and 3 of commutator, evaluated in the first-order of precision, show that above operators span a seven-dimensional approximate Lie algebra L7.Table 1
Approximate commutators of approximate symmetry algebra of (1) (a=2α-β, b=2β-α, c=α+β, d=3α/2).
v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 1 0 2 v 2 - d v 5 - v 3 - 3 v 4 + 3 ε c v 1 2 v 5 - v 6 - 3 v 7 v 2 - 2 v 2 0 0 6 v 3 - b v 5 0 0 6 v 6 v 3 d v 5 + v 3 0 0 - a v 6 0 0 0 v 4 3 v 4 - 3 ε c v 1 - 6 v 3 + b v 5 a v 6 0 - 6 v 6 0 0 v 5 - 2 v 5 0 0 6 v 6 0 0 0 v 6 v 6 0 0 0 0 0 0 v 7 3 v 7 - 6 v 6 0 0 0 0 0Table 2
Approximate commutators of approximate symmetry algebra of (1) (α=2β, a=3α).
v 1 v 2 v 3 v 4 v 5 v 6 v 7 v1 0 2 v 2 - a 2 v 5 - v 3 - 3 v 4 + 3 ε a v 1 2 v 5 - v 6 - 3 v 7 v2 - 2 v 2 0 0 - 6 v 3 + 5 2 v 5 0 0 6 v 6 v3 v 3 + 5 2 v 5 0 0 - a v 6 0 0 0 v4 3 v 4 - 3 ε a v 1 6 v 3 - 5 2 v 5 a v 6 0 - 6 v 6 0 0 v5 - 2 v 5 0 0 6 v 6 0 0 0 v6 v 6 0 0 0 0 0 0 v7 3 v 7 - 6 v 6 0 0 0 0 0Table 3
Approximate commutators of approximate symmetry algebra of (1) ( α=-β, a=3α).
v 1 v 2 v 3 v 4 v 5 v 6 v 7 v1 0 2 v 2 - a 2 v 5 - v 3 - 3 v 4 2 v 5 - v 6 - 3 v 7 v2 - 2 v 2 0 0 6 v 3 + a v 5 0 0 6 v 6 v3 a 2 v 5 + v 3 0 0 - a v 6 0 0 0 v4 3 v 4 - 6 v 3 - a v 5 a v 6 0 - 6 v 6 0 0 v5 - 2 v 5 0 0 6 v 6 0 0 0 v6 v 6 0 0 0 0 0 0 v7 3 v 7 - 6 v 6 0 0 0 0 0


#### 3.2.2. Approximate Symmetries: The Caseα=2β

Whenα=2β, the equation admits seven-dimensional approximate Lie algebra L7 as follows:(24)v1=x∂∂x+3t∂∂t-2u∂∂u+ε-9αtx∂∂x-272αt2∂∂t+3αx2+6αut+6βut∂∂u,v2=-6t∂∂x+∂∂u+ε-9αt2∂∂x,v3=∂∂x+ε3αt∂∂x,v4=∂∂u,v5=εv2,v6=εv3,v7=εv4.


#### 3.2.3. Approximate Symmetries and Optimal System: The Caseα=-β

Whenα=-β, the equation admits seven-dimensional approximate Lie algebra L7 as follows:(25)v1=x∂∂x+3t∂∂t-2u∂∂u+ε3αx2∂∂u,v2=-6t∂∂x+∂∂u+ε-3αt∂∂u,v3=∂∂x+ε3αt∂∂x,v4=∂∂u,v5=εv2,v6=εv3,v7=εv4.It is worth nothing that the seven-dimensional approximate Lie algebraL7=g is solvable and its finite sequence of ideals is as follows: (26)0⊂v6⊂v5,v6⊂v3,v5,v6⊂v2,v3,v5,v6⊂v2,v3,v4,v5,v6⊂v2,v3,v4,v5,v6,v7⊂gIn the following, we will construct the optimal system of above Lie algebraL7. The method used here for obtaining the one-dimensional optimal system of subalgebras is that given in [7]. This approach is taking a general element from the Lie algebra and reducing it to its simplest equivalent form by applying carefully chosen adjoint transformations that are defined as follows.Definition 6 (see [7]).
LetG be a Lie group. An optimal system of s-parameter subgroups is a list of conjugacy inequivalent s-parameter subalgebras with the property that any other subgroup is conjugate to precisely one subgroup in the list. Similarly, a list of s-parameter subalgebras forms an optimal system if every s-parameter subalgebra of g is equivalent to a unique member of the list under some element of the adjoint representation: h~=Ad(g(h)), g∈G.Theorem 7 (see [7]).
LetH and H~ be connected s-dimensional Lie subgroups of the Lie group G with corresponding Lie subalgebras h and h~ of the Lie algebra g of G. Then H=gHg-1 are conjugate subgroups if and only if h~=Ad(r(h)) are conjugate subalgebras.By Theorem7, the problem of finding an optimal system of subgroups is equivalent to that of finding an optimal system of subalgebras. To compute the adjoint representation, we use the Lie series(27)Adexp⁡μvivj=vj-μvi,vj+μ22vi,vi,vj-⋯=exp⁡ad-μvivj,where [vi,vj] is the commutator for the Lie algebra and μ is a parameter,and i,j=1,…,7. In this manner, we construct Table 4 with the (i,j)th entry indicating Ad(exp⁡(μvi))vj.Table 4
Adjoint representation of approximate symmetry of (1).
Ad v 1 v 2 v 3 v 4 v 5 v 6 v 7 v1 v 1 e - 2 μ v 2 e - μ v 3 + a 2 e - μ v 5 - a 2 v 5 e 3 μ v 4 e - 2 μ v 5 e - μ v 6 e 3 μ v 7 v2 v 1 + 2 μ v 2 v 2 v 3 v 4 - μ 6 v 3 + a v 5 v 5 v 6 v 7 - 6 μ v 6 v3 v1-μav52+v3 v 2 v 3 v 4 + μ ( a v 6 ) v 5 v 6 v 7 v4 v 1 - 3 μ v 4 v 2 + μ ( 6 v 3 + a v 5 ) v 3 - a μ v 6 v 4 v 5 + 6 μ v 6 v 6 v 7 v5 v 1 + 2 μ v 5 v 2 v 3 v 4 - 6 μ v 6 v 5 v 6 v 7 v6 v 1 - μ v 6 v 2 v 3 v 4 v 5 v 6 v 7 v7 v 1 - 3 μ v 7 v 2 + 6 μ v 7 v 3 v 4 v 5 v 6 v 7Theorem 8.
An optimal system of one-dimensional approximate symmetry algebra (caseα=-β) of equation (1) is provided by v7,v6,v1+bv6,v3+bv7,v2+bv5+cv7,v5+bv3+cv7,v4+bv2+cv5+dv7.Proof.
Considering the approximate symmetry algebra g of (1), whose adjoint representation was determined in the Table 4, our task is to simplify as many of the coefficients ai as possible through judicious applications of adjoint maps to Vi, so that Vi is equivalent Vi′ under the adjoint representation.
Given a nonzero vector(28)V1=a1v1+a2v2+a3v3+a4v4+a5v5+a6v6+a7v7.First, suppose that a1≠0. Scaling V1 if necessary, we can assume that a1=1. As for the 7th column of the Table 4, we have(29)V1′=Adexp⁡a43v4∘Adexp⁡-a22v2∘Adexp⁡a73v7∘Adexp⁡a3+2a2a4v3∘Adexp⁡-a52+aa34+aa2a412v5V1=v1+a6-aa3a43+2a4a5+3a2a7v6.The remaining approximate one-dimensional subalgebras are spanned by vectors of the above form with a1=0. If a4≠0, we have (30)V2=a2v2+a3v3+v4+a5v5+a6v6+a7v7.Next, we act on V2 to cancel the coefficients of v3 and v6 as follows: (31)V2′=Adexp⁡-a36a2v4∘Adexp⁡-a56a2+aa3236a2+a66v5V2.=a2v2+v4+a5-aa36v5+a7v7.If a1,a4=0 and a2≠0, the nonzero vector (32)V3=v2+a3v3+a5v5+a6v6+a7v7is equivalent to (33)V3′=Adexp⁡-a36v4∘Adexp⁡-a66+-aa3236+a6a56v7V3=v2+a5-aa36v5+a7v7.If a1,a2,a4=0 and a5≠0, we scale to make a5=1 and then (34)V4=a3v3+v5+a6v6+a7v7is equivalent to V4′ under the adjoint representation (35)V4′=Adexp⁡a66a7v2V4=a3v3+v5+a7v7.If a1,a2,a4,a5=0 and a3≠0, in the same way as before, the nonzero vector (36)V5=v3+a6v6+a7v7can be simplified as (37)V5′=Adexp⁡a66a7v2V2=v3+a7v7.If a1,a2,a3,a4,a5=0 and a7≠0, we can act (38)V6=a6v6+v7by Ad(exp⁡(((a6/6)v2))), to cancel the coefficient of v6, leading to (39)V6′=Adexp⁡a66v2V6=v7.The last remaining case occurs when a1,a2,a3,a4,a5,a7=0 and a6≠0, for which our earlier simplifications were unnecessary, because the only remaining vectors are the multiples of a6, on which the adjoint representation acts trivially.


## 3.1. Exact Symmetries

Let us write the approximate group generator in the form(12)X=X0+εX1=ξ0+εξ1∂∂x+τ0+ετ1∂∂y+η0+εη1∂∂u,where ξi, τi, and ηi(i=0,1) are unknown functions of x, t, and u.Solving the determining equation(13)X0kF0zF0z=0=0,for the exact symmetries of the unperturbed equation ut-6uux+uxxx=0, we can get(14)η0t=0,η0x=0,τ0x=0,τ0u=0,η0u+2ξ0x=0,ξ0xx=0,ξ0u=0,6η0+ξ0t+12uξ0x=0,3ξ0x-τ0t=0.Then, we obtain ξ0=c1x-6c2t+c3,τ0=3c1t+c2,η0=-2c1u+c2, where c1,c2,c3,c4 are arbitrary constants. Hence,(15)X0=c1x-6c2t+c3∂∂x+3c1t+c4∂∂t+-2c1u+c2∂∂u.In other words, ut-6uux+uxxx=0 admits the four-dimensional Lie algebra with the basis (16)X01=x∂∂x+3t∂∂t-2u∂∂u,X02=-6t∂∂x+∂∂u,X03=∂∂x,X04=∂∂u.


## 3.2. Approximate Symmetries and Optimal System

### 3.2.1. Approximate Symmetries: The Casesα and β Are Arbitrary

First we need to determine the auxiliary functionH by virtue of (15), by (17)H=1εX0kF0z+εF1zF0z+εF1z=0.Substituting the expression (15) of the generator X0 into above equation we obtain the auxiliary function:(18)H=c2α-2c2β+3c1α-6c1βu+2c3α-12c2αt+6c1αx-c3β+6c2βt-3c1βxux.Now, calculate the operators X1 by solving the inhomogeneous determining equation for deformations: (19)X1kF0zF0z=0+H=0.Above determining equation yields(20)c2α-2c2β-6c1αu-6c1βu+η1t=0,2c1α-2η1x=0,τ1x=0,η1u=0,η1u+2ξ1x=0,ξ1xx=0,τ1u=0,3ξ1x-τ1t=0,2c3α-c3β+6c1αx-3c1βx-12c2αt+6c2βt-6η1-ξ1t-12uξ1x=0.Solving this system, we obtain (21)X1=-3c1α+c1βtx+2c3α-c3β-6d1t-3c2α+c2βt2+d2∂∂x+-9c1α+c1βt22+d3∂∂t+3c1αx2+2c2β+6c1αu-c2α+6c1βut+d1∂∂u.Then, we obtain the following approximate symmetries of (1)(22)X=c1x-6c2t+c3+ε-3c1α+c1βtx+2c3α-c3β-6d1t-3c2α+c2βt2+d2∂∂x+3c1t+c2+ε-9c1α+c1βt22+d3∂∂t+-2c1u+c2+ε3c1αx2+2c2β+6c1αu-c2α+6c1βut+d1∂∂u,and we have(23)v1=x∂∂x+3t∂∂t-2u∂∂u+ε-3α+βtx∂∂x-92α+βt2∂∂t+3αx2+6αut+6βut∂∂u,v2=-6t∂∂x+∂∂u+ε-3α+βt2∂∂x+2β-αt∂∂u,v3=∂∂x+ε2α-βt∂∂x,v4=∂∂u,v5=εv2,v6=εv3,v7=εv4.Tables1, 2, and 3 of commutator, evaluated in the first-order of precision, show that above operators span a seven-dimensional approximate Lie algebra L7.Table 1
Approximate commutators of approximate symmetry algebra of (1) (a=2α-β, b=2β-α, c=α+β, d=3α/2).
v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 1 0 2 v 2 - d v 5 - v 3 - 3 v 4 + 3 ε c v 1 2 v 5 - v 6 - 3 v 7 v 2 - 2 v 2 0 0 6 v 3 - b v 5 0 0 6 v 6 v 3 d v 5 + v 3 0 0 - a v 6 0 0 0 v 4 3 v 4 - 3 ε c v 1 - 6 v 3 + b v 5 a v 6 0 - 6 v 6 0 0 v 5 - 2 v 5 0 0 6 v 6 0 0 0 v 6 v 6 0 0 0 0 0 0 v 7 3 v 7 - 6 v 6 0 0 0 0 0Table 2
Approximate commutators of approximate symmetry algebra of (1) (α=2β, a=3α).
v 1 v 2 v 3 v 4 v 5 v 6 v 7 v1 0 2 v 2 - a 2 v 5 - v 3 - 3 v 4 + 3 ε a v 1 2 v 5 - v 6 - 3 v 7 v2 - 2 v 2 0 0 - 6 v 3 + 5 2 v 5 0 0 6 v 6 v3 v 3 + 5 2 v 5 0 0 - a v 6 0 0 0 v4 3 v 4 - 3 ε a v 1 6 v 3 - 5 2 v 5 a v 6 0 - 6 v 6 0 0 v5 - 2 v 5 0 0 6 v 6 0 0 0 v6 v 6 0 0 0 0 0 0 v7 3 v 7 - 6 v 6 0 0 0 0 0Table 3
Approximate commutators of approximate symmetry algebra of (1) ( α=-β, a=3α).
v 1 v 2 v 3 v 4 v 5 v 6 v 7 v1 0 2 v 2 - a 2 v 5 - v 3 - 3 v 4 2 v 5 - v 6 - 3 v 7 v2 - 2 v 2 0 0 6 v 3 + a v 5 0 0 6 v 6 v3 a 2 v 5 + v 3 0 0 - a v 6 0 0 0 v4 3 v 4 - 6 v 3 - a v 5 a v 6 0 - 6 v 6 0 0 v5 - 2 v 5 0 0 6 v 6 0 0 0 v6 v 6 0 0 0 0 0 0 v7 3 v 7 - 6 v 6 0 0 0 0 0


### 3.2.2. Approximate Symmetries: The Caseα=2β

Whenα=2β, the equation admits seven-dimensional approximate Lie algebra L7 as follows:(24)v1=x∂∂x+3t∂∂t-2u∂∂u+ε-9αtx∂∂x-272αt2∂∂t+3αx2+6αut+6βut∂∂u,v2=-6t∂∂x+∂∂u+ε-9αt2∂∂x,v3=∂∂x+ε3αt∂∂x,v4=∂∂u,v5=εv2,v6=εv3,v7=εv4.


### 3.2.3. Approximate Symmetries and Optimal System: The Caseα=-β

Whenα=-β, the equation admits seven-dimensional approximate Lie algebra L7 as follows:(25)v1=x∂∂x+3t∂∂t-2u∂∂u+ε3αx2∂∂u,v2=-6t∂∂x+∂∂u+ε-3αt∂∂u,v3=∂∂x+ε3αt∂∂x,v4=∂∂u,v5=εv2,v6=εv3,v7=εv4.It is worth nothing that the seven-dimensional approximate Lie algebraL7=g is solvable and its finite sequence of ideals is as follows: (26)0⊂v6⊂v5,v6⊂v3,v5,v6⊂v2,v3,v5,v6⊂v2,v3,v4,v5,v6⊂v2,v3,v4,v5,v6,v7⊂gIn the following, we will construct the optimal system of above Lie algebraL7. The method used here for obtaining the one-dimensional optimal system of subalgebras is that given in [7]. This approach is taking a general element from the Lie algebra and reducing it to its simplest equivalent form by applying carefully chosen adjoint transformations that are defined as follows.Definition 6 (see [7]).
LetG be a Lie group. An optimal system of s-parameter subgroups is a list of conjugacy inequivalent s-parameter subalgebras with the property that any other subgroup is conjugate to precisely one subgroup in the list. Similarly, a list of s-parameter subalgebras forms an optimal system if every s-parameter subalgebra of g is equivalent to a unique member of the list under some element of the adjoint representation: h~=Ad(g(h)), g∈G.Theorem 7 (see [7]).
LetH and H~ be connected s-dimensional Lie subgroups of the Lie group G with corresponding Lie subalgebras h and h~ of the Lie algebra g of G. Then H=gHg-1 are conjugate subgroups if and only if h~=Ad(r(h)) are conjugate subalgebras.By Theorem7, the problem of finding an optimal system of subgroups is equivalent to that of finding an optimal system of subalgebras. To compute the adjoint representation, we use the Lie series(27)Adexp⁡μvivj=vj-μvi,vj+μ22vi,vi,vj-⋯=exp⁡ad-μvivj,where [vi,vj] is the commutator for the Lie algebra and μ is a parameter,and i,j=1,…,7. In this manner, we construct Table 4 with the (i,j)th entry indicating Ad(exp⁡(μvi))vj.Table 4
Adjoint representation of approximate symmetry of (1).
Ad v 1 v 2 v 3 v 4 v 5 v 6 v 7 v1 v 1 e - 2 μ v 2 e - μ v 3 + a 2 e - μ v 5 - a 2 v 5 e 3 μ v 4 e - 2 μ v 5 e - μ v 6 e 3 μ v 7 v2 v 1 + 2 μ v 2 v 2 v 3 v 4 - μ 6 v 3 + a v 5 v 5 v 6 v 7 - 6 μ v 6 v3 v1-μav52+v3 v 2 v 3 v 4 + μ ( a v 6 ) v 5 v 6 v 7 v4 v 1 - 3 μ v 4 v 2 + μ ( 6 v 3 + a v 5 ) v 3 - a μ v 6 v 4 v 5 + 6 μ v 6 v 6 v 7 v5 v 1 + 2 μ v 5 v 2 v 3 v 4 - 6 μ v 6 v 5 v 6 v 7 v6 v 1 - μ v 6 v 2 v 3 v 4 v 5 v 6 v 7 v7 v 1 - 3 μ v 7 v 2 + 6 μ v 7 v 3 v 4 v 5 v 6 v 7Theorem 8.
An optimal system of one-dimensional approximate symmetry algebra (caseα=-β) of equation (1) is provided by v7,v6,v1+bv6,v3+bv7,v2+bv5+cv7,v5+bv3+cv7,v4+bv2+cv5+dv7.Proof.
Considering the approximate symmetry algebra g of (1), whose adjoint representation was determined in the Table 4, our task is to simplify as many of the coefficients ai as possible through judicious applications of adjoint maps to Vi, so that Vi is equivalent Vi′ under the adjoint representation.
Given a nonzero vector(28)V1=a1v1+a2v2+a3v3+a4v4+a5v5+a6v6+a7v7.First, suppose that a1≠0. Scaling V1 if necessary, we can assume that a1=1. As for the 7th column of the Table 4, we have(29)V1′=Adexp⁡a43v4∘Adexp⁡-a22v2∘Adexp⁡a73v7∘Adexp⁡a3+2a2a4v3∘Adexp⁡-a52+aa34+aa2a412v5V1=v1+a6-aa3a43+2a4a5+3a2a7v6.The remaining approximate one-dimensional subalgebras are spanned by vectors of the above form with a1=0. If a4≠0, we have (30)V2=a2v2+a3v3+v4+a5v5+a6v6+a7v7.Next, we act on V2 to cancel the coefficients of v3 and v6 as follows: (31)V2′=Adexp⁡-a36a2v4∘Adexp⁡-a56a2+aa3236a2+a66v5V2.=a2v2+v4+a5-aa36v5+a7v7.If a1,a4=0 and a2≠0, the nonzero vector (32)V3=v2+a3v3+a5v5+a6v6+a7v7is equivalent to (33)V3′=Adexp⁡-a36v4∘Adexp⁡-a66+-aa3236+a6a56v7V3=v2+a5-aa36v5+a7v7.If a1,a2,a4=0 and a5≠0, we scale to make a5=1 and then (34)V4=a3v3+v5+a6v6+a7v7is equivalent to V4′ under the adjoint representation (35)V4′=Adexp⁡a66a7v2V4=a3v3+v5+a7v7.If a1,a2,a4,a5=0 and a3≠0, in the same way as before, the nonzero vector (36)V5=v3+a6v6+a7v7can be simplified as (37)V5′=Adexp⁡a66a7v2V2=v3+a7v7.If a1,a2,a3,a4,a5=0 and a7≠0, we can act (38)V6=a6v6+v7by Ad(exp⁡(((a6/6)v2))), to cancel the coefficient of v6, leading to (39)V6′=Adexp⁡a66v2V6=v7.The last remaining case occurs when a1,a2,a3,a4,a5,a7=0 and a6≠0, for which our earlier simplifications were unnecessary, because the only remaining vectors are the multiples of a6, on which the adjoint representation acts trivially.


## 3.2.1. Approximate Symmetries: The Casesα and β Are Arbitrary

First we need to determine the auxiliary functionH by virtue of (15), by (17)H=1εX0kF0z+εF1zF0z+εF1z=0.Substituting the expression (15) of the generator X0 into above equation we obtain the auxiliary function:(18)H=c2α-2c2β+3c1α-6c1βu+2c3α-12c2αt+6c1αx-c3β+6c2βt-3c1βxux.Now, calculate the operators X1 by solving the inhomogeneous determining equation for deformations: (19)X1kF0zF0z=0+H=0.Above determining equation yields(20)c2α-2c2β-6c1αu-6c1βu+η1t=0,2c1α-2η1x=0,τ1x=0,η1u=0,η1u+2ξ1x=0,ξ1xx=0,τ1u=0,3ξ1x-τ1t=0,2c3α-c3β+6c1αx-3c1βx-12c2αt+6c2βt-6η1-ξ1t-12uξ1x=0.Solving this system, we obtain (21)X1=-3c1α+c1βtx+2c3α-c3β-6d1t-3c2α+c2βt2+d2∂∂x+-9c1α+c1βt22+d3∂∂t+3c1αx2+2c2β+6c1αu-c2α+6c1βut+d1∂∂u.Then, we obtain the following approximate symmetries of (1)(22)X=c1x-6c2t+c3+ε-3c1α+c1βtx+2c3α-c3β-6d1t-3c2α+c2βt2+d2∂∂x+3c1t+c2+ε-9c1α+c1βt22+d3∂∂t+-2c1u+c2+ε3c1αx2+2c2β+6c1αu-c2α+6c1βut+d1∂∂u,and we have(23)v1=x∂∂x+3t∂∂t-2u∂∂u+ε-3α+βtx∂∂x-92α+βt2∂∂t+3αx2+6αut+6βut∂∂u,v2=-6t∂∂x+∂∂u+ε-3α+βt2∂∂x+2β-αt∂∂u,v3=∂∂x+ε2α-βt∂∂x,v4=∂∂u,v5=εv2,v6=εv3,v7=εv4.Tables1, 2, and 3 of commutator, evaluated in the first-order of precision, show that above operators span a seven-dimensional approximate Lie algebra L7.Table 1
Approximate commutators of approximate symmetry algebra of (1) (a=2α-β, b=2β-α, c=α+β, d=3α/2).
v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 1 0 2 v 2 - d v 5 - v 3 - 3 v 4 + 3 ε c v 1 2 v 5 - v 6 - 3 v 7 v 2 - 2 v 2 0 0 6 v 3 - b v 5 0 0 6 v 6 v 3 d v 5 + v 3 0 0 - a v 6 0 0 0 v 4 3 v 4 - 3 ε c v 1 - 6 v 3 + b v 5 a v 6 0 - 6 v 6 0 0 v 5 - 2 v 5 0 0 6 v 6 0 0 0 v 6 v 6 0 0 0 0 0 0 v 7 3 v 7 - 6 v 6 0 0 0 0 0Table 2
Approximate commutators of approximate symmetry algebra of (1) (α=2β, a=3α).
v 1 v 2 v 3 v 4 v 5 v 6 v 7 v1 0 2 v 2 - a 2 v 5 - v 3 - 3 v 4 + 3 ε a v 1 2 v 5 - v 6 - 3 v 7 v2 - 2 v 2 0 0 - 6 v 3 + 5 2 v 5 0 0 6 v 6 v3 v 3 + 5 2 v 5 0 0 - a v 6 0 0 0 v4 3 v 4 - 3 ε a v 1 6 v 3 - 5 2 v 5 a v 6 0 - 6 v 6 0 0 v5 - 2 v 5 0 0 6 v 6 0 0 0 v6 v 6 0 0 0 0 0 0 v7 3 v 7 - 6 v 6 0 0 0 0 0Table 3
Approximate commutators of approximate symmetry algebra of (1) ( α=-β, a=3α).
v 1 v 2 v 3 v 4 v 5 v 6 v 7 v1 0 2 v 2 - a 2 v 5 - v 3 - 3 v 4 2 v 5 - v 6 - 3 v 7 v2 - 2 v 2 0 0 6 v 3 + a v 5 0 0 6 v 6 v3 a 2 v 5 + v 3 0 0 - a v 6 0 0 0 v4 3 v 4 - 6 v 3 - a v 5 a v 6 0 - 6 v 6 0 0 v5 - 2 v 5 0 0 6 v 6 0 0 0 v6 v 6 0 0 0 0 0 0 v7 3 v 7 - 6 v 6 0 0 0 0 0


## 3.2.2. Approximate Symmetries: The Caseα=2β

Whenα=2β, the equation admits seven-dimensional approximate Lie algebra L7 as follows:(24)v1=x∂∂x+3t∂∂t-2u∂∂u+ε-9αtx∂∂x-272αt2∂∂t+3αx2+6αut+6βut∂∂u,v2=-6t∂∂x+∂∂u+ε-9αt2∂∂x,v3=∂∂x+ε3αt∂∂x,v4=∂∂u,v5=εv2,v6=εv3,v7=εv4.


## 3.2.3. Approximate Symmetries and Optimal System: The Caseα=-β

Whenα=-β, the equation admits seven-dimensional approximate Lie algebra L7 as follows:(25)v1=x∂∂x+3t∂∂t-2u∂∂u+ε3αx2∂∂u,v2=-6t∂∂x+∂∂u+ε-3αt∂∂u,v3=∂∂x+ε3αt∂∂x,v4=∂∂u,v5=εv2,v6=εv3,v7=εv4.It is worth nothing that the seven-dimensional approximate Lie algebraL7=g is solvable and its finite sequence of ideals is as follows: (26)0⊂v6⊂v5,v6⊂v3,v5,v6⊂v2,v3,v5,v6⊂v2,v3,v4,v5,v6⊂v2,v3,v4,v5,v6,v7⊂gIn the following, we will construct the optimal system of above Lie algebraL7. The method used here for obtaining the one-dimensional optimal system of subalgebras is that given in [7]. This approach is taking a general element from the Lie algebra and reducing it to its simplest equivalent form by applying carefully chosen adjoint transformations that are defined as follows.Definition 6 (see [7]).
LetG be a Lie group. An optimal system of s-parameter subgroups is a list of conjugacy inequivalent s-parameter subalgebras with the property that any other subgroup is conjugate to precisely one subgroup in the list. Similarly, a list of s-parameter subalgebras forms an optimal system if every s-parameter subalgebra of g is equivalent to a unique member of the list under some element of the adjoint representation: h~=Ad(g(h)), g∈G.Theorem 7 (see [7]).
LetH and H~ be connected s-dimensional Lie subgroups of the Lie group G with corresponding Lie subalgebras h and h~ of the Lie algebra g of G. Then H=gHg-1 are conjugate subgroups if and only if h~=Ad(r(h)) are conjugate subalgebras.By Theorem7, the problem of finding an optimal system of subgroups is equivalent to that of finding an optimal system of subalgebras. To compute the adjoint representation, we use the Lie series(27)Adexp⁡μvivj=vj-μvi,vj+μ22vi,vi,vj-⋯=exp⁡ad-μvivj,where [vi,vj] is the commutator for the Lie algebra and μ is a parameter,and i,j=1,…,7. In this manner, we construct Table 4 with the (i,j)th entry indicating Ad(exp⁡(μvi))vj.Table 4
Adjoint representation of approximate symmetry of (1).
Ad v 1 v 2 v 3 v 4 v 5 v 6 v 7 v1 v 1 e - 2 μ v 2 e - μ v 3 + a 2 e - μ v 5 - a 2 v 5 e 3 μ v 4 e - 2 μ v 5 e - μ v 6 e 3 μ v 7 v2 v 1 + 2 μ v 2 v 2 v 3 v 4 - μ 6 v 3 + a v 5 v 5 v 6 v 7 - 6 μ v 6 v3 v1-μav52+v3 v 2 v 3 v 4 + μ ( a v 6 ) v 5 v 6 v 7 v4 v 1 - 3 μ v 4 v 2 + μ ( 6 v 3 + a v 5 ) v 3 - a μ v 6 v 4 v 5 + 6 μ v 6 v 6 v 7 v5 v 1 + 2 μ v 5 v 2 v 3 v 4 - 6 μ v 6 v 5 v 6 v 7 v6 v 1 - μ v 6 v 2 v 3 v 4 v 5 v 6 v 7 v7 v 1 - 3 μ v 7 v 2 + 6 μ v 7 v 3 v 4 v 5 v 6 v 7Theorem 8.
An optimal system of one-dimensional approximate symmetry algebra (caseα=-β) of equation (1) is provided by v7,v6,v1+bv6,v3+bv7,v2+bv5+cv7,v5+bv3+cv7,v4+bv2+cv5+dv7.Proof.
Considering the approximate symmetry algebra g of (1), whose adjoint representation was determined in the Table 4, our task is to simplify as many of the coefficients ai as possible through judicious applications of adjoint maps to Vi, so that Vi is equivalent Vi′ under the adjoint representation.
Given a nonzero vector(28)V1=a1v1+a2v2+a3v3+a4v4+a5v5+a6v6+a7v7.First, suppose that a1≠0. Scaling V1 if necessary, we can assume that a1=1. As for the 7th column of the Table 4, we have(29)V1′=Adexp⁡a43v4∘Adexp⁡-a22v2∘Adexp⁡a73v7∘Adexp⁡a3+2a2a4v3∘Adexp⁡-a52+aa34+aa2a412v5V1=v1+a6-aa3a43+2a4a5+3a2a7v6.The remaining approximate one-dimensional subalgebras are spanned by vectors of the above form with a1=0. If a4≠0, we have (30)V2=a2v2+a3v3+v4+a5v5+a6v6+a7v7.Next, we act on V2 to cancel the coefficients of v3 and v6 as follows: (31)V2′=Adexp⁡-a36a2v4∘Adexp⁡-a56a2+aa3236a2+a66v5V2.=a2v2+v4+a5-aa36v5+a7v7.If a1,a4=0 and a2≠0, the nonzero vector (32)V3=v2+a3v3+a5v5+a6v6+a7v7is equivalent to (33)V3′=Adexp⁡-a36v4∘Adexp⁡-a66+-aa3236+a6a56v7V3=v2+a5-aa36v5+a7v7.If a1,a2,a4=0 and a5≠0, we scale to make a5=1 and then (34)V4=a3v3+v5+a6v6+a7v7is equivalent to V4′ under the adjoint representation (35)V4′=Adexp⁡a66a7v2V4=a3v3+v5+a7v7.If a1,a2,a4,a5=0 and a3≠0, in the same way as before, the nonzero vector (36)V5=v3+a6v6+a7v7can be simplified as (37)V5′=Adexp⁡a66a7v2V2=v3+a7v7.If a1,a2,a3,a4,a5=0 and a7≠0, we can act (38)V6=a6v6+v7by Ad(exp⁡(((a6/6)v2))), to cancel the coefficient of v6, leading to (39)V6′=Adexp⁡a66v2V6=v7.The last remaining case occurs when a1,a2,a3,a4,a5,a7=0 and a6≠0, for which our earlier simplifications were unnecessary, because the only remaining vectors are the multiples of a6, on which the adjoint representation acts trivially.


## 4. Approximately Invariant Solutions

In this section we use two different techniques to construct new approximate solutions of (1) when α=-β.


### 4.1. Approximately Invariant Solutions I

In the beginning of this section we compute an approximately invariant solution based on theX=v2. The approximate invariants for X are determined by (40)XJ=-6t∂∂x+∂∂u+ε-3αt∂∂uJ0+εJ1=oε.Equivalently(41)-6t∂∂x+∂∂uJ0=0,-3αt∂∂uJ1+-6t∂∂x+∂∂uJ0=0.The first equation has two functionally independent solutions J0=t and J0=x/6t+u. The simplest solutions of the second equation are, respectively, J1=0 and J1=-αx/2+t. Therefore, we have two independent invariants to x/6t+u+ε(-αx/2+t) and φ(t) with respect to X.Lettingx/6t+u+ε(3αtu)=φ(t), we obtain u=φ(t)-x/6t-ε(-αx/2+t) for the approximately invariant solutions.Therefore, we can obtain(42)ux,t≈-x6t+Γ0t+ε-t-αx2+Γ1t+Γ0tΓ1′t.Putting (42) into (1), we obtain (43)Γ0t=C1TΓ1t=C2t3+t+C2t,where C1 and C2 are arbitrary constants. Hence, we obtained the approximately invariant solution to (1) (44)ux,t≈-x6t+C1t+εC2t-αx2.In this manner, we compute approximate invariants with respect to the generators of Lie algebra and optimal system, as shown in Table5.Table 5
Approximate invariants with respect to the generators of Lie algebra and optimal system.
Operator Approximate Invariants v1 t x 3, x2u-εαx32 v2 t,x6t+u+ε-αx2+t v3 t, u v4 x, u v5 t, x6t+u v6 t, u v7 x, u v1+bv6 t x 3 + ε b x - 4 t, x2u-εαx32+2bx3u v3+bv7 t - ε ( b x ), u v4+bv2+cv5+dv7 t 2 2 + x 6 b + ε c b - d t, bu-t+ε3αt22+dt v2+bv5+cv7 t - ε u c, x6t+u+ε3αtu-cx272t3 v5+bv3+cv7 t - ε c x b, u


### 4.2. Approximately Invariant Solution II

Now, we apply a different technique to find approximately invariant solutions for (1). We will begin with one exact solution(45)u=wζ=-121cosh2⁡ζof the unperturbed equation ut-6uux+uxxx=0. Here, ζ=(x-t)/2 is an invariant of the group with the generator(46)X0=∂∂t+∂∂x.The function w given by (45) is invariant under the operator (46).Using the generatorsX1=∂/∂x, admitted by ut-6uux+uxxx=0, we will take the approximate symmetry(47)X=X0+εX1=∂∂t+∂∂x+ε∂∂xand use it looking for the approximately invariant solution of (1) in the form(48)u=w+εvt,x.Then the invariant equation testX(u-w-εv)|48=o(ε) can be written as(49)X0u-w+εX1u-w-X0vu=w=0.Note that X0 does not contain the differentiation in u; therefore (49) becomes (50)X1u-w-X0vu=w=0whence we obtain the following differential for v(t,x): (51)vt+vx=-wx.Because w=w(ζ), so (51) can be integrated by the variables (52)ζ=x-t2,ρ=t.Then, denoting by w′ the derivative of w with respect to ζ, we have (53)wt=-12w′,wx=12w′and (51) becomes (54)vρ=-12w′.The integration yields (55)v=-12ρw′+ψx.Returning to the variables t, x, we have (56)v=-twx+ψx.Inserting this v in (48) and substituting it into (1) we obtain ψ=(-1-3β/6)x.Thus, the approximate symmetry (47) provides the following approximately invariant solution (approximate travelling wave): (57)ux,t=-12·1cosh2⁡x-t/2+ε-t21cosh2⁡x-t/2·tanh⁡x-t2+-1-3β6x.


## 4.1. Approximately Invariant Solutions I

In the beginning of this section we compute an approximately invariant solution based on theX=v2. The approximate invariants for X are determined by (40)XJ=-6t∂∂x+∂∂u+ε-3αt∂∂uJ0+εJ1=oε.Equivalently(41)-6t∂∂x+∂∂uJ0=0,-3αt∂∂uJ1+-6t∂∂x+∂∂uJ0=0.The first equation has two functionally independent solutions J0=t and J0=x/6t+u. The simplest solutions of the second equation are, respectively, J1=0 and J1=-αx/2+t. Therefore, we have two independent invariants to x/6t+u+ε(-αx/2+t) and φ(t) with respect to X.Lettingx/6t+u+ε(3αtu)=φ(t), we obtain u=φ(t)-x/6t-ε(-αx/2+t) for the approximately invariant solutions.Therefore, we can obtain(42)ux,t≈-x6t+Γ0t+ε-t-αx2+Γ1t+Γ0tΓ1′t.Putting (42) into (1), we obtain (43)Γ0t=C1TΓ1t=C2t3+t+C2t,where C1 and C2 are arbitrary constants. Hence, we obtained the approximately invariant solution to (1) (44)ux,t≈-x6t+C1t+εC2t-αx2.In this manner, we compute approximate invariants with respect to the generators of Lie algebra and optimal system, as shown in Table5.Table 5
Approximate invariants with respect to the generators of Lie algebra and optimal system.
Operator Approximate Invariants v1 t x 3, x2u-εαx32 v2 t,x6t+u+ε-αx2+t v3 t, u v4 x, u v5 t, x6t+u v6 t, u v7 x, u v1+bv6 t x 3 + ε b x - 4 t, x2u-εαx32+2bx3u v3+bv7 t - ε ( b x ), u v4+bv2+cv5+dv7 t 2 2 + x 6 b + ε c b - d t, bu-t+ε3αt22+dt v2+bv5+cv7 t - ε u c, x6t+u+ε3αtu-cx272t3 v5+bv3+cv7 t - ε c x b, u


## 4.2. Approximately Invariant Solution II

Now, we apply a different technique to find approximately invariant solutions for (1). We will begin with one exact solution(45)u=wζ=-121cosh2⁡ζof the unperturbed equation ut-6uux+uxxx=0. Here, ζ=(x-t)/2 is an invariant of the group with the generator(46)X0=∂∂t+∂∂x.The function w given by (45) is invariant under the operator (46).Using the generatorsX1=∂/∂x, admitted by ut-6uux+uxxx=0, we will take the approximate symmetry(47)X=X0+εX1=∂∂t+∂∂x+ε∂∂xand use it looking for the approximately invariant solution of (1) in the form(48)u=w+εvt,x.Then the invariant equation testX(u-w-εv)|48=o(ε) can be written as(49)X0u-w+εX1u-w-X0vu=w=0.Note that X0 does not contain the differentiation in u; therefore (49) becomes (50)X1u-w-X0vu=w=0whence we obtain the following differential for v(t,x): (51)vt+vx=-wx.Because w=w(ζ), so (51) can be integrated by the variables (52)ζ=x-t2,ρ=t.Then, denoting by w′ the derivative of w with respect to ζ, we have (53)wt=-12w′,wx=12w′and (51) becomes (54)vρ=-12w′.The integration yields (55)v=-12ρw′+ψx.Returning to the variables t, x, we have (56)v=-twx+ψx.Inserting this v in (48) and substituting it into (1) we obtain ψ=(-1-3β/6)x.Thus, the approximate symmetry (47) provides the following approximately invariant solution (approximate travelling wave): (57)ux,t=-12·1cosh2⁡x-t/2+ε-t21cosh2⁡x-t/2·tanh⁡x-t2+-1-3β6x.


## 5. Approximate Conservation Laws

Approximate Lie symmetry can be used to construct the approximate conservation laws, but in this section, we will use partial Lagrangians to construct approximate conservation laws of (1); this is a more concise method.Definition 9 (see [10]).
An operatorY is a kth-order approximate Lie-Bäcklund symmetry:(58)Y=Y0+εY1+⋯+εkYk,where(59)Y=ξi∂∂xi+ηi∂∂uα,i=1,2,…,nand(60)Yb=ξbi∂∂xi+ηbα∂∂uα+ζb,iα∂∂uiα+ζb,i1i2α∂∂ui1i2α+⋯,b=0,…,k,where ξbi,ηbα∈A and the additional coefficients are(61)ζb,iα=DiWbα+ξbjuijα,ζb,i1i2α=Di1Di2Wbα+ξbjuji1i2α,i,j=1,…,nand Wbα is the Lie characteristic function defined by(62)Wbα=ηbα-ξbjujα.Definition 10 (see [10]).
The approximate Noether operator associated with an approximate Lie-Bäcklund symmetry operatorY is given by (63)Ni=ξi+Wαδδuα+∑s≥1Di1…DisWαδδuii1i2…isα,i=1,…,n,where (64)Ni=N0i+εN1i+⋯+εkNki,i=1,…,n,where Di is the total derivative operator defined as (65)Di=∂∂xi+ui∂∂u+uij∂∂uj+⋯+uii1i2…in∂∂ui1i2…in+⋯,i=1,2,…,n,and here Nbi,b=0,…,k are Noether operators and the Euler-Lagrange operators are (66)δδuα=∂∂uα+∑s≥1-1sDj1…Djs∂∂uij1j2…jsα,α=1,…,m,i=1,…,n.The Euler-Lagrange, approximate Lie-Bäcklund, and approximate Noether operators are connected by the operator identity (67)Y+Diξi=Wαδδuα+DiNi.Definition 11 (see [10]).
If there exists a functionL=L(x,u,u(1),…,u(l)∈A,l<r and nonzero functions fγβ∈A such that (1) which can be written as δL/δuβ=εfγβE1γ, where fγβ=fγβ(x,u,u(1),…,u(r-1),β,γ=1,…,m is an invertible matrix, then, provided that E1γ≠0, some L is called a partial Lagrangian; otherwise it is the standard Lagrangian. We term differential equations of the form(68)δLδuβ=εfγβE1γapproximate Euler-Lagrange-type equations.Definition 12 (see [5]).
An approximate Lie-Bäcklund symmetry operatorY is called an approximate Noether-type symmetry operator corresponding to a partial Lagrangian L∈A if and only if there exists a vector B=(B1,B2,…,Bn),Bi∈A defined by(69)Bi=B0i+εB1i+⋯+εkBkisuch that(70)YL+LDiξi=WαδLδuα+DiBi+oεk+1,where W=(W1,W2,…,Wm),Wβ∈A of Y is also the characteristic of the conservation law DiTi=o(εk+1), where(71)Ti=Bi-Lξi-Wβ∂L∂uiβ+⋯+oεk+1of the approximate Euler-Lagrange-type (68).Because (1) does not have a Lagrange function and if we put a transform u=vx, then (1) becomes (72)vxt-6vxvxx+vxxxx+εαvx+2xvxx-εβ2vx+xvxx=0.In order to write convenient, we can get (73)uxt-6uxuxx+uxxxx+εαux+2xuxx-εβ2ux+xuxx=0.Obviously, (74)uxt-6uxuxx+uxxxx=0,and the Lagrange function is (75)L=12uxx2+ux3-12uxut.And the approximate Euler-Lagrange-type equation is (76)δLδu=-εαux+2xuxx+εβ2ux+xuxx.So the approximate Noether symmetry operator is Y0+ϵY1 for L(77)Y0+εY1L+Diξ0i+εξ1iL=η0-ξ0juj+εη1-ξ1juj-ϵαux+2xuxx+εβ2ux+xuxx+DiB0i+εB1i,where(78)Y0+εY1=ξ01+εξ11∂∂t+ξ02+εξ12∂∂x+η0+εη1∂∂u+ζt∂∂ut+ζx∂∂ux+ζxx∂∂uxx,(79)ζt=η0t+η0u-ξ0t1ut-ξ0t2ux-ξ0u1ut2-ξ0u2uxut+ξ01utt+ξ02uxt+εη1t+η1u-ξ1t1ut-ξ1t2ux-ξ1u1ut2-ξ1u2ux2+ξ11utt+ξ12uxt,(80)ζx=η0x+η0u-ξ0x2ux-ξ0x1ut-ξ0u2ux2-ξ0u1uxut+ξ01uxt+ξ02uxx+εη1x+η1u-ξ1x2ux-ξ1x1ut-ξ1u2ux2-ξ1u2uxut+ξ11uxzt+ξ12uxx,(81)ζxx=η0xx+2η0xu-ξ0xx2ux+η0uu-2ξ0xu2ux2-ξ0xx1ut-ξ0uu1ux2ut-2ξ0xu1uxut-ξ0u1uxxut-2ξ0x1uxt-2ξ0u1uxuxt-ξ0uu2ux3-3ξ0u2uxuxx+η0u-2ξ0x2uxx+ξ01uxxt+ξ02uxxx+εη1xx+2η1xu-ξ1xx2ux+η1uu-2ξ1xu2ux2-ξ1xx1ut-ξ1uu1ux2ut-2ξ1xu1uxut-ξ1u1uxxut-2ξ1x1uxt-2ξ1u1uxuxt-ξ1uu2ux3-3ξ1u2uxuxx+η1u-2ξ1x2uxx+ξ11uxxt+ξ12uxxx.So (58) becomes(82)-12uxζt-12utζx+3ux2ζx+uxxζxx+Lξ0t1+ξ0u1ut+εξ1t1+ξ1u1ut+Lξ0t2+ξ0u2ut+εξ1t2+ξ1u2ut=η0-ξ01ut-ξ02ux+εη1-ξ11ut-ξ12ux-εαux+2xuxx+εβ2ux+xuxx+εB1t1+B1u1ut+εB1t2+B1u2ux+B0t1+B0u1ut+B0x2+B0u2ux.Put L,ζt,ζx,ζxx into (82), and let ε2=0. We obtain (83)-12η0t-B0u2=0,η0x=0,η0u=0,ξ01=0,ξ02=0,B0u1=0,B0x2+B0t1=0,-12η1x-B1u1=0,η1x=0,η1u=0,ξ11=0,ξ12=0,B0u1=0,B1x2+B1t1=0,-12η1t+α-2βη0-B1u2=0,η1xx+2α-βη0·x=0.In the following we will consider three cases of α and β.First Case. α≠2β and β≠2α(84)ξ01=0,ξ02=0,η0=0,ξ11=0,ξ12=0,η1=gt,B01=et,x,B02=ft,x,B11=it,x,B12=-gttu2+jt,x,where g and h are arbitrary function for t and e(t,x),f(t,x),i(t,x),j(t,x) are differential functions and et(t,x)+fx(t,x)=0 and it(t,x)+jx(t,x)=0.Thus the equation has the following approximate Noether symmetric operators(85)Y=εgt∂∂u.So the conservation vectors are (86)T1=et,x+εit,x+12εuxgt,T2=ft,x+ε-12gttu+jt,x-εgt3ux2-12ut+Dxεgtuxx.Second Case. α≠2β and β=2α(87)ξ01=0,ξ02=0,η0=ht,ξ11=0,ξ12=0,η1=gt,B01=et,x,B02=-httu2+ft,x,B11=it,x,B12=-gtt2u-3αht+jt,x,where g and h are arbitrary function for t and e(t,x),f(t,x),i(t,x),j(t,x) are differential functions and et(t,x)+fx(t,x)=0,it(t,x)+jx(t,x)=0.Thus the equation has the following approximate Noether symmetry operators(88)Y=ht∂∂u+εgt∂∂u.So the conservation vectors are (89)T1=et,x+εit,x+12uxht+εgt,T2=ft,x+-12httu+ε-12gttu-3αht+jt,x-εgt+ht3ux2-12ut+Dxεgt+htuxx.Third Case. α=2β and β≠2α; the conservation laws are the same as the first case.



---

*Source: 4743567-2018-09-09.xml*