# Lie Groups Analysis and Contact Transformations for Ito System

Authors: M. M. Al-Shomrani
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

Generalized Ito systems of four coupled nonlinear evaluation equations are proposed. New classes of exact invariant solutions by using Lie group analysis are obtained. Moreover, we investigate the existence of a one-parameter group of contact transformations for a generalized Ito system. Consequently, we study the relationship between one-parameter group of a contact transformation and a one-parameter Lie point transformation for a generalized Ito system.


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## Body

## 1. Introduction

A systematic investigation of continuous transformation groups was carried out by Lie (1882–1899). His original goal was the creation of a theory of integration for ordinary differential equations analogous to the Abelian theory for the solution of algebraic equations. He investigates the fundamental concept of the invariance group admitted by a given system of differential equations. Today, the mathematical approach whose object is the construction and analysis of the full invariance group admitted by a system of differential equations is called group analysis of differential equations. These groups now usually called Lie groups and the associated Lie algebras have important real-world applications.For the past two decades, the Lie group method has been applied to solve a wide range of problems and to explore many physically interesting solutions of nonlinear phenomena. Recently, several extensions and modifications of the classical Lie algorithm have been proposed in order to arrive at new solutions of partial differential equations (PDE). Lie symmetry analysis is one of the most powerful methods to get particular solutions of differential equations. It is based on the study of their invariance with respect to one-parameter Lie group of point transformations whose infinitesimal operators are generated by vector fields. Once the Lie groups that leave the differential equations invariant are known, we can construct an exact solution called a group invariant solution which is invariant under the transformation.In this work, we first find symmetry groups and obtain reduced forms and then seek some similarity solutions to the reduced forms of the following Ito coupled system. The application of one-parameter group reduces the number of independent variables, and consequently a generalized Ito system is reduced to set of ordinary differential equations (ODEs) which are solved analytically.Now we take into consideration a generalized Ito system of four coupled nonlinear evolution equations which was introduced recently by Tam et al., [1] and Karasu-Kalkanli et al. [2]:
(1.1)ut=vx,vt=2vxxx-6(uv)x-6(wp)x,wt=wxxx+3uwx,pt=pxxx+3upx.
Which is the generalization of the well-known integrable Ito system [3]:
(1.2)ut=vx,vt=-2vxxx-6(uv)x.
Now, we investigate the existence of a one-parameter group of contact transformations for a generalized Ito system (1.1) to obtain the Lie point transformations generators and use symmetry groups to find the same Lie point transformation generators which are obtained from contact transformations.


## 2. The Existence of Contact Transformations for a Generalized Ito System

Evolution equations model a wide variety of phenomena in the physical, biological, and economic sciences. Lie group theory provides a useful tool for the solution partial differential equations. Many books have been written on this aspect [4]. For Lie group theory to be useful for the solution of evolution-type partial differential equations, the Lie point transformation generators need to be determined [4]. Once the Lie point transformation generators have been determined, they can be used to obtain special solutions of the differential equations under consideration. A reduction in the number of variables and transformations to other simpler equations which may be easier to solve are also possible. The Lie theory has provided insight into many physical phenomena, which may otherwise not have been possible.


### 2.1. Preliminaries

We only summarize relevant aspects for the case of two independent variables (time,t, and one space variable, x). The reader is referred to [5]. The set of transformations in (t, x, u) space, namely,
(2.1)t=t(t,x,u,a),x=x(t,x,u,a),u=u(t,x,u,a),
where a is a real parameter, is a one-parameter group of Lie point transformations if it satisfies the group properties. The generator of the group of transformations (2.1) is given by
(2.2)X=ξ1(t,x,u)∂∂t+ξ2(t,x,u)∂∂x+η(t,x,u)∂∂u.The set of transformations in(t,x,u,ut,ux) space, namely,
(2.3)t-=t-(t,x,u,ut,ux,a),x-=x-(t,x,u,ut,ux,a),u-=u-(t,x,u,ut,ux,a),u-t=u-t(t,x,u,ut,ux,a),u-x=u-x(t,x,u,ut,ux,a),
where a is a real parameter, is a one-parameter group of contact transformations if it satisfies the group properties and u-t=(∂u-/∂t), u-x=(∂u-/∂x) hold.The generator of a group of contact transformations is(2.4)Y=ξ1(t,x,u,ut,ux)∂∂t+ξ2(t,x,u,ut,ux)∂∂x+η(t,x,u,ut,ux)∂∂u+ξ1(t,x,u,ut,ux)∂∂ut+ξ2(t,x,u,ut,ux)∂∂ux.The Lie characteristic function is defined by(2.5)W=η-utξ1-uxξ2,
where the functions ξ1, ξ2, and η can be given in terms of W as
(2.6)ξ1=-Wut,ξ2=-Wux,η=W-utWt-uxWx,
and the formulae for ξi can easily be written in terms of W as
(2.7)ξ1=Wt+utWu,ξ2=Wt+uxWu.
Higher-order prolongations can be calculated from the prolongation formula:
(2.8)ξi1i2⋯is=Di1⋯DisWuj(W)uji1i2⋯is,
where Di is the operator of total differentiation given by
(2.9)Di=∂∂xi+ui∂∂u+uij∂∂uij+⋯.IfW is linear in the first derivatives ut and ux, then the contact transformation generator (2.4) reduces to an extended Lie point transformation generator (2.2).


### 2.2. Application Contact Transformations for the Generalized Ito System

In this section, we determine contact transformations for the generalized Ito system (1.1), where
(2.10)F1=vx,(2.11)F2=2vxxx-6(uv)x-6(wp)x,(2.12)F3=wxxx+3uwx,(2.13)F4=pxxx+3upx.Lie point transformation generators were given by [4]. To determine contact transformations of (1.1), we solve the determining equations:
(2.14)X~(ut-vx)∣(2.10)-(2.13)=0,(2.15)X~(vt-(-2vxxx-6(υϖ)x-6(ωπ)x))∣(2.10)-(2.13)=0,(2.16)X~(wt-(wxxx+3υwx))∣(2.10)-(2.13){X~}=0,(2.17)X~(pt-(pxxx+3υpx))∣(2.10)-(2.13)=0,
where X~ is the prolongation of the operator (2.4) in terms of W.Consequently, we find that(2.18)φ1t-φ2x=0,φ2t-(-2φ2xxx-6η1vx+uφ2x+η2ux+vφ1x)-6(η3px+wφ4x+η4wx+pφ3x)=0,φ3t-(φ3xxx+3(η1wx+uφ3x))=0,φ4t-(φ4xxx+3(η1px+uφ4x))=0,
where φ1t, φ2t, φ3t, φ4t, φ1x, φ2x, φ3x, φ4x, φ2xxx, φ3xxx, and φ4xxx can be determined from the following relation
(2.19)φvi1i2i3⋯is=Di1Di2Di3⋯Dis(Wv)-Wujvvuji1i2i3⋯isv,
where v=1,2,3,4 and u1, u2, u3, u4  are u, v, w, p, respectively.By substitutingφ1t, φ2t, φ3t, φ4t, φ1x, φ2x, φ3x, φ4x, φ2xxx, φ3xxx, and φ4xxx into (2.18) and after some calculations, we obtain the Lie characteristic functions in the following form:
(2.20)W1=ut(k3-34k4t)+ux(k1-14k4x)+12k4u,W2=vt(k3-34k4t)+vx(k1-14k4x)+k4v,W3=wt(k3-34k4t)+wx(k1-14k4x)+k2w,W4=pt(k3-34k4t)+px(k1-14k4x)+(-k2+32k4)p,
where k1, k2,  k3,  and k4  are arbitrary constants. Then the Ito systems have the following infinitesimal:
(2.21)ξ1=(k3-34k4t),ξ2=(k1-14k4x),η1=12k4u,η2=k4v,η3=k2w,η4=(-k2+32k4)p.


### 2.3. Lie Groups Analysis

Many authors applied Lie group analysis to find exact solutions, for example, in [6] the authors used Lie symmetry analysis and the method of dynamical systems for the extended mKdV equation to obtain exact solutions, in [7] the authors applied Lie symmetry analysis and Painleve analysis for the new (2+1)-dimensional KdV equation, and in [8] the authors have some analytical solutions for groundwater flow and transport equation via using Lie group analysis. Various symmetry reduction is obtained and reduce the system of partial differential equations to the system of ordinary differential equations which we can obtain the complete solutions of the system of ordinary differential equations. In this section, by requiring the invariance of the equations in (1.1) under the one-parameter group of Lie transformation, we obtain a system of partial differential equations which allows us not only to find the generator of the group but also to use the invariant surface condition and arrive at the reduced equation in all the considered cases. Now requiring the invariance of (1.1) with respect to the one-parameter Lie group of infinitesimal transformations, we investigate the similarity solution for the generalized Ito system.Let us consider a one-parameter Lie group of infinitesimal transformations [9–11] of the form:
(2.22)x→X=x+εξ1(t,x,u,v,w,p)+O(ε2),t→T=t+εξ2(t,x,u,v,w,p)+O(ε2),u→U=u+εη1(t,x,u,v,w,p)+O(ε2),v→V=v+εη2(t,x,u,v,w,p)+O(ε2),w→W=w+εη3(t,x,u,v,w,p)+O(ε2),p→P=p+εη4(t,x,u,v,w,p)+O(ε2).ε≪1.The functionsξ1,  ξ2,  η1,  η2,  η3, and η4 are the infinitesimal of transformations for the variables t,  x,  u,  v,  w, and p, respectively. In order to obtain these infinitesimal functions we have to construct a third-extended vector field X~ that is defined by
(2.23)X~=ξ1∂∂t+ξ2∂∂x+η1∂∂u+∂∂ut+η1∂∂v+η2∂∂u+η3∂∂p+η4∂∂ρ+φ1x∂∂ux+φ1t∂∂ut+φ2x∂∂vx+φ2t∂∂vt+⋯
and the symmetry vector field X given by (2.2). The equations in (1.1) can be written in the form:
(2.24)H1=(ut-vx),H2=(vt-(-2vxxx-6(υϖ)x-6(ωπ)x)),H3=(wt-(wxxx+3υwx)),H4=(pt-(pxxx+3υpx)).The invariance of (2.24) under the infinitesimal transformations (2.23) needs applying the extended operator to the system of PDEs (2.24), and we have
(2.25)X~H1=φ1t-φ2x=0,X~H2=φ2t-[-2φ2xxx-6(η1ϖξ+υφ2x+η2υξ+ϖφ1x)-6(η3πξ+φ4xω+η4ωξ+φ3xπ)]=0,X~H3=φ3t-(φ3xxx+3(η1wx+φ3xu))=0,X~H4=φ4t-(φ4xxx+3(η1px+φ4xu))=0,
under H1=0,  H2=0,  H3=0, and H4=0. By using symbolic software Math Lie and equating the different coefficients of the various monomials in the first-, second- and third- order partial derivatives of u,  v,  w,  and p into (2.25) and after some calculation, we obtain the following system of partial differential equations for ξ1,  ξ2,  η1,  η2,  η3,  and η4 [12–15]:
(2.26)-3η1+3uξ2-3uξ1t+ξ2xxx-3ηxxw3=0,-3η1+3uξ2-3uξ1t+ξ2xxx-3ηxxp4=0,3η1-3uξ2+3uξ1t-ξ2xxx+3ηxxv2=0,ηt4-3uηx4-ηxxx4=0,ηt3-3uηx3-ηxxx3=0,6vηx1+ηt2+6uηx2+6pηx3+6wηx4+2ηxxx2=0,-ξ2+ηxw3=0,-ξ2+ηxp4=0,-ξ2+ηxv2=0,η3-wξ2+wξ1t-wξ1t-wηv2+pηp3+wηp4=0,η2-vξ2x+vξ1t+vηu1-vηv2=0,η4-pξ2x+pξ1t-pηv2+pηw3+wηw4=0,3ξ2x-3ξ1t=0,ξ2x-3ξ1t-ηu1-ηv2=0,ηt1-ηx2=0,ηv1=ηw1=ηp1=0,ηp2=ηw2=ηvv2=ηu2=0,ηu3=ηv3=ηxp3=ηpp3=ηww3=ηwp3=0,ηu4=ηv4=ηxw4=ηpp4=ηww4=ηxw4=0,ξ2p=ξ2w=ξ2v=ξ2t=ξ2=0,ξ1p=ξ1w=ξ1v=ξ1u=ξ1x=0.Now, we solve this system of linear partial differential (2.26) for the infinitesimal ξ1, ξ2, η1, η2, η3, and η4 and we obtain
(2.27)ξ1=(k3-34k4t),ξ2=(k1-14k4x),η1=12k4u,η2=k4v,η3=k2w,η4=(-k2+32k4)p,
where k1, k2, k3 and k4 are arbitrary constants. The above equations are the same as (2.21). We obtain from (2.21) that the Lie point transformation generators are
(2.28)𝒳1=∂∂x,𝒳2=w∂∂w-p∂∂p,𝒳3=∂∂t,𝒳4=-x4∂∂x-3t4∂∂t+u2∂∂u+v∂∂v+3p2∂∂p.The corresponding Lie algebra of infinitesimal symmetries of (1.1) is spanned by the infinitesimal generators 𝒳1, 𝒳2, 𝒳3, and 𝒳4. Thus, corresponding commutator table of {𝒳i;(i=1,2,3,4)} can be constructed Table 1.Table 1
𝒳
1
𝒳
2
𝒳
3
𝒳
4
𝒳
1
0
0
0
-
𝒳
4
/
4
𝒳
2
0
0
0
0
𝒳
3
0
0
0
-
3
𝒳
3
/
4
𝒳
4
𝒳
4
/
4
0
3
𝒳
3
/
4
0It is easy to check that{𝒳1,𝒳2,𝒳3,𝒳4} are closed under the Lie bracket. Thus, a basis for the Lie algebra is {𝒳1,𝒳2,𝒳3,𝒳4}, which is a 4-dimensional Lie group algebra.


### 2.4. Reduction to Ordinary Differential Equations (ODEs)

Theoretically, all of the similarity variables associated with Lie symmetries (2.23) can be derived by solving the following characteristic equation:
(2.29)dtξ1=dxξ2=duη1=dvη2=dwη3=dpη4.Consequently we get the following:(2.30)dt(k3-(3/4)k4t)=dx(k1-(1/4)k4x)=du(1/2)k4u=dvk4v=dwk2w=dp(-k2+(3/4)k4)p,
from
(2.31)dt(k3-(3/4)k4t)=dx(k1-(1/4)k4x).Solving (2.31), we obtain the similarity variable is
(2.32)Z=(4k1-k4x)(4k3-3k4t)1/3.
From (2.30), we get
(2.33)dt(k3-(3/4)k4t)=du(1/2)k4u,dt(k3-(3/4)k4t)=dvk4vdt(k3-(3/4)k4t)=dwk2w,dt(k3-(3/4)k4t)=dp(-k2+(3/4)k4)p.By solving (2.33), we obtain the similarity solutions take the form:
(2.34)u(t,x)=(k3-34k4t)-2/3F1(Z),v(t,x)=(k3-34k4t)-4/3F2(Z),w(t,x)=(k3-34k4t)(-4k2/3k4)F3(Z),p(t,x)=(k3-34k4t)((-4k2/3k4)-2)F4(Z),
where Z=(4k1-k4x)/(4k3-3k4t)1/3 and F1(Z), F2(Z), F3(Z), and F4(Z) are arbitrary functions.Substituting (2.34) into the equations in (1.1), we finally obtain the system of nonlinear ordinary differential equations for F1(Z), F2(Z), F3(Z), and F4(Z) takes the form:
(2.4)F2′(Z)+2F1′(Z)+2F1(Z)=0,2k42F2′′′(Z)+6(F1F2)'+6(F3F4)'-ZF2′-4F2=0,k43F3′′′(Z)+3k4F1F3′+k4ZF3′-4k2F2=0,k43F4′′′(Z)+3k4F1F4′+k4ZF4′+2(-2k2+3k4)F4=0.Solving a system of an ordinary differential equations (2.4), we have four cases of solutions for F1(Z), F2(Z), F3(Z), and F4(Z).Case 1.
(2.36)
F
1
(
Z
)
=
-
Z
,
F
2
(
Z
)
=
3
2
Z
2
,
F
3
(
Z
)
=
A
1
Z
,
F
4
(
Z
)
=
B
1
Z
2
,
where k4=2k2, A1 and B1 are arbitrary constants with A1B1=2.Case 2.
(2.37)
F
1
(
Z
)
=
-
Z
,
F
2
(
Z
)
=
3
2
Z
2
,
F
3
(
Z
)
=
A
2
Z
2
,
F
4
(
Z
)
=
B
2
Z
,
where k4=k2, A2 and B2are arbitrary constants with A2B2=2.Case 3.
(2.38)
F
1
(
Z
)
=
-
Z
,
F
2
(
Z
)
=
3
2
Z
2
,
F
3
(
Z
)
=
2
d
Z
3
-
8
k
2
2
9
d
,
F
4
(
Z
)
=
d
,
where K4=(2/3)K2 and d is arbitrary constant.Case 4.
(2.39)
F
1
(
Z
)
=
1
3
Z
,
F
2
(
Z
)
=
-
1
2
Z
2
,
F
3
(
Z
)
=
c
Z
,
F
4
(
Z
)
=
0
,
where, K4=2K2 and C is arbitrary constant.Substituting from (2.36)–(2.39) into (2.34) we obtain the solutions for the generalized Ito system (1.1) in the followingFamily 1.
(2.40)
u
(
t
,
x
)
=
-
Z
(
k
3
-
3
4
k
4
t
)
-
2
/
3
,
v
(
t
,
x
)
=
3
2
Z
2
(
k
3
-
3
4
k
4
t
)
-
4
/
3
,
w
(
t
,
x
)
=
A
1
Z
(
k
3
-
3
4
k
4
t
)
(
-
4
k
2
/
3
k
4
)
,
p
(
t
,
x
)
=
B
1
Z
2
(
k
3
-
3
4
k
4
t
)
(
(
-
4
k
2
/
3
k
4
)
-
2
)
,
where k4=2k2, A1 and B1 are arbitrary constants with A1B1=2.Family 2.
(2.41)
u
(
t
,
x
)
=
-
Z
(
k
3
-
3
4
k
4
t
)
-
2
/
3
,
v
(
t
,
x
)
=
3
2
Z
2
(
k
3
-
3
4
k
4
t
)
-
4
/
3
,
w
(
t
,
x
)
=
A
2
Z
2
(
k
3
-
3
4
k
4
t
)
(
-
4
k
2
/
3
k
4
)
,
p
(
t
,
x
)
=
B
2
Z
(
k
3
-
3
4
k
4
t
)
(
(
-
4
k
2
/
3
k
4
)
-
2
)
,
where k4=k2, A2 and B2 are arbitrary constants with A2B2=2.Family 3.
(2.42)
u
(
t
,
x
)
=
-
Z
(
k
3
-
3
4
k
4
t
)
-
2
/
3
,
v
(
t
,
x
)
=
3
2
Z
2
(
k
3
-
3
4
k
4
t
)
-
4
/
3
,
w
(
t
,
x
)
=
(
2
d
Z
3
-
8
k
2
2
9
d
)
(
k
3
-
3
4
k
4
t
)
(
-
4
k
2
/
3
k
4
)
,
p
(
t
,
x
)
=
d
(
k
3
-
3
4
k
4
t
)
(
(
-
4
k
2
/
3
k
4
)
-
2
)
,
where k4=(2/3)k2 and d is arbitrary constant.Family 4.
(2.43)
u
(
t
,
x
)
=
1
2
Z
(
k
3
-
3
4
k
4
t
)
-
2
/
3
,
v
(
t
,
x
)
=
-
1
2
Z
2
(
k
3
-
3
4
k
4
t
)
-
4
/
3
,
w
(
t
,
x
)
=
c
Z
(
k
3
-
3
4
k
4
t
)
(
-
4
k
2
/
3
k
4
)
,
p
(
t
,
x
)
=
0
,
where k4=2k2 and c is arbitrary constant andZ=((4k1-k4x)/(4k3-3k4t)1/3).


## 2.1. Preliminaries

We only summarize relevant aspects for the case of two independent variables (time,t, and one space variable, x). The reader is referred to [5]. The set of transformations in (t, x, u) space, namely,
(2.1)t=t(t,x,u,a),x=x(t,x,u,a),u=u(t,x,u,a),
where a is a real parameter, is a one-parameter group of Lie point transformations if it satisfies the group properties. The generator of the group of transformations (2.1) is given by
(2.2)X=ξ1(t,x,u)∂∂t+ξ2(t,x,u)∂∂x+η(t,x,u)∂∂u.The set of transformations in(t,x,u,ut,ux) space, namely,
(2.3)t-=t-(t,x,u,ut,ux,a),x-=x-(t,x,u,ut,ux,a),u-=u-(t,x,u,ut,ux,a),u-t=u-t(t,x,u,ut,ux,a),u-x=u-x(t,x,u,ut,ux,a),
where a is a real parameter, is a one-parameter group of contact transformations if it satisfies the group properties and u-t=(∂u-/∂t), u-x=(∂u-/∂x) hold.The generator of a group of contact transformations is(2.4)Y=ξ1(t,x,u,ut,ux)∂∂t+ξ2(t,x,u,ut,ux)∂∂x+η(t,x,u,ut,ux)∂∂u+ξ1(t,x,u,ut,ux)∂∂ut+ξ2(t,x,u,ut,ux)∂∂ux.The Lie characteristic function is defined by(2.5)W=η-utξ1-uxξ2,
where the functions ξ1, ξ2, and η can be given in terms of W as
(2.6)ξ1=-Wut,ξ2=-Wux,η=W-utWt-uxWx,
and the formulae for ξi can easily be written in terms of W as
(2.7)ξ1=Wt+utWu,ξ2=Wt+uxWu.
Higher-order prolongations can be calculated from the prolongation formula:
(2.8)ξi1i2⋯is=Di1⋯DisWuj(W)uji1i2⋯is,
where Di is the operator of total differentiation given by
(2.9)Di=∂∂xi+ui∂∂u+uij∂∂uij+⋯.IfW is linear in the first derivatives ut and ux, then the contact transformation generator (2.4) reduces to an extended Lie point transformation generator (2.2).


## 2.2. Application Contact Transformations for the Generalized Ito System

In this section, we determine contact transformations for the generalized Ito system (1.1), where
(2.10)F1=vx,(2.11)F2=2vxxx-6(uv)x-6(wp)x,(2.12)F3=wxxx+3uwx,(2.13)F4=pxxx+3upx.Lie point transformation generators were given by [4]. To determine contact transformations of (1.1), we solve the determining equations:
(2.14)X~(ut-vx)∣(2.10)-(2.13)=0,(2.15)X~(vt-(-2vxxx-6(υϖ)x-6(ωπ)x))∣(2.10)-(2.13)=0,(2.16)X~(wt-(wxxx+3υwx))∣(2.10)-(2.13){X~}=0,(2.17)X~(pt-(pxxx+3υpx))∣(2.10)-(2.13)=0,
where X~ is the prolongation of the operator (2.4) in terms of W.Consequently, we find that(2.18)φ1t-φ2x=0,φ2t-(-2φ2xxx-6η1vx+uφ2x+η2ux+vφ1x)-6(η3px+wφ4x+η4wx+pφ3x)=0,φ3t-(φ3xxx+3(η1wx+uφ3x))=0,φ4t-(φ4xxx+3(η1px+uφ4x))=0,
where φ1t, φ2t, φ3t, φ4t, φ1x, φ2x, φ3x, φ4x, φ2xxx, φ3xxx, and φ4xxx can be determined from the following relation
(2.19)φvi1i2i3⋯is=Di1Di2Di3⋯Dis(Wv)-Wujvvuji1i2i3⋯isv,
where v=1,2,3,4 and u1, u2, u3, u4  are u, v, w, p, respectively.By substitutingφ1t, φ2t, φ3t, φ4t, φ1x, φ2x, φ3x, φ4x, φ2xxx, φ3xxx, and φ4xxx into (2.18) and after some calculations, we obtain the Lie characteristic functions in the following form:
(2.20)W1=ut(k3-34k4t)+ux(k1-14k4x)+12k4u,W2=vt(k3-34k4t)+vx(k1-14k4x)+k4v,W3=wt(k3-34k4t)+wx(k1-14k4x)+k2w,W4=pt(k3-34k4t)+px(k1-14k4x)+(-k2+32k4)p,
where k1, k2,  k3,  and k4  are arbitrary constants. Then the Ito systems have the following infinitesimal:
(2.21)ξ1=(k3-34k4t),ξ2=(k1-14k4x),η1=12k4u,η2=k4v,η3=k2w,η4=(-k2+32k4)p.


## 2.3. Lie Groups Analysis

Many authors applied Lie group analysis to find exact solutions, for example, in [6] the authors used Lie symmetry analysis and the method of dynamical systems for the extended mKdV equation to obtain exact solutions, in [7] the authors applied Lie symmetry analysis and Painleve analysis for the new (2+1)-dimensional KdV equation, and in [8] the authors have some analytical solutions for groundwater flow and transport equation via using Lie group analysis. Various symmetry reduction is obtained and reduce the system of partial differential equations to the system of ordinary differential equations which we can obtain the complete solutions of the system of ordinary differential equations. In this section, by requiring the invariance of the equations in (1.1) under the one-parameter group of Lie transformation, we obtain a system of partial differential equations which allows us not only to find the generator of the group but also to use the invariant surface condition and arrive at the reduced equation in all the considered cases. Now requiring the invariance of (1.1) with respect to the one-parameter Lie group of infinitesimal transformations, we investigate the similarity solution for the generalized Ito system.Let us consider a one-parameter Lie group of infinitesimal transformations [9–11] of the form:
(2.22)x→X=x+εξ1(t,x,u,v,w,p)+O(ε2),t→T=t+εξ2(t,x,u,v,w,p)+O(ε2),u→U=u+εη1(t,x,u,v,w,p)+O(ε2),v→V=v+εη2(t,x,u,v,w,p)+O(ε2),w→W=w+εη3(t,x,u,v,w,p)+O(ε2),p→P=p+εη4(t,x,u,v,w,p)+O(ε2).ε≪1.The functionsξ1,  ξ2,  η1,  η2,  η3, and η4 are the infinitesimal of transformations for the variables t,  x,  u,  v,  w, and p, respectively. In order to obtain these infinitesimal functions we have to construct a third-extended vector field X~ that is defined by
(2.23)X~=ξ1∂∂t+ξ2∂∂x+η1∂∂u+∂∂ut+η1∂∂v+η2∂∂u+η3∂∂p+η4∂∂ρ+φ1x∂∂ux+φ1t∂∂ut+φ2x∂∂vx+φ2t∂∂vt+⋯
and the symmetry vector field X given by (2.2). The equations in (1.1) can be written in the form:
(2.24)H1=(ut-vx),H2=(vt-(-2vxxx-6(υϖ)x-6(ωπ)x)),H3=(wt-(wxxx+3υwx)),H4=(pt-(pxxx+3υpx)).The invariance of (2.24) under the infinitesimal transformations (2.23) needs applying the extended operator to the system of PDEs (2.24), and we have
(2.25)X~H1=φ1t-φ2x=0,X~H2=φ2t-[-2φ2xxx-6(η1ϖξ+υφ2x+η2υξ+ϖφ1x)-6(η3πξ+φ4xω+η4ωξ+φ3xπ)]=0,X~H3=φ3t-(φ3xxx+3(η1wx+φ3xu))=0,X~H4=φ4t-(φ4xxx+3(η1px+φ4xu))=0,
under H1=0,  H2=0,  H3=0, and H4=0. By using symbolic software Math Lie and equating the different coefficients of the various monomials in the first-, second- and third- order partial derivatives of u,  v,  w,  and p into (2.25) and after some calculation, we obtain the following system of partial differential equations for ξ1,  ξ2,  η1,  η2,  η3,  and η4 [12–15]:
(2.26)-3η1+3uξ2-3uξ1t+ξ2xxx-3ηxxw3=0,-3η1+3uξ2-3uξ1t+ξ2xxx-3ηxxp4=0,3η1-3uξ2+3uξ1t-ξ2xxx+3ηxxv2=0,ηt4-3uηx4-ηxxx4=0,ηt3-3uηx3-ηxxx3=0,6vηx1+ηt2+6uηx2+6pηx3+6wηx4+2ηxxx2=0,-ξ2+ηxw3=0,-ξ2+ηxp4=0,-ξ2+ηxv2=0,η3-wξ2+wξ1t-wξ1t-wηv2+pηp3+wηp4=0,η2-vξ2x+vξ1t+vηu1-vηv2=0,η4-pξ2x+pξ1t-pηv2+pηw3+wηw4=0,3ξ2x-3ξ1t=0,ξ2x-3ξ1t-ηu1-ηv2=0,ηt1-ηx2=0,ηv1=ηw1=ηp1=0,ηp2=ηw2=ηvv2=ηu2=0,ηu3=ηv3=ηxp3=ηpp3=ηww3=ηwp3=0,ηu4=ηv4=ηxw4=ηpp4=ηww4=ηxw4=0,ξ2p=ξ2w=ξ2v=ξ2t=ξ2=0,ξ1p=ξ1w=ξ1v=ξ1u=ξ1x=0.Now, we solve this system of linear partial differential (2.26) for the infinitesimal ξ1, ξ2, η1, η2, η3, and η4 and we obtain
(2.27)ξ1=(k3-34k4t),ξ2=(k1-14k4x),η1=12k4u,η2=k4v,η3=k2w,η4=(-k2+32k4)p,
where k1, k2, k3 and k4 are arbitrary constants. The above equations are the same as (2.21). We obtain from (2.21) that the Lie point transformation generators are
(2.28)𝒳1=∂∂x,𝒳2=w∂∂w-p∂∂p,𝒳3=∂∂t,𝒳4=-x4∂∂x-3t4∂∂t+u2∂∂u+v∂∂v+3p2∂∂p.The corresponding Lie algebra of infinitesimal symmetries of (1.1) is spanned by the infinitesimal generators 𝒳1, 𝒳2, 𝒳3, and 𝒳4. Thus, corresponding commutator table of {𝒳i;(i=1,2,3,4)} can be constructed Table 1.Table 1
𝒳
1
𝒳
2
𝒳
3
𝒳
4
𝒳
1
0
0
0
-
𝒳
4
/
4
𝒳
2
0
0
0
0
𝒳
3
0
0
0
-
3
𝒳
3
/
4
𝒳
4
𝒳
4
/
4
0
3
𝒳
3
/
4
0It is easy to check that{𝒳1,𝒳2,𝒳3,𝒳4} are closed under the Lie bracket. Thus, a basis for the Lie algebra is {𝒳1,𝒳2,𝒳3,𝒳4}, which is a 4-dimensional Lie group algebra.


## 2.4. Reduction to Ordinary Differential Equations (ODEs)

Theoretically, all of the similarity variables associated with Lie symmetries (2.23) can be derived by solving the following characteristic equation:
(2.29)dtξ1=dxξ2=duη1=dvη2=dwη3=dpη4.Consequently we get the following:(2.30)dt(k3-(3/4)k4t)=dx(k1-(1/4)k4x)=du(1/2)k4u=dvk4v=dwk2w=dp(-k2+(3/4)k4)p,
from
(2.31)dt(k3-(3/4)k4t)=dx(k1-(1/4)k4x).Solving (2.31), we obtain the similarity variable is
(2.32)Z=(4k1-k4x)(4k3-3k4t)1/3.
From (2.30), we get
(2.33)dt(k3-(3/4)k4t)=du(1/2)k4u,dt(k3-(3/4)k4t)=dvk4vdt(k3-(3/4)k4t)=dwk2w,dt(k3-(3/4)k4t)=dp(-k2+(3/4)k4)p.By solving (2.33), we obtain the similarity solutions take the form:
(2.34)u(t,x)=(k3-34k4t)-2/3F1(Z),v(t,x)=(k3-34k4t)-4/3F2(Z),w(t,x)=(k3-34k4t)(-4k2/3k4)F3(Z),p(t,x)=(k3-34k4t)((-4k2/3k4)-2)F4(Z),
where Z=(4k1-k4x)/(4k3-3k4t)1/3 and F1(Z), F2(Z), F3(Z), and F4(Z) are arbitrary functions.Substituting (2.34) into the equations in (1.1), we finally obtain the system of nonlinear ordinary differential equations for F1(Z), F2(Z), F3(Z), and F4(Z) takes the form:
(2.4)F2′(Z)+2F1′(Z)+2F1(Z)=0,2k42F2′′′(Z)+6(F1F2)'+6(F3F4)'-ZF2′-4F2=0,k43F3′′′(Z)+3k4F1F3′+k4ZF3′-4k2F2=0,k43F4′′′(Z)+3k4F1F4′+k4ZF4′+2(-2k2+3k4)F4=0.Solving a system of an ordinary differential equations (2.4), we have four cases of solutions for F1(Z), F2(Z), F3(Z), and F4(Z).Case 1.
(2.36)
F
1
(
Z
)
=
-
Z
,
F
2
(
Z
)
=
3
2
Z
2
,
F
3
(
Z
)
=
A
1
Z
,
F
4
(
Z
)
=
B
1
Z
2
,
where k4=2k2, A1 and B1 are arbitrary constants with A1B1=2.Case 2.
(2.37)
F
1
(
Z
)
=
-
Z
,
F
2
(
Z
)
=
3
2
Z
2
,
F
3
(
Z
)
=
A
2
Z
2
,
F
4
(
Z
)
=
B
2
Z
,
where k4=k2, A2 and B2are arbitrary constants with A2B2=2.Case 3.
(2.38)
F
1
(
Z
)
=
-
Z
,
F
2
(
Z
)
=
3
2
Z
2
,
F
3
(
Z
)
=
2
d
Z
3
-
8
k
2
2
9
d
,
F
4
(
Z
)
=
d
,
where K4=(2/3)K2 and d is arbitrary constant.Case 4.
(2.39)
F
1
(
Z
)
=
1
3
Z
,
F
2
(
Z
)
=
-
1
2
Z
2
,
F
3
(
Z
)
=
c
Z
,
F
4
(
Z
)
=
0
,
where, K4=2K2 and C is arbitrary constant.Substituting from (2.36)–(2.39) into (2.34) we obtain the solutions for the generalized Ito system (1.1) in the followingFamily 1.
(2.40)
u
(
t
,
x
)
=
-
Z
(
k
3
-
3
4
k
4
t
)
-
2
/
3
,
v
(
t
,
x
)
=
3
2
Z
2
(
k
3
-
3
4
k
4
t
)
-
4
/
3
,
w
(
t
,
x
)
=
A
1
Z
(
k
3
-
3
4
k
4
t
)
(
-
4
k
2
/
3
k
4
)
,
p
(
t
,
x
)
=
B
1
Z
2
(
k
3
-
3
4
k
4
t
)
(
(
-
4
k
2
/
3
k
4
)
-
2
)
,
where k4=2k2, A1 and B1 are arbitrary constants with A1B1=2.Family 2.
(2.41)
u
(
t
,
x
)
=
-
Z
(
k
3
-
3
4
k
4
t
)
-
2
/
3
,
v
(
t
,
x
)
=
3
2
Z
2
(
k
3
-
3
4
k
4
t
)
-
4
/
3
,
w
(
t
,
x
)
=
A
2
Z
2
(
k
3
-
3
4
k
4
t
)
(
-
4
k
2
/
3
k
4
)
,
p
(
t
,
x
)
=
B
2
Z
(
k
3
-
3
4
k
4
t
)
(
(
-
4
k
2
/
3
k
4
)
-
2
)
,
where k4=k2, A2 and B2 are arbitrary constants with A2B2=2.Family 3.
(2.42)
u
(
t
,
x
)
=
-
Z
(
k
3
-
3
4
k
4
t
)
-
2
/
3
,
v
(
t
,
x
)
=
3
2
Z
2
(
k
3
-
3
4
k
4
t
)
-
4
/
3
,
w
(
t
,
x
)
=
(
2
d
Z
3
-
8
k
2
2
9
d
)
(
k
3
-
3
4
k
4
t
)
(
-
4
k
2
/
3
k
4
)
,
p
(
t
,
x
)
=
d
(
k
3
-
3
4
k
4
t
)
(
(
-
4
k
2
/
3
k
4
)
-
2
)
,
where k4=(2/3)k2 and d is arbitrary constant.Family 4.
(2.43)
u
(
t
,
x
)
=
1
2
Z
(
k
3
-
3
4
k
4
t
)
-
2
/
3
,
v
(
t
,
x
)
=
-
1
2
Z
2
(
k
3
-
3
4
k
4
t
)
-
4
/
3
,
w
(
t
,
x
)
=
c
Z
(
k
3
-
3
4
k
4
t
)
(
-
4
k
2
/
3
k
4
)
,
p
(
t
,
x
)
=
0
,
where k4=2k2 and c is arbitrary constant andZ=((4k1-k4x)/(4k3-3k4t)1/3).


## 3. Conclusion

In this paper, we proved the existence of a one-parameter group of contact transformations for a generalized Ito system (1.1). Moreover, we obtained the relation between the Lie point transformations generators and contact transformations for a generalized Ito system. Also, we used the symmetry groups to find the same Lie point transformation generators which are obtained from contact transformations.Finally, applying one-parameter group, we explored several new solutions for the Ito system through the Lie symmetry analysis which have not been reported in the literature for this model.



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*Source: 342680-2012-11-29.xml*