Lie Groups Analysis and Contact Transformations for Ito System

and Applied Analysis 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, t t t, x, u, a , x x t, x, u, a , u u t, x, u, a , 2.1 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 X ξ1 t, x, u ∂ ∂t ξ2 t, x, u ∂ ∂x η t, x, u ∂ ∂u . 2.2 The set of transformations in t, x, u, ut, ux space, namely, t t t, x, u, ut, ux, a , x x t, x, u, ut, ux, a , u u t, x, u, ut, ux, a , ut ut t, x, u, ut, ux, a , ux ux t, x, u, ut, ux, a , 2.3 where a is a real parameter, is a one-parameter group of contact transformations if it satisfies the group properties and ut ∂u/∂t , ux ∂u/∂x hold. The generator of a group of contact transformations is 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 . 2.4 The Lie characteristic function is defined by W η − utξ1 − ux ξ2, 2.5 where the functions ξ1, ξ2, and η can be given in terms of W as ξ1 −Wut, ξ2 −Wux, η W − utWt − uxWx, 2.6 4 Abstract and Applied Analysis and the formulae for ξi can easily be written in terms of W as ξ1 Wt utWu, ξ2 Wt uxWu. 2.7 Higher-order prolongations can be calculated from the prolongation formula: ξi1i2···is Di1 · · ·DisWuj W uji1i2···is , 2.8 where Di is the operator of total differentiation given by Di ∂ ∂xi ui ∂ ∂u uij ∂ ∂uij · · · . 2.9 If W 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 F1 vx, 2.10 F2 2vxxx − 6 uv x − 6 ( wp ) x, 2.11 F3 wxxx 3uwx, 2.12 F4 pxxx 3upx. 2.13 Lie point transformation generators were given by 4 . To determine contact transformations of 1.1 , we solve the determining equations: X̃ ut − vx | 2.10 − 2.13 0, 2.14 X̃ vt − −2vxxx − 6 υ x − 6 ωπ x | 2.10 − 2.13 0, 2.15 X̃ wt − wxxx 3υwx | 2.10 − 2.13 0, 2.16 X̃ ( pt − ( pxxx 3υpx | 2.10 − 2.13 0, 2.17 where X̃ is the prolongation of the operator 2.4 in terms of W . Abstract and Applied Analysis 5 Consequently, we find that φt1 − φ2 0, φ2 − ( −2φxxx 2 − 6ηvx uφ2 ηux vφx1 ) − 6 ( ηpx wφx4 η wx pφ3 ) 0, φ3 − ( φ 3 3 ( ηwx uφ3 )) 0, φt4 − ( φ 4 3 ( ηpx uφx4 )) 0, 2.18and Applied Analysis 5 Consequently, we find that φt1 − φ2 0, φ2 − ( −2φxxx 2 − 6ηvx uφ2 ηux vφx1 ) − 6 ( ηpx wφx4 η wx pφ3 ) 0, φ3 − ( φ 3 3 ( ηwx uφ3 )) 0, φt4 − ( φ 4 3 ( ηpx uφx4 )) 0, 2.18 where φt1, φ t 2, φ t 3, φ t 4, φ x 1 , φ x 2 , φ x 3 , φ x 4 , φ xxx 2 , φ xxx 3 , and φ xxx 4 can be determined from the following relation φi1i2i3···is v Di1Di2Di3 · · ·Dis W −Wv uj u v ji1i2i3···is , 2.19 where v 1, 2, 3, 4 and u1, u2, u3, u4 are u, v, w, p, respectively. By substituting φt1, φ t 2, φ t 3, φ t 4, φ x 1 , φ x 2 , φ x 3 , φ x 4 , φ xxx 2 , φ xxx 3 , and φ xxx 4 into 2.18 and after some calculations, we obtain the Lie characteristic functions in the following form: W1 ut ( k3 − 34 k4t ) ux ( k1 − 14k4x ) 1 2 k4u, 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, 2.20 where k1, k2, k3, and k4 are arbitrary constants. Then the Ito systems have the following infinitesimal: ξ1 ( k3 − 34k4t ) , ξ2 ( k1 − 14k4x ) , η1 1 2 k4u, η 2 k4v, η3 k2w, η4 ( −k2 32k4 ) p. 2.21 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 6 Abstract and Applied 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: 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 εη1t, x, u, v,w, p O ( ε2 ) , v −→ V v εη2t, x, u, v,w, p O ( ε2 ) , w −→ W w εη3t, x, u, v,w, p O ( ε2 ) , p −→ P p εη4t, x, u, v,w, p O ( ε2 ) . ε 1. 2.22 The functions ξ1, 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 X̃ that is defined by X̃ ξ1 ∂ ∂t ξ2 ∂ ∂x η1 ∂ ∂u ∂ ∂ut η1 ∂ ∂v η2 ∂ ∂u η3 ∂ ∂p η4 ∂ ∂ρ φx1 ∂ ∂ux φt1 ∂ ∂ut φ2 ∂ ∂vx φ2 ∂ ∂vt · · · 2.23 and the symmetry vector field X given by 2.2 . The equations in 1.1 can be written in the form: H1 ut − vx , H2 vt − −2vxxx − 6 υ x − 6 ωπ x , H3 wt − wxxx 3υwx , H4 ( pt − ( pxxx 3υpx )) . 2.24 Abstract and Applied Analysis 7 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 haveand Applied Analysis 7 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 X̃H1 φt1 − φ2 0, X̃H2 φ2− [ −2φxxx 2 − 6 ( η1 ξ υφ2 η 2υξ φx1 ) −6 ( η3πξ φx4ω η 4ωξ φ3π )] 0, X̃H3 φ3 − ( φ 3 3 ( η1wx φ3u )) 0, X̃H4 φt4 − ( φ 4 3 ( η1px φx4u )) 0, 2.25 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-, secondand thirdorder 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 : − 3η1 3uξ2 − 3uξ1t ξ2xxx − 3η3 xxw 0, − 3η1 3uξ2 − 3uξ1t ξ2xxx − 3η4 xxp 0, 3η1 − 3uξ2 3uξ1t − ξ2xxx 3η2 xxv 0, η4 t − 3uη4 x − η4 xxx 0, η3 t − 3uη3 x − η3 xxx 0, 6vη1 x η 2 t 6uη 2 x 6pη 3 x 6wη 4 x 2η 2 xxx 0, − ξ2 η3 xw 0, −ξ2 η4 xp 0, − ξ2 η2 xv 0, η3 −wξ2 wξ1t −wξ1t −wη2 v pη3 p wη4 p 0, η2 − vξ2x vξ1t vη1 u − vη2 v 0, η4 − pξ2x pξ1t − pη2 v pη3 w wη4 w 0, 3ξ2x − 3ξ1t 0, ξ2x − 3ξ1t − η1 u − η2 v 0, η1 t − η2 x 0, η1 v η1 w η1 p 0, η2 p η 2 w η 2 vv η 2 u 0, η 3 u η 3 v η 3 xp η 3 pp η 3 ww η 3 wp 0, η4 u η 4 v η 4 xw η 4 pp η 4 ww η 4 xw 0, ξ2p ξ2w ξ2v ξ2t ξ2 0, ξ1p ξ1w ξ1v ξ1u ξ1x 0. 2.26 8 Abstract and Applied Analysis Table 1 X1 X2 X3 X4 X1 0 0 0 −X4/4 X2 0 0 0 0 X3 0 0 0 −3X3/4 X4 X4/4 0 3X3/4 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 ξ1 ( k3 − 34k4t ) , ξ2 ( k1 − 14k4x ) , η1 1 2 k4u, η 2 k4v, η3 k2w, η4 ( −k2 32k4 ) p, 2.27 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


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
Which is the generalization of the well-known integrable Ito system 3 :

1.2
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.

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.

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,

2.4
The Lie characteristic function is defined by where the functions ξ 1 , ξ 2 , and η can be given in terms of W as and the formulae for ξ i can easily be written in terms of W as Higher-order prolongations can be calculated from the prolongation formula: where D i is the operator of total differentiation given by If W is linear in the first derivatives u t and u x , then the contact transformation generator 2.4 reduces to an extended Lie point transformation generator 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 11 12

2.13
Lie point transformation generators were given by 4 .To determine contact transformations of 1.1 , we solve the determining equations: where X is the prolongation of the operator 2.4 in terms of W.
Consequently, we find that

and ϕ xxx
4 can be determined from the following relation into 2.18 and after some calculations, we obtain the Lie characteristic functions in the following form:

2.20
where k 1 , k 2 , k 3 , and k 4 are arbitrary constants.Then the Ito systems have the following infinitesimal:

2.21
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
The functions ξ 1 , ξ 2 , η 1 , η 2 , η 3 , and η 4 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 and the symmetry vector field X given by 2.2 .The equations in 1.1 can be written in the form:

2.24
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

2.27
where k 1 , k 2 , k 3 and k 4 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
The corresponding Lie algebra of infinitesimal symmetries of 1.1 is spanned by the infinitesimal generators X 1 , X 2 , X 3 , and X 4 .Thus, corresponding commutator table of {X i ; i 1, 2, 3, 4 } can be constructed Table 1.
It is easy to check that {X 1 , X 2 , X 3 , X 4 } are closed under the Lie bracket.Thus, a basis for the Lie algebra is {X 1 , X 2 , X 3 , X 4 }, which is a 4-dimensional Lie group algebra.

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
Consequently we get the following:

2.33
By solving 2.33 , we obtain the similarity solutions take the form:
Substituting 2.34 into the equations in 1.1 , we finally obtain the system of nonlinear ordinary differential equations for F 1 Z , F 2 Z , F 3 Z , and F 4 Z takes the form: Solving a system of an ordinary differential equations 2.4 , we have four cases of solutions for F 1 Z , F 2 Z , F 3 Z , and

2.36
where k 4 2k 2 , A 1 and B 1 are arbitrary constants with A 1 B 1 2.

2.37
where k 4 k 2 , A 2 and B 2 are arbitrary constants with A 2 B 2 2.

2.38
where K 4 2/3 K 2 and d is arbitrary constant.
where, K 4 2K 2 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 following Family 1.

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.