Research Article | Open Access
Lie Groups Analysis and Contact Transformations for Ito System
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.
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.,  and Karasu-Kalkanli et al. : Which is the generalization of the well-known integrable Ito system : 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 . 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 . 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.
We only summarize relevant aspects for the case of two independent variables (time, , and one space variable, ). The reader is referred to . The set of transformations in (, , ) space, namely, where 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
The set of transformations in space, namely, where is a real parameter, is a one-parameter group of contact transformations if it satisfies the group properties and , hold.
The generator of a group of contact transformations is
The Lie characteristic function is defined by where the functions , , and can be given in terms of as and the formulae for can easily be written in terms of as Higher-order prolongations can be calculated from the prolongation formula: where is the operator of total differentiation given by
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
Consequently, we find that where , , , , , , , , , , and can be determined from the following relation where and , , , are , , , , respectively.
By substituting , , , , , , , , , , and into (2.18) and after some calculations, we obtain the Lie characteristic functions in the following form: where , , , and are arbitrary constants. Then the Ito systems have the following infinitesimal:
2.3. Lie Groups Analysis
Many authors applied Lie group analysis to find exact solutions, for example, in  the authors used Lie symmetry analysis and the method of dynamical systems for the extended mKdV equation to obtain exact solutions, in  the authors applied Lie symmetry analysis and Painleve analysis for the new (2+1)-dimensional KdV equation, and in  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.
The functions , , , , , and are the infinitesimal of transformations for the variables , , , , , and , respectively. In order to obtain these infinitesimal functions we have to construct a third-extended vector field that is defined by and the symmetry vector field given by (2.2). The equations in (1.1) can be written in the form:
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 under , , , and . 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 , , , and into (2.25) and after some calculation, we obtain the following system of partial differential equations for , , , , , and [12–15]:
Now, we solve this system of linear partial differential (2.26) for the infinitesimal , , , , , and and we obtain where , , and are arbitrary constants. The above equations are the same as (2.21). We obtain from (2.21) that the Lie point transformation generators are
It is easy to check that are closed under the Lie bracket. Thus, a basis for the Lie algebra is , 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:
Consequently we get the following: from
By solving (2.33), we obtain the similarity solutions take the form: where and , , , and are arbitrary functions.
Solving a system of an ordinary differential equations (2.4), we have four cases of solutions for , , , and .
Case 1. where , and are arbitrary constants with .
Case 2. where , and are arbitrary constants with .
Case 3. where and is arbitrary constant.
Case 4. where, and is arbitrary constant.
Family 1. where , and are arbitrary constants with .
Family 2. where , and are arbitrary constants with .
Family 3. where and is arbitrary constant.
Family 4. where and is arbitrary constant and.
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.
The authors would like to thank the Deanship of Scientific Research, Northern Borders University For their financial support under Project no. 432/32.
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Copyright © 2012 M. M. Al-Shomrani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.