Two-Component Generalization of a Generalized the Short Pulse Equation

     In this article, we introduce a two-component generalization for a new generalization type of the short pulse equation was recently found by Hone and his collaborators. The coupled of nonlinear equations is analyzed from the viewpoint of Lieâ€™s method of a continuous group of point transformations. Our results show the symmetries that the system of nonlinear equations can admit, as well as the admitting of the three-dimensional Lie algebra. Moreover, the Lie brackets for the independent vectors field are presented. Similarity reduction for the system is also discussed.


Introduction
The short pulse equation appears in many fields of sciences, and it becomes a source of interest in nonlinear optics, among others. The derivation of the equation came out in the differential geometry in the work of [1,2]. The equation is given by , (1) where is a function of two independent variables and and the subscripts denote to the partial derivatives regarding independent variables. Sakovich [3] brought in a generalized version of equation (1), and later he gave a generalization for the generalized equation [4]. The super extensions of equation (1) are inspected in [5]. Pietrzyk el al. [6] suggested a vector generalization of the short pulse type equation, which is later studied in [7]. A long the same line, Mastsuno [8] came up with a multicomponent type of equation (1), and the author also succeeded in getting some type of solutions. Feg [9] also derived an integrable coupled short pulse equations, and the multi-component generalization of modified type of equation (1) that obtained as a reduction of Feg's system is considered in [10]. The Lax representations of Mastuno's and Feg's systems were examined by Popowicz [11]. More recently,

ISSN: 0067-2904
Allami Iraqi Journal of Science, 2019, Vol. 60, No.8, pp: 1760-1765 1761 Hone el al. [12] classified a general form of partial differential equations of order two and they found a new equation that generalizing equation (1). This is given by , (2) where is a field of one spatial dimension plus time . In the current work, we propose a generalization for equation (2) to two-component system of nonlinear equations, that is , (3) , (4) where and are two fields of one spatial dimension plus time . Clearly, the system consists of six nonlinear terms , and , and four linear terms and . As far as we know, the two-component system (3)-(4) that generalizes the generalized type of the short pulse equation that has introduced in the current work seen nowhere in the literature. We shall look at the system, which represents the main problem, in the coming sections. The remaining of the paper is lined up in such a way. In section two, the Lie group analysis is overviewed in order to make the work is a self-contained. We proceed, in section three, to analyse the two-component system of equations by Lie's method of a continuous group of point transformations. Section four is stressed to discuss the similarity reduction of the main problem. The conclusions are given in the last section.

Lie Symmetry Analysis
The Lie's method of a continuous group of point transformations is a well-known approach that has been applied to several types of differential equations. The start point of this approach goes back to Sophus Lie, and from since the method has been noticed and developed. The approach can be briefly summarized, following the same descriptions in [13] and also can be found in [14][15][16][17][18], as follows Consider a system of equations ( ) where are independent variables and are dependent variables, and for refer to the partial derivatives regarding independent variables, is the number of equations, and is the number of the highest derivatives. A general form of group of point transformations [13], reads , where the parameter is considered to be too small The linearization of the Lie group around the identity ( shapes the infinitesimals form of Lie group, this is given by [13], , (7) , (8) and the infinitesimals and are therefore taken to be and , with the initial conditions . The infinitesimal generator of (6) is expressed by and the extension of the infinitesimal generator (the Prolongations) to include the derivatives is [13], The groups (7)- (8) is admitted by (5) if the following hold [13], When a group of symmetries acts upon a system of equations it maps it to another system of new variables and the new system preserves the original system form as well as it maps its solutions; which leads to getting a new solution from the known one. The similarity variables associated with the Lie symmetries can be obtained by solving the characteristics equation [13], and the general solution can be expressed as , where are arbitrary constants. In the next section, we analyse the two-component system under study by Lie symmetry analysis.

Lie symmetry analysis for the Main problem
We deal with the main problem (3) subject to the conditions ( , and the corresponding infinitesimals generator is the linear first order differential operator (11) and the extended of transformation to include derivatives is . (12) From acting the second prolongation (12) on the system of equations (9)-(10), this is written by ) whenever one can obtain the following nonlinear partial differential equations , and the Lie bracket (or Commutator) is given by [20], [ ] . The Commentator of the Lie algebra is listed in Table-1.

Allami
Iraqi Journal of Science, 2019, Vol. 60, No.8, pp: 1760-1765 1764 Based on the relations [15], ∑ , and , . The adjoint construction for the Lie algebra is listed in Table-2   Table 2-The adjoint table of the Lie algebra  Ad We are now at the position to state the following, if is a solution for the twocomponents system of equations (3) , are also solutions satisfy the system of equations (3)-(4).

Similarity reduction of the main problem
We focus, in this section, on the reduction of the main problem (3)-(4) relying on similarity variables; these variables come from solving characteristics equations, and as a result a coupled of ordinary differential equations are formed.
In order to gain similarity variables related to the Lie symmetries (15) we solve the characteristics equations Take and solve it to have . In the same way, one can have and , and that implies and . Substituting into the coupled of nonlinear partial differential equations (9)-(10) (or (3)-(4)) one can get the following nonlinear system of equations as a reduction (16) where , and . That leads to state the following, if is a solution for the nonlinear equations (16)-(17) then {u(x,t),v(x,t)} is a solution for nonlinear equations (3)-(4).

Conclusions
To sum up, in the present work, we have proposed two-component generalization of a generalized the short pulse equation. The system of nonlinear equations have introduced here does not appear to have been considered before in the literature. Based on the Lie analysis we have characterized all possible symmetry groups that the two-component system of equations can admit; in terms of the space translation, the time translation and the scaling. The symmetry algebra of the two-component system of nonlinear equations is generated by the three vector fields. The Lie brackets for the vector fields are given. The similarity variable is used to get the reduction of the main problem to the coupled of nonlinear ordinary differential equations. To be clear, the main problem results are settled in sections three and four.