Journal of Applied Mathematics and Physics
Vol.2 No.7(2014), Article ID:47035,14 pages DOI:10.4236/jamp.2014.27075

Lie Symmetries, One-Dimensional Optimal System and Optimal Reduction of (2 + 1)-Coupled nonlinear Schrödinger Equations

A. Li1, Chaolu Temuer2

1Inner Mongolia University, Hohhot, China

2Shanghai Maritime University, Shanghai, China

Email: 342388241@qq.com

Copyright © 2014 by authors and Scientific Research Publishing Inc.

Received 27 March 2014; revised 27 April 2014; accepted 7 May 2014

Abstract

For a class of (1 + 2)-dimensional nonlinear Schrödinger equations, the infinite dimensional Lie algebra of the classical symmetry group is found and the one-dimensional optimal system of an 8-dimensional subalgebra of the infinite Lie algebra is constructed. The reduced equations of the equations with respect to the optimal system are derived. Furthermore, the one-dimensional optimal systems of the Lie algebra admitted by the reduced equations are also constructed. Consequently, the classification of the twice optimal symmetry reductions of the equations with respect to the optimal systems is presented. The reductions show that the (1 + 2)-dimensional nonlinear Schrödinger equations can be reduced to a group of ordinary differential equations which is useful for solving the related problems of the equations.

Keywords:Nonlinear Schrödinger Equations, Lie Aymmetry Group, Lie algebra, Optimal System

1. Introduction

We plan to consider the (1 + 2)-dimensional coupled nonlinear Schrödinger (2D-CNLS) equations with cubic nonlinearity

(1)

where, are complex-valued functions. The 2D-CNLS equations which describes the evolution of the wave packet on a two-dimensional water surface under gravity was derived by Benny and Roskes [1] and Davey and Stewartson [2] . The solutions of the equation have been studied by several authors [3] -[12] . The multisoliton solutions were obtained by Anker and Freeman [8] . They showed that the two-soliton resonant interaction occurs and a triple soliton structure is produced. A similarity reductions of the 2D-CNLS equation is also studied in [9] . Nakamura [10] found explode-decay mode solutions by using the bilinear method. However, the algebra properties of the Lie algebra admitted by (1) has not been studied so far. The optimal system of the Lie algebra yields the optimal classification of the invariant solutions set to the 2D-CNLS which is essential to distinguish the inequivalent classes of the invariant solutions of the equation.

In this paper, we show the optimal reduction classifications of the 2D-CNLS equations (1) through studying one-dimensional optimal system of the Lie algebra of the equations.

Outline of the paper is following. In §2, the complete infinite-dimensional Lie algebra of the Lie symmetry group of the 2D-CNLS equations is derived which covered the results obtained in [9] . In §3, the onedimensional optimal system of an 8-dimensional subalgebra, presented in [9] , of the is constructed. In §4 the first reductions of the 2D-CNLS equation (1) with respect to the optimal system obtained in §3 are given. In §5 we construct one-dimensional optimal systems of Lie algebras of the reduced equations obtained in §4 which yields the second reductions of (1). Consequently, the 2D-CNLS equation (1) can be reduced to a group of scale ordinary differential equations, which is essential to solve different exact solutions of the 2D-NLS equation (1).

2. The Lie Algebra of the 2D-CNLS Equations (1)

In this section, we present the Lie algebra of point symmetries of 2D-CNLS (1). To obtain the Lie algebra, we consider the one parameter Lie symmetry group of infinitesimal transformations in given by

(2)

where is the group parameter. Hence the corresponding generator of the Lie algebra of the symmetry group is

Transforming 2D-CNLS equations (1) to real case by transformations , where, , and are real functions, one has real form of the 2D-CNLS equations (1) in four unknown functions and.

Assuming that the 2D-CNLS equations (1) is invariant under the transformations (2), then its real form transformed system is invariant under the Lie symmetry group with generator

, in which

. By invariance criterion in [11] -[13] , we have the DTEs of the Lie symmetry group as follows

for functions. Solving this system by characteristic set algorithm given in [14] [15] , we obtain the infinitesimal functions of generator, i.e.,

(3)

where, , , are arbitrary functions of their argument. Hence the 2D-CNLS (1) admits infinite dimensional Lie algebra. It is notice that in [9] only a special subset of (3) were found. Namely, if taking here a linear independent representatives of the vectors as

respectively and by transforming , we recover the basis of the 8-dimensional Lie algebra given in [9] as follows

(4)

If taking other linear independents case of vector, we obtain other subalgebras of

. In this paper, we take the case (4) as example to show the investigation procedure for finite sub-algebras properties of the infinite dimensional algebra.

The commutators of the generators (4) are given in the Table 1, where the entry in the row and

column is defined as,

The table is fundamental for our constructing the optimal system of the with basis (4).

3. One-Dimensional Optimal System of (8

In this section, we give an one-dimensional optimal system of the Lie algebra spaned by (4). Finding one-dimensional optimal system of one-dimensional subalgebras of a Lie algebra is a subalgebra classification problem. It is essentially the same as the problem of classifying the orbit of the adjoint representation, since each one-dimensional subalgebra is determined by nonzero vector in the Lie algebra. Hence it is equivalent to classification of subalgebras under the adjoint representation of the Lie algebra. The adjoint representation is given by the Lie series

where is the commutator given in Table 1, is a parameter, and. This yields following adjoint commutator Table 2 for (4) in which the entry gives.

The following is the deduction procedure of one-dimensional optimal system of (4) by using the method given in [15] -[20] .

Let be an element of spanned by (4), which we shall try to simplify using suitable adjoint maps and find its equivalent representative. A key observation here is that the function

is an invariant of the full adjoint action, that means

(the corresponding symmetry group of the Lie algebra). The detection of such an invariant is important since it places restrictions on how far we can expect to simplify. For example, if, then we cannot simultaneously make and all zero through adjoint maps; if, we cannot make either

or zero!

To begin the classification process, we first concentrate on the coefficients of. Acting simultaneously adjoints of and, one has

with coefficients

(5)

Table 1. The commutators of (4).

Table 2. The adjoint commutator of (4).

There are now three cases, depending on the sign of the invariant.

Case 1. If, then we choose to be either real root of the quadratic equation

and (which is always well defined). Then, while

, so is equivalent to a multiple of. Acting further by adjoint maps generated respectively by and we can arrange that the coefficients of

and in vanish. Therefore, every element with is equivalent to a multiple of for some. No further simplifications are possible.

Case 2. If (implies), set to make. Acting on by the group generated by, we can make the coefficients of and agree, so is equivalent to a scalar multiple of. Further use of the groups generated by

and show that is equivalent to a scalar multiple of for some.

Case 3. If, there are two subcases. If not all of the coefficients vanish, then we can choose and in (5) so that, but, so is equivalent to a multiple of

. Suppose. Then we can make the coefficients of

and zero using the groups generated by and, while the group generated by independently scales the coefficients of and. Thus such a is equivalent to a multiple of either

for some . If, so is equivalent to a multiple of

, suppose. Then we can make the coefficients of and

zero using the groups generated by, and, while the group generated by independently scales the coefficients of and. Thus such a is equivalent to a multiple of either. If

, then the group generated by and can be reduce to a vector of the formfor some.

The last remaining case occurs when, for which our earlier simplifications were unnecessary. If, then using groups generated by and we can arrange to become a multiple of

for some. If, but, , then we can make the coefficients of and zero using the groups generated by and, while is equivalent to a multiple of for some. If, but, we can first act by

and get a nonzero coefficients in front of which is reduced to the previous case. If

but, then we can arrange to become multiple of

for some. The only remaining vectors are the multiple of.

In summary, an optimal system of one-dimensional subalgebras of with base (4) is provided by generators

(6)

4. First Optimal Reductions of 2D-CNLS (1) with (6)

In this section, we give a classification of symmetry reductions of 2D-CNLS (1) by using optimal system (6).

Since the similarity, we will introduce the details of computation for in (6) and directly give the computation results without showing the details of the procedure for the remaining cases in (6).

The differential invariants (and hence the similarity variables) for the generator can be obtained by solving the characteristic system

(7)

The system yields the similarity variables as follows

Hence we let

(8)

and substitute them into the equations (1), then the equations are reduced to

(9)

Using the rest elements in (6), we can obtain the rest reductions of 2D-CNLS Equations (1) presented in following Table3 Here

5. Further Optimal Reductions of (1) through Reductions of the Equations in Table 3

In fact, the equations in Table 3 can be reduced further in the similar way which results in the second time reductions of 2D-CNLS (1). We take the second case in Table 3 as example to show the procedure of the second time reduction of the equation (1).

Using characteristic set algorithm given in [14] [15] , the symmetry algebra generators of the second equations (9) with similarity variables of case B in Table 3 is determined as follows

(10)

Using the same procedure in last section, we can also find an one-dimensional optimal system of one-dimensional subalgebras of the Lie algebra spanned by (10). The optimal system consists of

(11)

where, are arbitrary constants. We take as example to show the further

Table 3 . The first reductions of the 2D-CNLS (1) by optimal system (6).

reduction procedure.

The characteristic system

(12)

yields the corresponding similarity variables

Hence we let

(13)

and substitute them into the underline equations, then the second equation in Table 3 is reduced to

(14)

This is a result of twice reductions of (1) by and successively. In the same manner, we can obtain the other reductions of the equation with using the other elements in (11) which are listed in the following Table4 In fact, (13) and (14) are listed as second case in Table4

Solving the second equation in (14), we have

where are arbitrary constants. Substituting this into the first equation of (14), we get a scale reduction of 2D-CNLS (1) as follows

Table 4. The second reductions of the 2D-CNLS (1) with X2.

where

where and are arbitrary constants.

For and, we also have optimal systems and the corresponding reductions which are given in

Table 5, Table 6, Table 7 and Table 8 respectively.

6. Conclusion

In this paper, the infinite dimensional Lie algebra of 2D-NLS equations (1) is determined. The optimal system of a sub-algebra of the infinite dimensional Lie algebra is constructed using method given in [12] -[14] . As a result, the first reductions of the 2D-NLS equation (1) is presented by infinitesimal invariant method [14] [15] . The corresponding optimal systems of the Lie algebras admitted by the first reduced equations are also constructed. Consequently, the second time reductions classifications of the 2D-NLS equations (1) are obtained by these optimal systems. The twice reduction procedure shows that the 2D-NLS equation (1) can be reduced to a group of ordinary differential equations, which is helpful to explicitly solve the 2D-NLS equations (1).

Table 5. The second reductions of 2D-CNLS (1) with X3.

where

Table 6. The second reductions of 2D-CNLS (1) with X4.

where

Table 7. The second reductions of 2D-CNLS (1) with X5.

where

Table 8. The second reductions of 2D-CNLS (1) with X1, X6, X7, X8 and X9.

where

Acknowledgements

This research was supported by the Natural science foundation of China (NSF), under grand number 11071159.

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