^{1}

^{*}

^{2}

For a class of (1 + 2)-dimensional nonlinear Schrodinger 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 Schrodinger equations can be reduced to a group of ordinary differential equations which is useful for solving the related problems of the equations.

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

where

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

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

where

Transforming 2D-CNLS equations (1) to real case by transformations

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

follows

follows

for functions

where

respectively and by transforming

recover the basis of the 8-dimensional Lie algebra

If taking other linear independents case of vector

The commutators of the generators (4) are given in the

column is defined as

The table is fundamental for our constructing the optimal system of the

In this section, we give an one-dimensional optimal system of the Lie algebra

where

following adjoint commutator

The following is the deduction procedure of one-dimensional optimal system of (4) by using the method gi-

ven in [

Let

suitable adjoint maps and find its equivalent representative. A key observation here is that the function

(the corresponding symmetry group of the Lie algebra

since it places restrictions on how far we can expect to simplify

simultaneously make

or

To begin the classification process, we first concentrate on the coefficients

taneously adjoints of

with coefficients

. The commutators of (4)

0 | 0 | 0 | 0 | 0 | ||||

0 | 0 | 0 | 0 | 0 | ||||

0 | 0 | 0 | 0 | |||||

0 | 0 | |||||||

0 | 0 | 0 | 0 | 0 | ||||

0 | 0 | 0 | 0 | 0 | ||||

0 | 0 | 0 | 0 | |||||

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

. The adjoint commutator of (4)

Ad | ||||||||
---|---|---|---|---|---|---|---|---|

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

Case 1. If

further by adjoint maps generated respectively by

of

Case 2. If

group generated by

multiple of

Case 3. If

choose

and

ently scales the coefficients of

zero using the groups generated by

the coefficients of

for some

The last remaining case occurs when

sary. If

make the coefficients of

equivalent to a multiple of

for some

In summary, an optimal system of one-dimensional subalgebras of

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

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

The system yields the similarity variables as follows

Hence we let

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

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

In fact, the equations in

Using characteristic set algorithm given in [

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

where

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

No. | Generators in (6) | The first reductions | Invariance variables |
---|---|---|---|

1 | |||

2 | |||

3 | |||

4 | |||

5 | |||

6 | |||

7 | |||

8 | |||

9 |

reduction procedure.

The characteristic system

yields the corresponding similarity variables

Hence we let

and substitute them into the underline equations, then the second equation in

This is a result of twice reductions of (1) by

sively. 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

Solving the second equation in (14), we have

where

. The second reductions of the 2D-CNLS (1) with X^{2}

No. | Generators in (11) | The second time reductions of 2D-CNLS (1) | Invariance variables |
---|---|---|---|

1 | |||

2 | |||

3 | |||

4 |

where

where

For

In this paper, the infinite dimensional Lie algebra of 2D-NLS equations (1) is determined. The optimal system of a sub-algebra

. The second reductions of 2D-CNLS (1) with X^{3}

No. | Generators in (11) | The second time reductions of 2D-CNLS (1) | Invariance variables |
---|---|---|---|

1 | |||

2 | |||

3 | |||

4 |

where

. The second reductions of 2D-CNLS (1) with X^{4}

No. | Generators in (11) | The second time reductions of 2D-CNLS (1) | Invariance variables |
---|---|---|---|

1 | |||

2 | |||

3 | |||

4 |

where

. The second reductions of 2D-CNLS (1) with X^{5}

No. | Generators in (11) | The second time reductions of 2D-CNLS (1) | Invariance variables |
---|---|---|---|

1 | |||

2 | |||

3 | |||

4 |

where

. The second reductions of 2D-CNLS (1) with X^{1}, X^{6}, X^{7}, X^{8} and X^{9}

No. | Generators in (6) | Generators of the first reduced eqs. | The second time reductions of 2D-CNLS (1) | Invariance variables |
---|---|---|---|---|

1 | ||||

2 | infinite dimensional | |||

3 | ||||

4 | ||||

5 | ||||

where

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