Group analysis and variational principle for nonlinear ( 3 + 1 ) schrodinger equation

In recent years, a number of works of the symmetry methods are found to be very efficient in applications to differential equations in Physics and Engineering. A subject of a special interest is a study of invariance properties of the equations with respect to local Lie groups point transformations of dependent and independent variables. The importance of the conservation laws lies in the fact that there are situations where numerical schemes have been devised keeping in view the conservation form of the DEs. Also, the conservation law can be used for serving a priori estimates and to obtain integrals of motion, where for certain types of solutions, the conserved density, when integrated, provides us with a constant of motion of the system. Actually, finding the conservation laws of a system is often the first step towards finding its solution. Rund1 and Logan2 have studied the invariance of fundamental functional integral and deduced the first integrals or conservation law for the corresponding system of DEs. The nonlinear (3+1) Schrödinger equation [3, 4] is described by nonlinear couple partial differential equations. It is well know that a Material Science Research India Vol. 7(1), 115-122 (2010)


INTRODUCTION
In recent years, a number of works of the symmetry methods are found to be very efficient in applications to differential equations in Physics and Engineering.A subject of a special interest is a study of invariance properties of the equations with respect to local Lie groups point transformations of dependent and independent variables.The importance of the conservation laws lies in the fact that there are situations where numerical schemes have been devised keeping in view the conservation form of the DEs.Also, the conservation law can be used for serving a priori estimates and to obtain integrals of motion, where for certain types of solutions, the conserved density, when integrated, provides us with a constant of motion of the system.Actually, finding the conservation laws of a system is often the first step towards finding its solution.Rund 1 and Logan 2 have studied the invariance of fundamental functional integral and deduced the first integrals or conservation law for the corresponding system of DEs.The nonlinear (3+1) Schrödinger equation [3,4] is described by nonlinear couple partial differential equations.It is well know that a part of one parameter symmetry groups of these equations turns out to be their variational symmetries.According to Noether theorem [5,6] such as invariance of the elementary action is a necessary and sufficient condition of the existence of conservation laws for the Schrödinger equation.Let us consider Schrödinger equation with nonlinear term: ...(1)
But solving it with arbitrary functions requires analysis.We omit the determining system from which we are able to obtain the following results in the form of the coordinates ...( 4 ... (7) from ( 7), we have where,,TRN are constants.

Case 2
When then the infinitesimal takes the form ... (8) The invariance surface condition may be solved to yield the functional form: This leads to a reduction of the system (2) in the following form ... (10) and This leads to a reduction of the system (2) in the following form We make use of dilation group and after some manipulations on the system (10) we have the equivalent system: ... (10) Where using again dilation group on the system (11) one gets where Consequently, system (12) has the solution when ,ABare constants.

Case 3
In this case the infinitesimal takes the form Applying the rules as before (such as case 1, 2), we get:

variational principle
In order to study variational principle for our problem, ... (18) The system (18) satisfied the consistency conditions for the existence of functional integral.Consequently, a functional integral can be written by using the formula given by Tonti [8,9]

1
) and where, λ 1 , λ 2 , λ 3 and λ 4 arbitrary constants and are arbitrary functions.Let the function ) S (t, x, y, z,) satisfies the equation Then we have the following cases: Case When and then the infinitesimal take the form and ) S (t, x, y, z,) satisfies the equation: By applying the invariance surface condition, we obtainWhere λ 1 , λ 2 , L 1 , L 2 are constants and F, H, µ are functions to be determined by substituting ( 4 ) into the system (2-i,2-ii) ,that is ,by solving the following reduced system ...(6) In order to solve system (6), we change the variables Consequently, the system (6) takes the form: and system(13) in(2), we obtain, ...(14) After some calculations on (14), we have the solution in the form: ) and changing the variables then (2) can be written as the follows and System (16) has the solution In this case µ(θ) is an arbitrary function.
as: ),(vuJ Since or since ...(19) with L being the Lagrangian function, we obtain for which the Euler-Lagrange equation is order to prove the invariance of the fundamental functional integral where L is the Lagrangian function and Ω represents the domain of integration to be invariant under the one-parameter group of transformations (3), we can use the following theorem: Theorem 1 [ 2] If the fundamental functional integral defined by (19) is invariant under the ƒ{rparameter family of transformation (3), then the Lagrangian L and its derivatives satisfy the ƒ{ridentities: for where and Then, Lagrangian and its derivatives must satisfy the condition: L On Substituting for L and its derivatives in equation (21), we get a polynomial in etc Collecting it in descending order of various powers of a n d equating to zero the different powers of w e obtain a system of first order PDEs.On solving this system, we get the following expressions for ...(22)where and 4care arbitrary constants, but arbitrary functions of t[12- 19].By using the following theorem Theorem 2 (Noether's Identity)[5,10] Under the hypothesis of theorem 1, the following ƒ{rconservation laws concerned with finding potentials that admit Lie point symmetries for the nonlinear (3+1) Schrödinger equation, i.e., we give a classification of potentials admitting point symmetries.Finally, we apply Noether's theorem to show which of this point symmetries are variational and we obtain the corresponding conservation laws