fulltext

On the invariant variational principle for Sturm-Liouville equation,

linear and nonlinear Dirac system

A. H. KHATER(1) (2), D. K. CALLEBAUT(2), A. G. RAMADY(1) and S. F. ABDUL-AZIZ(1)

(1_{) Department of Mathematics, Faculty of Science, Cairo University - Beni-Suef, Egypt}

(2_{) Physics Department, U.I.A. University of Antwerp}

Universiteitsplein 1, B-2610 Antwerp (Wilrijk), Belgium (ricevuto il 5 Agosto 1997; approvato il 6 Novembre 1997)

Summary. — The functional integrals corresponding to Sturm-Liouville equation

and linear as well as nonlinear Dirac system are derived. The invariance of the resulting functional integrals are studied under a one-parameter group of transformations (Lie group). Finally, conservation laws are obtained through the applications of the invariant variational principle technique and classes of solutions are given in some cases.

PACS 11.30 – Symmetry and conservation laws.

PACS 11.10.Ef – Lagrangian and Hamiltonian approach. PACS 03.65.Pm – Relativistic wave equations.

PACS 02.30.Wd – Calculus of variations and optimal control.

1. – Introduction

The calculus of variation usually deals with the direct problem of obtaining the Euler-Lagrange equations for a given functional (with various applications [1-4]), but it does not consider the inverse problem (IP) of finding a functional from which a given equation (linear or nonlinear) can be deduced and when this is possible [5].

The search for a variational principle is equivalent to the IP of the calculus of variations: that is, finding the functional integral whose stationary points are described by the descriptive equations. In fact, a functional integral corresponding to a given differential equation (DE) or a system of DEs was formulated through practice only.

Within the context of modern functional analysis [6-8], however, the solution of the IP becomes fairly straightforward. The IP was solved in principle by Vainberg (1954), and his results were published in English in 1964. Answering the question of the existence of a variational principle is equivalent to determining whether or not the given operator is potential [6], where a potential operator is the gradient of a functional integral, i.e., the operator N is called the potential operator of the functional integral J(u) if dJ(u) 4

s

VN(u) dJ(u) dV . dJ(u) is the variation of the functional integral J(u) for a variation

du of the vector u in the direction tangent to a line [9]. If the operator is a potential

operator, the variational principle is given by a straightforward calculation.

Unfortunately Vainberg’s very general and abstract theorem was still inaccessible to many applied mathematicians and engineers. Tonti [9] recognized the importance of Vainberg’s work and developed a formalism and procedure to derive many operational formulas to determine whether a given operator is potential. Finlayson [ 10 , 11 ] used Tonti’s formalism and extended the concept of an adjoint operator to nonlinear (NL) equations. After this work appeared, results were available for fourth-order ordinary differential equations (ODEs), second-order partial differential equations (PDEs), and systems of second-order PDEs.

Atherton and Homsy [12] generalized the work of Tonti by deriving the consistency conditions for the existence of a functional integral for nonlinear differential equations (NLDEs) in an arbitrary number of independent variables and of arbitrary order, discussed their use, and presented applications.

There is a precise test [13] to find out whether a given DE (problem) has a variational formulation or not. This test is the symmetry of the operator, if this is linear, or the symmetry of its Gateaux derivative, if the operator is nonlinear.

The essential point, not usually stressed, is that the symmetry of a linear operator, like that of a matrix, is not an absolute notion: it is relative to a bilinear form. Using the integrating operator, Tonti [13] showed that one may find many functionals whose minimum is the variational formulation, for every linear or NLDEs (problems), whose solution exists and is unique.

Ibragimov [14] considered the 4-dimensional free particle linear Dirac equation and its conjugate equation. He derived the Lagrangian and the vector conservation laws.

Moreover NL Dirac systems for constructing models of extended particles have been investigated by various authors [15-20]. The complete algebras as well as conserved currents associated to infinitesimal symmetries for linear and nonlinear Dirac system have been established in [15]. Steeb et al. [16] demonstrated for a NL Dirac equation how the knowledge of infinitesimal symmetry generators can be used for deriving similarity solutions and conserved current, by adopting the Hamilton-Cartan formalism (jet bundle formalism). They discussed the integrability of the latter equations applying results in soliton theory [17]. The existence of stationary states for Dirac fields with singular nonlinearities is proved by Balan et al. [18]. Esteban and Sere [19] used a general variational technique to prove the existence of stationary solutions for some NL Dirac equations, while the split-step spectral schemes for the numerical integration of [ 1 11]-dimensional NL Dirac systems was considered by de Frutos and Sanz-Serna [20].

Influenced by the work of Lie [21] and Klein [22] on the transformation properties of DEs under continuous groups of transformations, Noether proved two fundamental results which she published in the form of two theorems.

Bhutani and Mital [23], Bhutani et al. [24], and Bhutani and Vijayakumar [25] have examined the existence and hence formulated the variational principles for the NL shallow membrane equation, the NLDEs of nuclear engineering system, the Emden-Fowler equation and Emden’s equation, respectively. Also the variational principles have eventually been utilized for writing down their first integrals via Noether’s theorem.

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. Indeed finding the conservation laws of a system is often the first step towards finding its solution.

Recently, using the invariant variational principle technique [ 26 , 27 ], solutions for some NLODEs and conservation laws for some NLPDEs are obtained. The inverse scattering problem for a linear Dirac system is studied by Khater et al. [28].

The main aim of this paper is threefold, the first is the formulation of the functional integrals corresponding to Sturm-Liouville equation and linear as well as nonlinear Dirac system by using Tonti [ 5 , 9 ], the second is the invariance of the resulting functional integrals under a one-parameter group of transformations (Lie group) and the third is concerned with classes of solutions and conservation laws through the applications of Noether’s identity.

2. – Necessary preliminaries

2.1. Existence and formulation of variational principles. – The consistency conditions for the existence of a functional integral and the method for writing it down whenever it exists for a single NLODE of second order are summarized as follows: for the general second-order NLDOE of the form:

N(x , y , y8, y9) 40 ,

(2.1)

the consistency conditions for the existence of a functional integral are given by [9] ¯N ¯_{y 8} 4 d dx

g

¯N ¯_{y 9}

h

. (2.2)

Furthermore, for any operator N( y) for which the above conditions are satisfied, a functional integral J( y) can be written down using the formula given by Tonti [9] as

J( y) 4



V y

y



0 1 N(ly) dy

z

dV , (2.3)

or through its equivalent variational statement

dJ( y) 4



V

N( y) dy dV ;

(2.4)

in eqs. (2.3), (2.4),

s

VdV respresents the integration over the domain V and

s

0 1

dl represents the integration over the scalar variable l .

2.2. Conservation laws via Noether’s theorem. – The fundamental functional integral [29] corresponding to second-order variational problems can be written as

J(u) 4



L

(

x , u(x), ¯u(x), ¯2_u(x)

₎

_dx1_Rdxm_,

(2.5)

where x 4 (x1_{, R , x}m

), u(x) 4

(

u1_{(x), R , u}n_(x)

₎

C_n4(V), where C_n4(V) is the set of all continuous functions with continuous fourth-order partial derivatives, L is the Lagrangian density and ¯u(x) and ¯2_{u(x) denote the collection of first and second}

partial derivatives un_k a4 ¯uk ¯xa , u n n_k ab4 ¯2uk ¯xb ¯xa ,

respectively and a r-parameter family of transformations . / ´ xa_{4 W}a(x , u , e) , (a , b 41, R, m; k41, R, n) , uk 4 ck(x , u , e) . (2.6)

The following theorems are of fundamental importance [29]:

Theorem 2.1. If the fundamental functional integral (2.5) is invariant under the r-parameter family of transformations (2.6), then the Lagrangian L and its derivatives

satisfy the r-identities ¯L ¯xa tas1 ¯L ¯uk z k s1 ¯L ¯un_k a

g

dzks dxa 2 u n_k n dtsn dxa

h

1 ¯L ¯un n k ab 3 3

g

d 2_z s k dxa_dxb 2 u n n k bn dtns dxa 2 u n n k na dtsn dxa 2 u n_k n d2_t s n dxa_dxb

h

1 L dtas dxa 4 0 , for s 41, R, r, where tas(x , y) f ¯Wa ¯es

N

_{e 40} and z k s(x , u) f ¯ck ¯es

N

_{e 40}.

Theorem 2.2 (Noether’s identity). Under the hypothesis of theorem 2.1, the

following r-conservation laws hold true: d dxa

y

Lt a s1

g

¯L ¯un_k a 2 d dxb ¯L ¯un n_k ba

h

Csk1 ¯L ¯un n_k ab dCsk dxb

z

4 0 for s 41, R, r, where Cskfzks2 u n_k ntns. 3. – Applications

In view of the strong connection between the Sturm-Liouville operator and the Dirac system, we will consider the variational formulation for the Sturm-Liouville equation before considering the variational formulation for the Dirac system.

3.1. The Sturm-Liouville equation. – Consider the Sturm-Liouville equation 2 y 9 1 q(x) y 4 ly .

(3.1)

The existence of the variational principle is proved as follows: for any linear operator

N( y) of the form

(

see eq. (2.1)

)

N(x , y , y 9) 42y 91q(x) y2ly40

to be a potential operator, it must satisfy the consistency condition (2.2). It is clear that, the condition (2.2) is satisfied for eq. (3.2). Using eq. (2.3), the functional integral J for eq. (3.1) can be written as

J( y) 4 (1O2)



y

(

_{y 91 (l1q) y}

)

dx .

On choosing the BCs on y8 to be such that the boundary terms vanish, we get the functional integral in the form

J 4 (21O2)



(

_{y 8}2

1 (q 2 l) y2

)

dx . (3.3)

Thus, the Lagrangian, leading to eq. (3.1), is given by

L 4 (21O2)



(

y 82

1 (q 2 l) y2

)

, (3.4)

for which the Euler-Lagrange equation becomes ¯L ¯y 2 d dx

g

¯L ¯_{y 8}

h

4 0 ,

yielding eq. (3.1). In order to prove the invariance of the fundamental functional integral

(

s

L dx

)

, we look for a one-parameter infinitesimal group of transformations

(

see eq. (2.6)

)

of the form [30] . / ´ x 4x1et(x, y)1O(e2) , y 4y1ez(x, y)1O(e2) . (3.5)

On using theorem 2.1, the necessary condition for the fundamental functional integral

(

s

L dx

)

to be invariant under the one-parameter group of transformations (3.5) is given by (3.6) ¯L ¯x t 1 ¯L ¯y z 1 ¯L ¯_{y 8}

k

¯z ¯x 1

g

¯z ¯y 2 ¯t ¯x

h

y 82 ¯t ¯y y 8 2

l

1 L

k

¯t ¯x 1 ¯t ¯y y 8

l

4 0 .

On substituting for L and its derivatives in eq. (3.6) and collecting in descending order the coefficients of various powers of y8 and setting these coefficients equal to zero, we get the following system of first-order PDEs for t(x , y) and z (x , y):

¯t ¯y 4 0 , (3.7) ¯z ¯x 4 0 , (3.8) ¯t ¯x 2 2 ¯z ¯y 4 0 , (3.9) y2 tq 81txy2(q 2l)12[ yz(q2l) ] 40 . (3.10)

On solving the system of eqs. (3.7)-(3.9) we get the following expressions for t and z :

t 4 [c1x 1c2] , z 4

c1

2 y 1c3, (3.11)

where ci, i 41, 2, 3 are arbitrary constants.

Substituting eq. (3.11) into eq. (3.10), we obtain

y2[q8(c1x 1c2) 12c1(q 2l) ]12c2y(q 2l) 40 , (3.12) thus c34 0 and q8(c1x 1c2) 12c1(q 2l) 40 . (3.13) Equation (3.13) yields q 2l4 c4 (c1x 1c2)2 , (3.14)

which is a condition on q, where c4 is an arbitrary constant. Thus the one-parameter

infinitesimal group of transformations (3.5) takes the form

.

/

´

x 4x1e[c1x 1c2] 1O(e2) , y 4y1e c1 2 y 1O(e 2 ) . (3.15)

Further, using theorem 2.2, the first integral can be written as

g

L 2y 8 ¯L

¯_{y 8}

h

t 1 ¯L

¯_{y 8} z 4const , (3.16)

which for the case under consideration reduces to (c1x 1c2) y 822 c1yy 82

c4

(c1x 1c2)

y2

4 c5,

where c5 is an arbitrary constant. If we choose c54 0 and integrate, we get an exact

solution of eq. (3.1) in the form

y 4 (c1x 1c2)a, (3.17) where a 4 1 2 6

o

1 4 1 c4 c12 .

3.2. The linear Dirac system. – The linear Dirac system can be expressed as [31] 2 y 824

(

2 v(x) 2 c 1 l

)

y11 p(x) y2,

(3.18)

y 814 p(x) y11

(

v(x) 1c1l

)

y2.

Differentiating the second equation and using the first in it, we obtain (3.20) _{y 9}12 y1

{

p8(x)1

(

(v 1c)22 l2

)

1 p2(x) 2 p(x) v8(x) l 1c1v(x)

}

2 2

g

v8(x) l 1c1v(x)

h

y 814 0 4 N( y1) .

Now we test the applicability of the consistency condition of the above equation. It is clear that N( y1) is a potential operator if it satisfies the consistency condition (2.2). In

order to obtain the functional integral and the exact solution of eq. (3.20) we address the following two cases:

Case (i) v(x) 4const

Clearly the condition (2.2) is satisfied. Using eq. (2.3), the functional integral J for eq. (3.20) can be written as

J( y1) 4 (21O2)



] y1821

(

p 81p21 (v 1 c)22 l2

)

y12( dx .

(3.21)

Thus, the Lagrangian, leading to eq. (3.20), is given by

L 4 (21O2)] y1821

(

p 81p21 (v 1 c)22 l2

)

y12( ,

(3.22)

for which the Euler-Lagrange equation ¯L ¯y1 2 d dx

g

¯L ¯_{y 8}1

h

4 0 ,

yields eq. (3.20). In order to prove the invariance of the fundamental functional integral

(

s

L dx

)

, we look for a one-parameter infinitesimal group of transformations (3.5). On using theorem 2.1, we get the following system of first-order PDEs for t(x , y) and z (x , y): ¯t ¯y1 4 0 , (3.23) ¯z ¯x 4 0 , (3.24) ¯t ¯x 2 2 ¯z ¯y1 4 0 , (3.25) 2 ( 1 O2 ) ty12( p 912pp8)2y1z

(

p 81p21 (v 1 c)22 l2

)

2 (3.26) 2 ( 1 O2 ) c1y12

(

p 81p21 (v 1 c)22 l2

)

4 0 .

On solving the system of eqs. (3.23)-(3.25), we get the following expressions for t and z :

t 4b1x 1b2, z 4 (1O2) b1y11 b3,

(3.27)

where bi, i 41, 2, 3, are arbitrary constants. Substituting eq. (3.27) into eq. (3.26), we

obtain y1](b1x 1b2)( p 912pp8)12b1

(

p 81p21 (v 1 c)22 l2

)

( 1 1 2 b3

(

p 81p21 (v 1 c)22 l2

)

4 0 , thus, b34 0 and (b1x 1b2)( p 912pp8)12b1

(

p 81p21 (v 1 c)22 l2

)

4 0 , (3.28) which yields p 912pp 8 p 81p2 1 (v 1 c)22 l2 1 2 c1 c1x 1c2 4 0 , integrating, we have p 81p2 1 (v 1 c)22 l24 1 b5(b1x 1b2)2 , (3.29)

where b5is on arbitrary constant.

Further, using (3.16) the first integral takes the form 1 2 y18 2_(b 1x 1c2) 2y 81

g

b1 2 y1

h

2 1 2 y 2 1

(

p 81p21 (v 1 c)22 l2

)

(b1x 1b2) 4c1, choosing c1

4 0 , and using eq. (3.29) we have 1 2 y18 2_(b 1x 1b2) 2 b1 2 y1y 812 1 2 y 2 1 1 b5(b1x 1b2) 4 0 , which has the solution

y14 B(b1x 1b2)( 1 6A)O2, (3.30) where A 4

o

1 1 4 b5b12 , B is an arbitrary constant .

Then eq. (3.30) is an exact solution of eq. (3.20). Substituting (3.30) into (3.19) the second component y2is y24 B v 1c1l

k

b1

g

1 6A 2

h

(c1x 1b2) (216A)O2 2 p(x)(b1x 1b2)( 1 6A)O2

l

. (3.31)

Case (ii) v(x) is a function of x

For the linear operator (3.20), it is clear that condition (2.2) is not satisfied. Consequently, we look for an integrating factor of the form f (x) which, when multiplied by eq. (3.20), transforms it to a potential operator. Thus, we can write eq. (3.20) in the form (3.32) N(x , y1, y 81, y 91) 4 4 f (x)

m

y 912 y1

g

p 81

(

(v 1c)22 l2

)

1 p22 pv 8 v 1c1l

h

2 v 8 v 1c1l y 81

n

4 0 .

On using the consistency condition (2.2) for eq. (3.20), we find that f (x) 4mO

(

v(x) 1 c 1l

)

, where m is an arbitrary constant. Thus, the existence of the alternative potential principle [29] for eq. (3.32) is proved. Now, as in the previous case, the Lagrangian L, leading to eq. (3.32), is given by

L 4 21 2

g

y182 v 1c1l 1

g

p 81p2 v 1c1l 2 pv 8 v 1c1l 1 (v 1 c 2 l)

h

y1 2

h

_. (3.33)

Using theorem 2.1 we get the following system of first-order PDEs: ¯t ¯y1 4 0 , (3.34) ¯z ¯x 4 0 , (3.35) ¯z ¯x 1 v 8 v 1c1l t 22 ¯z ¯y1 4 0 , (3.36) 1 2 y 2 1t

g

v 81 p 912pp 8 v 1c1l 2 v 8( p 81p2₎ (v 1c1l)2 2 p 91p 8 v 8 (v 1c1l)2 1 2 pv 82 (v 1c1l)3

h

1 (3.37) 1

g

p 81p 2 v 1c1l 2 pv 8 v 1c1l 1 (v 1 c 2 l)

hk

1 2 y 2 1 ¯t ¯x 1 y1z

l

4 0 .

On solving the system of eqs. (3.34)-(3.36), we get the following expressions for t and z : t 4 1 v 1c1l [h1Q 1h2] , z 4 h1 2 y11 h3, (3.38)

Substituting form eq. (3.38) into eq. (3.37), we obtain y1

m

h1(v 1c1l) T1 1 2 (h1Q 1h2) T 8

n

1 h3(v 1c1l) T40 , (3.39) where T 4

g

p 81p 2 (v 1c1l)2 2 pv 8 (v 1c1l)3 1 v 1c2l v 1c1l

h

, thus h34 0 and h1(v 1c1l) T1 1 2 (h1Q 1h2) T 840 , (3.40) which yields 2 h1 v 1c1l c1Q 1c2 1 T 8 T 4 0 .

Integrating, then we obtain

T 4 1

h4(h1Q 1h2)2

, (3.41)

where h4is an arbitrary constant.

The first integral in this case is given by

h1Q 1h2 2(v 1c1l) y18 2 2 h1 2 y1y 812 1 2 (h1Q 1h2)(v 1c1l) Ty 2 14 h5,

choosing h54 0 , and using eq. (3.41) we have

h1Q 1h2 v 1c1l y18 2 2 h1y1y 812 v 1c1l h4(h1Q 1h2) y124 0 ,

which has the exact solution

y14 M(h1Q 1h2)aO2h, (3.42) where a 4h16

o

h121 4 h4

and M is an arbitrary constant,

y24 M v 1c1l

gg

a 2

h

(h1Q 1h2) aO2h12 1_{2 p(h} 1Q 1h2)aO2h

h

.

3.3. The nonlinear Dirac system. – The nonlinear 2-dimensional Dirac system can be expressed as [15-17]:

.

`

/

`

´

2 ia ¯y2 ¯x1 2 ia ¯y1 ¯x2 1 y11 ka3

(

Ny1N22 Ny2N2

)

y14 0 , ia ¯y1 ¯x1 1 ia ¯y2 ¯x2 1 y21 ka3

(

Ny1N22 Ny2N2

)

y24 0 , (3.43)

where x14 x , x24 c * t , a 4 ˇOmc * , c * is the speed of light, i 4k21 , ˇ is Planck’s

constant divided by 2 p , and m is the rest mass. On substituting

y14 u11 iv1, y24 u21 iv2,

where u1, v1, u2 and v2 are real functions of x1 and x2 in eq. (3.43), we get four

equations which we express in the form

.

`

/

`

´

av2 11 av211 u11 ka3u1N 40 , 2 au122 au211 v11 ka3v1N 40 , 2 av112 av221 u21 ka3u2N 40 , au111 au221 v21 ka3v2N 40 , (3.44) where N 4 (u1₎2

1 (v1)22 (u2)22 (v2)2and uji4 ¯uiO¯xj, vji4 ¯viO¯xj. Using the same

method of the above subsection the functional integral can be written as:

J(u1, v1_{, u}2_{, v}2 ) 4 1 2



V

{

a(2v2u112 v1u211 u2v111 u1v212 2 v1u2 12 v2u221 u1v121 u2v22) 1N

g

1 1 1 2 ka 3_N

h

}

_{dV ,}

then the Lagrangian is given by (3.45) _{L 4} 1 2 a(2v 2_u1 12 v1u211 u2v111 1 u1v1 22 v1u122 v2u221 u1v121 u2v22) 1 1 2 N

g

1 1 1 2 ka 3_N

h

_,

for which the Euler-Lagrange equations ¯L ¯ui 2 ¯ ¯x1

g

¯L ¯u1i

h

2 ¯ ¯x2

g

¯L ¯u2i

h

, ¯L ¯vi 2 ¯ ¯x1

g

¯L ¯vi 1

h

2 ¯ ¯x2

g

¯L ¯vi 2

h

, yield the system of eqs. (3.44), where i 41, 2.

Now, we look for a one-parameter group of transformations of the form

.

`

`

/

`

`

´

x14 x11 ez(x1, x2, u1, v1, u2, v2) 1O(e2) , x24 x21 et(x1, x2, u1, v1, u2, v2) 1O(e2) , u1 4 u11 ez1(x1, x2, u1, v1, u2, v2) 1O(e2) , u2 4 u21 ez2(x1, x2, u1, v1, u2, v2) 1O(e2) , v1 4 v11 eh1(x1, x2, u 1 , v1, u2, v2_{) 1O(e}2) , v2_{4 v}2_{1 eh}2(x1, x2, u 1 , v1, u2, v2_{) 1O(e}2) , (3.46)

under which

(

ss

L dx dt

)

is invariant. The necessary condition for the fundamental functional integral

(

ss

L dx dt

)

to be invariant under the one-parameter group of transformation (3.46) is given by (3.47) t ¯L ¯x2 1 z ¯L ¯x1 1 z1 ¯L ¯u1 1 z2 ¯L ¯u2 1 h1 ¯L ¯v1 1 h2 ¯L ¯v2 1 1 ¯L ¯u21

g

dz1 dx2 2 u21 dt dx2 2 u11 dz dx2

h

1 ¯L ¯u11

g

dz1 dx1 2 u21 dt dx1 2 u11 dz dx1

h

1 1 ¯L ¯u22

g

dz2 dx2 2 u22 dt dx2 2 u12 dz dx2

h

1 ¯L ¯u12

g

dz2 dx1 2 u22 dt dx1 2 u12 dz dx1

h

1 1 ¯L ¯v1 2

g

dh1 dx2 2 v21 dt dx2 2 v11 dz dx2

h

1 ¯L ¯v1 1

g

dh1 dx1 2 v21 dt dx1 2 v11 dz dx1

h

1 1 ¯L ¯v2 2

g

dh2 dx2 2 v22 dt dx2 2 v12 dz dx2

h

1 ¯L ¯v2 1

g

dh2 dx1 2 v22 dt dx1 2 v12 dz dx1

h

1 1 L

g

dt dt 1 dz dx

h

4 0 . On substituting for L and its derivatives in eq. (3.47), we get the following system of

first-order PDEs determining t , z , z1, h1and h2: 2 h22 v2 ¯z1 ¯u1 1 v 2 ¯z ¯x1 2 v1 ¯z2 ¯u1 1 u 2 ¯h1 ¯u1 1 1 v1 ¯z ¯x2 1 u1 ¯h2 ¯u1 1 2 a L ¯z ¯u1 4 0 , 2 h12 v1 ¯z1 ¯u1 1 v 1 ¯t ¯x2 2 v2 ¯z2 ¯u1 1 u 1 ¯h1 ¯u1 1 1 v2 ¯t ¯x1 1 u2 ¯h2 ¯u1 1 2 a L ¯t ¯u1 4 0 , z22 v2 ¯z1 ¯v1 2 u 1 ¯z ¯x2 2 v1 ¯z2 ¯v1 1 u 2 ¯h1 ¯v1 1 1 u2 ¯z ¯x1 1 u1 ¯h2 ¯v1 1 2 a L ¯z ¯v1 4 0 , z12 v1 ¯z1 ¯v1 2 u 1 ¯t ¯x2 2 v2 ¯z2 ¯v1 1 u 1 ¯h1 ¯v1 1 1 u2 ¯t ¯x1 1 u2 ¯h2 ¯v1 1 2 a L ¯t ¯v1 4 0 , 2 h12 v2 ¯z1 ¯u2 1 v 2 ¯z ¯x2 2 v1 ¯z2 ¯u2 1 u 2 ¯h1 ¯u2 1 1 v1 ¯z ¯x1 1 u1 ¯h2 ¯u2 1 2 a L ¯z ¯u2 4 0 , 2 h22 v1 ¯z1 ¯u2 1 v 2 ¯t ¯x2 2 v2 ¯z2 ¯u2 1 u 1 ¯h1 ¯u2 1 1 v1 ¯t ¯x1 1 u2 ¯h2 ¯u2 1 2 a L ¯t ¯u2 4 0 , z12 v2 dz1 dv2 2 u 2 dz dx2 2 v1 dz2 dv2 1 u 2 dh1 dv2 2 2 u1 dz dx1 1 u1 dh2 dv2 1 2 a L dz dv2 4 0 , z22 v1 ¯z1 ¯v2 2 u 2 ¯t ¯x2 2 v2 ¯z2 ¯v2 1 u 1 ¯h1 ¯v2 2 2 u1 ¯t ¯x1 1 u2 ¯h2 ¯v2 1 2 a L ¯t ¯v2 4 0 , 2 a(z1u 1 2z2u21h1v12h2v2)(11ka3N)2v1 ¯z1 ¯x2 2v2¯z1 ¯x1 2v2¯z2 ¯x2 2 2 v1 ¯z2 ¯x2 1u1 ¯h1 ¯x2 1u2 ¯h1 ¯x1 1u2 ¯h2 ¯x2 1u1 ¯h2 ¯x1 1 2 a L

g

¯t ¯x4 1 ¯z ¯x1

h

40 .

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´

(3.48)

On solving the system of eqs. (3.48), for t , z , z1, h1and h2, we have t 4k1, z 4k2, z14 k3v11 k4v21 k5u2, z24 k3v21 k4v11 k5u1, h14 2 k3u11 k4u22 k5v2, h24 2 k3u22 k5v11 k4u1,

where ki are arbitrary constants, i 41, 2, 3, 4, 5.

Accordingly, for the case under consideration, the conservation laws given by theorem 2.2 assume the following form:

¯ ¯x2

{

Lt 1 ¯L ¯u1 2 (z12 tu212 zu11) 1 ¯L ¯u2 2 (z22 tu222 zu12) 1 1 ¯L ¯v1 2 (h12 tv212 zv11) 1 ¯L ¯v2 2 (h22 tv222 zv12)

}

1 1 ¯ ¯x1

{

Lz 1 ¯L ¯u11 (z12 tu212 zu11) 1 ¯L ¯u12 (z22 tu222 zu12) 1 1 ¯L ¯v11 (h12 tv212 zv11) 1 ¯L ¯v12 (h22 tv222 zv12)

}

4 0 .

Now the conservation law corresponding to eq. (3.43), yields after simplifications ¯ ¯x4

{

2 k3

(

(v1)21 (v2)21 (u1)21 (u2)2

)

2 2 k4(v1v22 u2u1) 2 2 2 k5(u2v11 u1v2) 1u11(k2v12 k1v2) 1v11(k1u22 k2u1) 1 1 v12(k1u12 k2u2) 1u12(k2v22 k1v1) 1 k1 a N

g

1 1 1 2 ka 3_N

h

}

1 1 ¯ ¯x4

{

2 k4

(

(v1)21 (v2)22 (u1)22 (u2)2

)

2 2 k3(v1v21 u2u1) 2 2 2 k5(u2v21 u1v1) 1u21(k2v12 k1v2) 2v21(k1u22 k2u1) 2 2 v22(k1u12 k2u2) 2u22(k2v22 k1v1) 1 k2 a N

g

1 1 1 2 ka 3_N

h

}

4 0 . 4. – Conclusions

We deal with the inverse problem and use it to obtain exact solutions as well. Functional integrals are obtained from which the Sturm-Liouville equation, the linear Dirac system and the nonlinear Dirac system may be derived by variation. The invariance of those functional integrals is investigated under a one-parameter group of

transformations (Lie group). Conservation laws are obtained (cf. Noether’s theorem). It is illustrated that the variational formulation and the conservation laws may be useful in obtaining exact solutions. In fact this is exemplified in the case of the Sturm-Liouville equation and in the case of the linear Dirac system, where it was possible to obtain a class of exact solutions, even when the function v(x) is arbitrary.

R E F E R E N C E S

[1] CALLEBAUT D. K. and KHATER A. H., VIII European Conference on Controlled Fusion & Plasma Physics, Prague-Czechoslovakia (1977), p. 149.

[2] CALLEBAUTD. K. and KHATERA. H., Bull. Am. Phys. Soc., 23 (1978) 319. [3] CALLEBAUTD. K. and KHATERA. H., Bull. Am. Phys. Soc., 24 (1979) 948.

[4] BERNSTEINI. B., FRIEMANE. A., KRUSKALM. D. and KULSRUDR. M., Proc. R. Soc. London, Ser. A, 17 (1958) 244.

[5] TONTIE., Acad. R. Belg. Bull. Cl. Sci., 55 (1969) 262.

[6] VAINBERGM. M., Variational Methods for the Study of Nonlinear Operators (Holden-Day, San Francisco) 1964.

[7] NASHEDM. Z., in Nonlinear Functional Analysis and Applications, edited by L. B. RALL (Academic Press, New York) 1971, pp. 103-310.

[8] TAPIA R. A., in Nonlinear Functional Analysis and Applications, edited by L. B. RALL (Academic Press, New York) 1971, pp. 45-102.

[9] TONTIE., Acad. R. Belg. Bull. Cl. Sci., 55 (1969) 137. [10] FINLAYSONB. A., Phys. Fluids, 15 (1972) 963.

[11] FINLAYSONB. A., The Methods of Weighted Residuals and Variational Principles (Academic Press, New York) 1972.

[12] ATHERTONR. W. and HOMSYG. M., Stud. Appl. Math., 54 (1975) 31. [13] TONTI_{E., Int. J. Eng. Sci., 22, No. 11O12 (1984) 1343.}

[14] IBRAGIMOV N. H., Transformation Groups Applied to Mathematical Physics (D. Reidel Publishing Company, Dordrecht, Holland) 1985.

[15] KERSTENP. H. M., J. Math. Phys., 24 (1983) 2374.

[16] STEEBW. H., ERIGW. and STRAMPPW., J. Math. Phys., 23 (1982) 145. [17] STEEBW. H., OEVELW. and STRAMPPW., J. Math. Phys., 25 (1984) 2331.

[18] BALABANM., CAZENAVET. and VAZQUEZL., Commun. Math. Phys., 133 (1990) 53. [19] ESTEBANM. J. and SEREE., Commun. Math. Phys., 171 (1995) 323.

[20] DEFRUTOSJ. and SANZ-SERNAJ. M., J. Comp. Phys., 83 (1989) 407.

[21] LIES., Vorlesungen über Differentialgleichungen mit Bekannten Infinitesimalen Transfor-mationen (Teubner, Leipzig) 1912.

[22] KLEINF., Nachr. Acad.-Wiss. Göttingen, Math. Phys. Kl. II (1918) 171. [23] BHUTANIO. P. and MITALP., Int. J. Eng. Sci., 23 (1985) 353.

[24] BHUTANIO. P., CHANDRASEKARANG. and MITALP., Int. J. Eng. Sci., 26 (1988) 243. [25] BHUTANIO. P. and VIJAYAKUMARK., J. Aust. Math. Soc. Ser. B, 32 (1991) 457.

[26] KHATERA. H., MOUSSAM. H. M., CALLEBAUTD. K. and ABDUL-AZIZS. F., submitted to J. Math. Phys.

[27] KHATERA. H., MOUSSAM. H. M., CALLEBAUTD. K. and ABDUL-AZIZS. F., submitted to J. Math. Phys.

[28] KHATERA. H., ABDALLAA. A., CALLEBAUTD. K. and RAMADYA. G., submitted to Adv. Appl. Math.

[29] LOGANJ. D., Invariant Variational Principles (Academic Press) 1977.

[30] BLUMANG. W. and COLEJ. D., Similarity Methods for Differential Equations (Springer-Verlag, New York) 1989.