pseudospherical 3- planes in ℝ ? and evolution equations of type

IOSR Journal of Mathematics (IOSR-JM)

e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 2 Ver. VI (Mar - Apr. 2015), PP 28-38 www.iosrjournals.org

DOI: 10.9790/5728-11262838 www.iosrjournals.org 28 | Page

[

Pseudospherical 3- Planes In and Evolution Equations Of Type

= ,, . . ,

,, ,

,

M.F. El-Sabbagh1, K.R. Abdo

2

1(Mathematics Department/ Faculty of Science, Minia University, Egypt) 2(Mathematics Department/ Faculty of Science ,fayoum University)

Abstract:In this paper, evolution equations with two or more spatial variables, which may describe pseudospherical planes in higher dimensions, are considered. Necessary and sufficient conditions for equations

of type = , , ,

, , ,

,

to describe a 3-dimensionalpseudospherical plane of 5are given . Such equations are characterized. Keywords:Evolution equations, Pseudospherical surfaces, Riemannian manifold, Solitons and differential equations.

I. Introduction It has been observed that exactly solvable nonlinear differential equations with two independent

variables are obtained as compatibility conditions for linear systems. Moreover, obtaining a spectral linear

problem associated with a nonlinear equation[14-16] has been useful in order to solve the initial value problem

by the inverse scattering method, [1]. In [2,13] the notion of a differential equation which describe

pseudospherical surfaces (surfaces with constant negative Gaussian curvature inR5) was given . Studies are made in this direction concerning non linear evolution equations of type

uxt = u, ux , uxx , . . ,ku

xk in 2,11

ut = u, ux , uxx , . . ,ku

xk in 1,10

And utt = u, ux , , uxx , ut in [8]. Then, we [6,7] generalized these studies to evolution equations with three independent variables which are

related to pseudospherical planes (P.S.P) in R5 (3-dim planes of R5 with constant negative sectional curvature ) where equations of types

= , , , . . ,

, , , ,

(1)

and

= , , , . . ,

, , , ,

(2)

Are studied . Here we provide a similar study for another class of evolution equations with two spatial variables

plus the time variable which have the from

= , , . . ,

, , ,

, (3)

II. Basic notations and Preliminaries A differential equation -for a real function , , describes a 3-dimensional pseudospherical plane

in 5(simply P.S.P.) if it is the necessary and sufficient condition for the existence of differentiable functions , 1 6 and1 3, depending on and its derivatives, such that the 1-forms

= 1 + 2 + 3 satisfy the structure equations of a 3-plane of constant sectional curvature 1 in 5 i.e. equations (4).

1 = 4 2 + 5 3 2 = 4 1 + 6 33 = 5 1 6 2

4 = 1 25 = 1 3

6 = 2 3

(4)



where we have written

4 = 12 5 = 13, and 6 = 23 with = , , = 1,2,3 , = 0

We shall define such 3-dimensional P.S.P to be a two-parameters 3-dimensional P.S.P 31 =41= and 22 = 42= , with and constant parameters.

To study equation (3).we first write

0 = , 1 = , 2 = , , =

,

1 = , 2 = , . . =

and p =

Thus equation (3) becomes

= 0 , 1 , . , , 1, , , (5) In Particular, we shall consider equation (5) with the following assumptions

, = , = 0

, = , = 0

0, = for 1 , 1 (6)

where the comma denotes partial differentiation with respect to the shown variable. Now consider the

following ideal of forms on the space of variables ,, , 0 , 1 , . . , , 1, 2, , ,:

= +1 , 0 1

= +1 , 0 1

= + 1

= + 1

= = 0

(7)

Note that assumptions (6) mean that has no () terms. Now, if we apply Cartan-Kahler theory, for equation (5) and using the notation above we can obtain the following result which relates solutions of the

differential equation (3) with integral manifolds of the ideal formed by the forms in (7).

Lemma 2.1

Let = 0 , 1 , . , ,1, , , ,be a differential equation which describe an (, ) 3-dirnensional P.S.P with the associated 1-forms = 1 + 2 + 3, = 1,2, . . . ,6 where and are real differentiable (C) functions defined on an open connected subset ++1 with no explicit dependence on , and . Then

11, = 12, = 21, = 22, = 51, = 52, = 61, = 62, = 0 1

11,

= 13,

= 21,

= 23,

= 33,

= 43,

=

51, = 53, = 61, = 63, = 0

33, 1

= 43, 1

= 0

12, = 13, = 22, = 23, = 33, = 43, =

52, = 53, = 62, = 63, = 0

33,1 = 43,1 = 0

13, = 23, = 33, = 43, = 53, = 63, = 0

33, = 43, = 0, 33,1 = 43,1 = 0

33,1

= 43,1

= 0 , 13, = 11,1 = 12,123, = 21,1 = 22,1

, 53, = 51,1 = 52,163, = 61,1 = 62,1

11,2 + 12,

2 + 21,2 + 22,

2 + 51,2 + 52,

2 + 61,2 + 62,

2 0

(8)

In U, and

+111,

1

=1

+ +112,

1

=0

= 22 21 + 51 + 52 (9)



11, 11,0 + +113,

1

=0

= 23 4321 + 5133 53 (10)

12, 12,0 + +113,

1

=0

= 23 4322 + 5233 53 (11)

+121,

1

=1

+ +122,

1

=0

= 11 12 + 61 62 (12)

21, 21,0 + +123, = 11 43 13

1

=0

+ 6133 63 (13)

22, 22,0 + +123,= 1243 13

1

=1

+ 6233 63 (14)

1152 1251 = 2261 2162(15)

+133, = 1153 1351

2

=0

+ 2163 2361 (16)

+133,= 1253 1352

2

=1

+ 2263 2362 (17)

1122 1221 = 0 (18)

+143, = 1123 1321

2

=0

(19)

+143,= 1223 1322

2

=1

(20)

+151,+ +152, = 11 12

1

=0

1

=1

(21)

51, 51,0 + +153, = 1133 13

1

=0

(22)

52, 52,0 + +153,= 1233 13

1

=1

(23)

+161,+

1

=1

+162, = 2132 2231

1

=0

(24)

61, 61,0 + +163, = 2133 23

1

=0

(25)

62, 62,0 + +163,= 2233 23

1

=1

(26)

with the assumptions (6).

Proof:

In the space of variables ,, , 0 , 1 , , , 1, , , we consider the ideal generated by , ,k , p , defined by equations (7) with given by equation(53) then

= = k = p = = = 0, when restricted to each integral manifold of .

Hence, for 0 , 1 , . , , 1, , , satisfying (5), we have

= +1 , = 0,1, . . , 1 = +1 , = 0,1, . . , 1

= 1 = 1



= 0 =

Where from assumptions (6) we have

= ,

= 0 = +1

= 0 = +1

= 0 =

= 0 = 0

(27)

And also we have

, = , = 0

, = , = 0

0, = 1 , 1

11,12,21,22,51,52,61,62, 0 1

11, 13,

21,

23,

33,

43,

51,

53,

61,

63,

0

12,13,22,23,33,43,52,53,62,63, 0

28

At the beginning by using assumptions (6) and (8) we have

The 1-forms satisfy the structure equations (4) therefore,

11,

=0

+ 11,

=1

+ 11, + 12,

=0

+ 12,

=1

+ 12,

+ 13,

=0

+ 13,

=1

+ 13,

= 22 21 + 51 52 + 23 43 21 + 5133 53 + 23 43 22 + 5233 53 ()

From the above equation we can obtain the following equations by simple calculations and by using equations (27)

12, 12,0 + +113,

1

=0

= 23 4322 + 5233 53

11, 11,0 + +113,

1

=0

= 23 4321 + 5133 53

+111,

1

=1

+ +112,

1

=0

= 22 21 + 51 + 52

In similar way by using assumptions (6) and (8) we have the 1-forms satisfy the structure equations (4) then

21, + 21,

=1

+

=0

21, + 22,

=0

+ 22, + 22, + 23, + 23,

=1

=0

=1

+ 23,

= 12 + 11 + 61 62 + 13 + 43 11 + 6133 63 + 13 + 43 12 + 6233 63 ()


22, 22,0 + +123,= 1243 13

1

=1

+ 6233 63



21, 21,0 + +123, = 11 43 13

1

=0

+ 6133 63

+121,

1

=1

+ +122,

1

=0

= 11 12 + 61 62

In similar way by using assumptions(6) and(8) we have the 1-forms satisfy the structure equations (4) then

31, + 31,

=1

+ 31, +

=0

32,

=0

+ 32, + 32, + 33, + 33,

=1

+ 33,

=0

=1

= 51 12 + 52 11 61 22 + 62 21 + 52 13 + 53 12 6223 + 22 63 + 51 13 + 53 11 6123 6321 ()


+133,= 1253 1352

2

=1

+ 2263 2362

31, = 31,= 32, = 32,

= 0where31 = ,32 =

+133, = 1153 1351

2

=0

+ 2163 2361

31, = 31,= 32, = 32,

= 0where31 = ,32 =

1152 1251 = 2261 2162 31, = 31,

= 32, = 32,= 0where31 = ,32 =

Similarly by using assumptions(6) and(8) we have the 1-forms satisfy the structure equations (4) then

41, + 41,

=1

+

=0

41, + 42,

=0

+ 42, + 42, + 43, + 43,

=1

=0

=1

+ 43,

= 11 22 12 21 + 12 23 1322 + 11 23 13 21 () From the above equation we can obtain the following equations by simple calculations and by using equations (27)

41, = 41,= 42, = 42,

= 0where41 = ,42 = we have

+143,= 1223 1322

2

=1

41, = 41,= 42, = 42,

= 0where41 = ,42 = then we have

+143, = 1123 1321

2

=0

41, = 41,= 42, = 42,

= 0where41 = ,42 = then we have

11221221 = 0 Similarly by using assumptions (6) and (8) we have the 1-forms satisfy the structure equations (4) then

51, + 51,

=1

+ 51, +

=0

52,

=0

+ 52, + 52, + 53, + 53, + 53,

=1

=0

=1

= 1132 12 31 + 12 33 1332 + 11 33 13 31 ()




52, 52,0 + +153,= 1233 13

1

=1

51, 51,0 + +153, = 1133 13

1

=0

+151,+ +152, = 11 12

1

=0

1

=1

Finally by using assumptions (6) and (8) we have the 1-forms satisfy the structure equations (4) then

61, + 61,

=1

+ 61, +

=0

62,

=0

+ 62, + 62, + 63, + 63,

=1

+ 63,

=0

=1

= 21 32 22 31 + 22 33 2332 + 2133 23 31 ()


62, 62,0 + +163,= 2233 23

1

=1

61, 61,0 + +163, = 2133 23

1

=0

+161,+

1

=1

+162, = 2132 2231

1

=0

To justify the last equation of (8) we observe that if 11, ,12, ,21, ,22, ,51, , 52, , 61, , and62, vanish

simulataneously, it follows that equation (5) can not be the necessary and sufficient condition for the forms to satisfy the structure equations of a 3-dim. p.s.p. Therefore, this condition is added, and that completes the proof

of the lemma.

III. Characterization of this type of equations: Now, by the lemma (2.1), necessary conditions for an equation of the type

= 0 , 1 , . , ,1, , , To describe a 3-dim.P.S.P, is that the functions satisfy (8) to (26). Therefore we shall assume these conditions in order to characterize all such equations. We consider quantities 1 0, 2 0, 1

0 and2 0 where

1, = 1151, 5111, , 2, = 2161, 6121,1, = 51,11 11,51 , 1, = 51,11, 11,51,2, = 61,21 21,61 , 2, = 61,21, 21,61,

1 = 512 11

2 , 2 = 612 21

2

(29)

Also, we consider the following

1, = 1252, 5212, , 2,

= 2262, 6222,

1, = 52,12 12,52 , 1,

= 52,12, 12,52,2,

= 62,22 22,62 , 2, = 62,22, 22,62,

1 = 52

2 122 , 2

= 622 22

2

(30)

Now we state the following theorem

Theorem 3.1

Let , 1 6 , 1 3,be differentiable functions of , 0 , 1 , , 1 , such that quations (8)

hold and 31 = 41 = , 32 = 42 = are parameters. Suppose 1, ,1, , 2,and2,

as given before are non



zero.Then theequation = 0 , 1 , . , ,1, , , describes a two-parameters 3-dimensional P.S.P with associated 1-forms = 1 + 2 + 3If and only if the function is given by

=1

(1,)2 +1 1, 33,

2

=0

33,

2

=0

+11,+1 + 1,1,1 + 1,01,

1, 33,+1

2

=0

1,1,1 + 33,+11,

2

=0

+1

21, 33,+1

2

=0

+ 1(1,1

331,)

+1

(2,)2 +1 2, 33,

2

=0

33,

2

=0

+12,+1 + 2,2,1 + 2,02,

2, 33,+1 2,2,1

2

=0

+ 33,+12, +1

22,

2

=0

33,+1

2

=0

+ 2(2,1

332,)

+1

(1, )2

+1 1, 33,

2

=1

33,

2

=1

+11,+1 + 1,

1,

1 + 1,0

1,

1, 33,

+1 1,

1,1

2

=1

+ 33, +11,

+1

21,

2

=1

33, +1

2

=1

+ 1 (1,1

331, )

+1

(2, )2

+1 2, 33,

2

=1

33,

2

=1

+12,+1 + 2,

2,

1 + 2,0

2,

2, 33,

+1 2,

21

2

=1

+ 33, +12,

2

=1

+1

22,

33, +1

2

=1

+ 2 (2,1

332, ) + 51, 4321 23 + 61, 13 1143 + 52, 4322 23

+ 62, 13 1243 (31) Where 1 1 , 1 1 Moreover

13 =1

1, 11, 33,

2

=0

+1 + 11 1,1 + 2,1 23 11 , 61, 6111, 63 (21 ,11, 1121,) 32

53 =1

1, 51, 33,

2

=0

+1 + 51 1,1 + 2,1 23 51 ,61, 6151, 63 (21 ,51, 5121,) 33

23 =1

2, 21, 33,

2

=0

+1 + 21 1,1 + 2,1 13 21 , 51, 5121, 53 (11 ,21, 2111,) 34

63 =1

2, 61, 33,

2

=0

+1 + 61 1,1 + 2,1 13 61 , 51, 5161, 53 (11 ,61, 6111,) 35

It is noted that by similar construction, one may obtain

13 =1

1,

12, 33,

2

=1

+1 + 12 1,1 + 2,1

23 12 ,62, 6212, 63 (22 , 12, 1222,) 36



53 =1

1,

52, 33,

2

=1

+1 + 52 1,1 + 2,1

23 52 , 62, 6252, 63 (22 ,52, 5222,) 37

23 =1

2,

22, 33,

2

=1

+1 + 22 1,1 + 2,1

13 22 ,52, 5222, 53 (1222, 22 ,12,) 38

63 =1

2,

62, 33,

2

=1

+1 + 62 1,1 + 2,1

13 62 ,52, 5262, 53 (12 ,62, 6212,) 39

Proof

Suppose the equation = describes a 3-dim. P.S.P. Then it follows from the lemma that equations (9) (26) are satisfied. Now, consider eqns. (16), (17) and their derivatives with respect to , it follows from (8) that

1153 1351 + 2163 2361 = 33,+1

2

=0

11,53 + 51,13 21,63 + 61,23 = 1,1 + 2,1

(40)

1223 1352 + 2263 2362 = 33, +1

=

12,53 + 52,13 22,63 + 62,23 = 1,1 + 2,1

(41)

Using notations (29), (30) and from (40) one gets 53 , 53 ,53 and 53as given by(32) (35). Also, by the same way, (41) gives, the formulas (36) (39) for these functions. Thus we have the following result:

Now, consider the derivative with respect to +1 of eqns (10), (13), (22), and (25) as well as the derivative with

respect to +1 of eqns (11), (14), (23), and (26) . Then it follows from (8) that

+111, + 13, = 0

+112, + 23, = 0

+151, + 53, = 0

+161, + 63, = 0

(42)

And

+1

12, + 13,

= 0

+1

22, + 23,

= 0

+1

52, + 53,

= 0

+1

62, + 63,

= 0

(43)

Therefore (42), and (43) give +1+1 = 0and +1 +1= 0 . Thusis of the form :

= +1 + +1 + (44)

Where is independent of +1 , is independent of +1, while is independent of both +1 , and +1. From

eqns (42), (43), and (44) we get

52,11, 13,51, = 0 , 63,21, 23,61, = 0 (45)

53,12, 13,

52, = 0 , 63,

22, 23,

62, = 0 (46)

53,11 13,51 +11, = 0

63,21 23,61 +12, = 0 (47)

53,

12 13,

52

+11,

= 0

63,22 23,

62

+12,

= 0 (48)

Where 1 1 , 1 1 Now from equations (44), (47) and (48) one gets:

=1

1, 53, 11 13, 51 +

1

2, 63, 21 23, 61 (49)

=1

1,

53, 12 13,

52 +

1

2,

63, 22 23,

62 (50)



Hence, by (32) (35) and (36) (39) we get

=1

(1,)2 1, 33,

2

=0

33,

2

=0

+1 1,+1 + 1, 1,1

+1

(2,)2 2, 33,

2

=0

33,

2

=0

+1 2,+1 + 2,2,1 (51)

=1

(1, )2

1, 33,

2

=1

33,

2

=1

+1 1, +1 + 1,

1,

1

+1

(2, )2

2, 33,

2

=1

33,

2

=1

+1 2, +1 + 2,

2,

1 (52)

Now, it follows from (44) (50) that equations (10) and (22) are equivalent to thefollowing

1, + 1,0 +1

1

=0

53,11 13,51 331 + 5351 1311 331 + 51 4321 23

= 0(53)

1,0 +1

1

=0

53,11, 13,51, + 5351, 1311, 1

2331, + 51, 4321 23

= 0(54)

Also equations (13) and (25) are equivalent to:

2, + 2,0 +1

1

=0

63,21 23,61 332 + 6361 2321 332 + 61 13 1143

= 0(55)

2,0 +1

1

=0

63,21, 23,61, + 6361, 2321, 1

2332, + 61, 13 1143

= 0(56)

Similarly, eqns (11) and (23) are equivalent to the following :

1, + 1,0

+1

1

=1

53,12 13,

52 331

+ 5253 1312 331 + 52 4322 23

= 0 (57)

1,0 +1

1

=1

53,12, 13,

52, + 52,53 1312,

1

2331,

+ 52, 4322 23

= 0(58)

Also, equations (14) and (26) are equivalent to:

2, + 2,0

+1

1

=1

63,22 23,

62 332

+ 6362 2223 332 + 62 13 1243

= 0(59)

2,0 +1

1

=1

63,22, 23,

62, + 6362, 2322,

1

2332,

+ 62, 13 1243

= 0(60)



Therefore, using (32) (35) and (36) (39) in ((55) (60), we get :

=1

(1,)2 1,01, 1, 33,+1 1,

2

=0

1,1 + 33,+11,

2

=0

+1

21, 33,+1

2

=0

+ 1(1,1 331,)

+1

(2,)2 2,02, 2, 33,+1 2,

2

=0

2,1 + 33,+12,

2

=0

+1

22, 33,+1

2

=0

+ 2(2,1 332,)

+1

(1, )2

1,0 1,

1, 33,

+1 1,

1,1

2

=1

+ 33, +11,

2

=1

+1

21,

33, +1

2

=1

+ 1 (1,1

331, )

+1

(2, )2

2,0 2,

2, 33,

+1 2,

21

2

=1

+ 33, +12,

2

=1

+1

22,

33, +1

2

=1

+ 2 (2,1

332, ) + 51, 4321 23

+ 61, 13 1143 + 52, 4322 23

+ 62, 13 1243 (61)

Thus, form (44),(51),(52), and (61) we obtain as given by (31) in the theorem. Conversely, given functions

11 ,51 , 21 ,61 , 12 ,52 ,22 and 62 of , 0 , 1 , , 1 , and the functions 13 , 53 ,23 , 63 and as

given by (32) (35), or(36) (39) and (31), then straightforward computations show that the equation =

describes two-parameters 3-dimensional p.s.p with associated 1-forms = 1 + 2 + 3 , 1

6 which satisfy equations (4). This completes the proof of the theorem.

As a matter of fact, this model of evolution equations with more than two spatial variables, which we are

considering here, fits many equations of physical interest namely the higher dimension sine-Gordon equations,

3,12

IV. Conclusion In this paper, we extended the notion of P.S.P to higher dimensions i.e. 3-dim plane of constant

sectional curvature-1 imbedded in R5 and we studied the change in the results and properties.

Acknowledgements All gratitude is to my supervisor Prof. Dr. Mostafa El- Sabbagh, Department of Mathematics, Faculty of

Sciences, University of Minia, Egypt for his valuable guidance and encouragement .



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pseudospherical 3- planes in ℝ ? and evolution equations of type

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