стр. 75 |

Making the change of variables (24) in equation (1), we get

u?1 ?1 ? u?2 ?2 = V (? + ?)(f?(?)g(?))?1 u. (66)

?

Provided that equation (1) admits SV, by virtue of equation (25), there exist

functions g1 (f + g) and g2 (f ? g) such that

V (? + ?)(f?(?)g(?))?1 = g1 (f + g) ? g2 (f ? g).

?

Since f + g = ?1 and f ? g = ?2 , equation (66) takes the form

u?1 ?1 ? u?2 ?2 = (g1 (?1 ) ? g2 (?2 ))u

or

X = ??1 ? ??2 ? g1 (?1 ) + g2 (?2 ).

2 2

Xu = 0,

Separation of variables in the two-dimensional wave equation with potential 307

It is evident that the operators Qi = ??i ? gi (?i ), i = 1, 2, commute with the

2

operator X, i.e., they are symmetry operators of equation (1) and, moreover, the

relations

Qi u = Qi ?1 (?1 )?2 (?2 ) = ??1 (?1 )?2 (?2 ) = ?u, i = 1, 2

hold.

Thus, each solution of equation (1) with separated variables is an eigenfunction of

some second-order symmetry operator admitted by equation (1).

Now, let us turn to partial differential equations related to equation (1). First, we

consider the wave equation

2u + U (y0 ? y1 )u = 0.

2 2

(67)

It occurs [11] that equation (67) is reduced to the form (1) by the change of

variables

1 1

t = exp y1 ch y0 , t = exp y1 sh y0

2 2

and, moreover, the potentials V (? ), U (? ) are connected by the relation

1

(68)

U (? ) = V (? ).

4?

Consequently, to obtain all potentials U (y0 ? y1 ) such that equation (67) ad-

2 2

mits a nontrivial SV, one has to substitute potentials V (x) listed in Theorem 2 into

formula (68). The solution with separated variables has the form (7), where

y1 ? y0 = exp{R((?1 ? ?2 )/2)}.

y1 + y0 = exp{P ((?1 + ?2 )/2)},

The explicit form of the functions P and R is given in Theorems 3–8.

Another related equation is the following equation of hyperbolic type

vx0 x0 ? vx1 x1 c2 (x1 ) = 0, (69)

which is widely used in various areas of mathematical physics (see, e.g. [19] and

references therein).

Equation (69) is reduced to the form (1) by the change of variables

u(t, x) = [c(x1 )]?1/2 v(x0 , x1 ) [c(x1 )]?1 dx1 ,

t = x0 , x=

and, moreover,

V (x) = ?c3/2 (x1 )(c1/2 (x1 ))·· (70)

.

dx1

x= c(x1 )

Thus, to describe all functions c(x1 ) that provide separability of equation (69),

it suffices to integrate the ordinary differential equation (70). Let us show how to

reduce the nonlinear equation (70) to a linear one.

On making in (70) the change of the variable

c(x1 ) = (y(x1 ))?1 ,

?

308 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

we get

3

...

y (y)?1 + 2V (y)y 3 .

y= ?? ?

2

The above equation with the change of the variable y = z 2 (y) is reduced to the

?

form

zyy ? V (y)z = 0. (71)

So, the general solution of the nonlinear equation (70) is given by the formula

c(x1 ) = z ?2 (y(x1 )), (72)

where z(y) is a general solution of the linear differential equation (71) and the function

y(x1 ) is determined by the quadrature

y(x1 )

z ?2 (? )d? = x1 + C, C ? R1 . (73)

Consequently, the problem of description of all functions c(x) such that equa-

tion (69) admits a nontrivial SV is reduced to the integration of the linear ordinary

differential equation (71), where V is given by (6). Solutions with separated variables

have the form

?= c(x1 )?1 (?1 (x0 , x1 ))?2 (?2 (x0 , x1 )),

where the functions ?i , are determined by the equalities

?1 ? ?2

1 dx1 ?1 + ?2 1 dx1

x0 ?

x0 + =P , =R ,

2 c(x1 ) 2 2 c(x1 ) 2

and the explicit form of P and R is given in Theorems 3–8.

Let us also note that, by using the corollary of Theorem 9 and formulas (71)–(73),

it is not difficult to obtain the results of Bluman and Kumei [19]. In that paper, they

have pointed out all the functions c(x1 ) that provide the extension of the symmetry

group admitted by equation (69).

The third related equation is the nonlinear wave equation

Utt ? [c?2 (U )Ux ]x = 0. (74)

By substitution U = Vx , equation (74) is reduced to the form

Vtt ? c?2 (Vx )Vxx = 0.

Applying the Legendre transformation

x0 = Vt , x1 = Vx , vx0 = t, vx1 = x, v + V = tVt + xVx ,

we get equation (69). Consequently, the method of SV in the linear equation (1) makes

it possible to construct exact solutions of the nonlinear wave equation (74).

In conclusion, we suggest a possible generalization of the definition of SV in order

to take into consideration nonlinear partial differential equations,

U x, u, u, u, . . . , u = 0, (75)

12 N

Separation of variables in the two-dimensional wave equation with potential 309

where x = (x0 , x1 , . . . , xn?1 ) and the symbol u denotes the collection of k-th order

k

derivatives of the function u(x).

When speaking about a solution of equation (75) with separated variables ?i =

?i (x, u), i = 1, n, we mean the ansatz

(76)

F (x, u, ?1 (?1 ), . . . , ?n (?n )) = 0,

which reduces equation (75) to n ordinary differential equations

(N ?1)

(N )

(77)

?i = fi ?i , ?i , ?i , . . . , ?i

? ,? .

In the above formulas, ?i ? C N (Rn+1 , R1 ), fi are some sufficiently smooth func-

tions, and ? = (?1 , . . . , ?n?1 ) are real parameters.

We say that equation (75) admits SV in the coordinates ?i (x, u), i = 1, n, if the

substitution of ansatz (76) into (75) with subsequent elimination of the N -th order

(N )

derivatives ?i , i = 1, n, yields an identity with respect to the variables ?i , ?i , . . .,

?

(N ?1)

, i = 1, n, ? (considered as independent ones).

?i

An application of the above approach to SV in nonlinear equations will be the topic

of our future publications.

Here, we present without derivation some results on separation of variables in

a two-dimensional nonlinear wave equation obtained with the use of the above descri-

bed approach.

We have succeeded in separating variables in the following PDE:

22 u = ?1 (ch u + (sh 2u) arctg eu ) + ?2 sh 2u;

1)

22 u = ?1 eu + ?2 e?2u ;

2)

22 u = ?1 (sh u ? (sh 2u) arctg eu ) + ?2 sh 2u;

3)

u

22 u = ?1 2 sin u + (sin 2u) ln tg

4) + ?2 sin 2u;

2

22 u = ?1 u + ?2 u ln u,

5)

where ?1 and ?2 are arbitrary constants.

Below, we adduce ansatzes for u(x) which provide a separation of equations 1–5

and corresponding reduced ordinary differential equations.

1) u(x) = ln tg(?1 (x0 ) + ?2 (x1 )),

?2 = C cos 4?1 + A?1 + B1 , ?2 = C cos 4?2 ? A?2 + B2 ,

?1 ?2

where C, A, B1 and B2 are arbitrary constants satisfying the relations A = ?1 /2,

B1 ? B2 = ?2 /2;

2) u(x) = ln(?1 (x0 ) + ?2 (x1 )),

?2 = ?2A?3 + B?2 ? C?2 + D2 ,

?2 = 2A?3 + B?2 + C ?1 + D1 ,

?1 ? ?2

1 1 2 2

where A, B, C, D1 and D2 are arbitrary constants satisfying relations A = ?1 ,

D2 ? D1 = ?2 /2;

3) u(x) = ln th (?1 (x0 ) + ?2 (x1 )),

?2 = C ch 4?1 + A?1 + B1 , ?2 = C ch 4?2 ? A?2 + B2 ,

?1 ?2

310 R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych

where C, A, B1 and B2 are arbitrary constants satisfying the relations A = ?1 /2,

B1 ? B2 = ?2 /2;

4) u(x) = 2 arctg exp(?1 (x0 ) + ?2 (x1 )),

?2 = C sh 2?1 + 2A?1 + 2B1 , ?2 = C sh 2?2 ? 2A?2 + 2B2 ,

?1 ?2

where C, A, B1 , and B2 are arbitrary constants satisfying the relations A = ?1 ,

B1 ? B2 = ?2 ;

5) u(x) = exp(?1 (x0 ) + ?2 (x1 )),

?2 = C1 e?2?1 + A?1 + B1 , ?2 = C2 e?2?2 ? A?2 + B2 ,

?1 ?2

where C1 , C2 , A, B1 , and B2 are arbitrary constants satisfying the relations A = ?1 ,

B1 ? B2 = ?2 ? ?1 .

1. Bocher M., Die Reihentwickelunger der Potentialtheorie, Leipzig, Teubner, 1894, 105 p.

2. Darboux G., Lecons sur les systemes orthogonaux et les coordonnees curvilignes, Paris, 1910,

270 p.

3. Eisenhart L.P., Separable systems of St?kel, Ann. Math., 1934, 35, № 2, 284–305.

a

4. Степанов В.В., Об уравнении Лапласа и некоторых трижды ортогональных системах, Мат. сб.,

1942, 11, 204–238.

5. Олевский М.Н., Триортогональные системы в пространствах постоянной кривизны, в которых

уравнение ?2 u+ ?u = 0 допускает полное разделение переменных, Мат. сб., 1950, 27, 379–426.

6. Miller W., Symmetry and separation of variables, Massachusetts, Addison-Wesley, 1977, 340 p.

7. Fushchych W.I., Serov N.I., The symmetry and some exact solutions of many-dimensional nonlinear

d’Alembert, Liouville and eikonal equations, J. Phys. A: Math. Gen., 1983, 16, № 15, 3645–3656.

8. Fushchych W.I., Zhdanov R.Z., On some new exact solutions of the nonlinear d’Alembert–Hamilto-

nian system, Phys. Lett. A, 1989, 141, № 3–4, 113–115.

9. Fushchych W.I., Zhdanov R.Z., Yegorchenko I.A., On the reduction of the nonlinear multidimen-

sional wave equations and compatibility of the d’Alembert–Hamiltonian system, J. Math. Anal. and

Appl., 1991, 161, № 2, 352–360.

10. Koornwinder T.H., A precise definition of separation of variables, Lect. Notes Math., 1980, 810,

240–263.

11. Zhdanov R.Z., Revenko I.V., Fushchych W.I., On the new approach to the variable separation in the

wave equation with potential, Dokl. AN Ukrainy, Ser. A, 1993, № 1, 9–11.

12. Kalnins E., Miller W., Lie theory and separation of variables, II: The EPD equation, J. Math. Phys.,

1976, 17, № 3, 369–377.

13. Kalnins E., Miller W., Lie theory and separation of variables, III: The equation ftt ? fss = ? 2 f ,

J. Math. Phys., 1974, 15, № 9, 1025–1032.

14. Шаповалов В.Н., Разделение переменных в линейном дифференциальном уравнении второго

порядка, Диффереренц. уравнения, 1980, 16, № 10, 1864–1874.

15. Овсянников Л.В., Групповой анализ дифференциальных уравнений, М., Наука, 1978, 400 c.

16. Olver P., Applications of Lie groups to differential equations, New York, Springer, 1986, 497 p.

17. Камке Э., Справочник по обыкновенным дифференциальным уравнениям, М., Наука, 1976,

576 c.

18. Erdelyi A. et al, Higher transcendental functions, New York, McGraw and Hill, 1953, Vol. 1, 253 p.,

Vol. 2, 320 p.

19. Bluman G., Kumei S., On invariance properties of the wave equation, J. Math. Phys., 1987, 28,

№ 2, 307–318.

W.I. Fushchych, Scientific Works 2003, Vol. 5, 311–325.

New solutions of the wave equation

by reduction to the heat equation

P. BASARAB-HORWATH, L. BARANNYK, W.I. FUSHCHYCH

In this article we make a new connection between the linear wave equation and the

linear heat equation. In this way we are able to construct new solutions of the linear

wave equation, using symmetries and conditional symmetries of the heat equation.

1. Introduction

The linear wave equation in (1 + n)-dimensional timespace R(1, n)

?2u ?2u ?2u

2u = 2 ? ?x2 ? · · · ? ?x2 = ?m u

2

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