ñòð. 79 |

Substituting these into (12), one obtains exact solutions of (10). We use these exact

solutions for A together with theorem 3 (iii) (with B = 0) as follows. The equation (8)

is conditionally invariant under

(16)

?y2 + Av?v

and this gives us an ansatz for v to be substituted into (8), and this, in turn, gives us

an exact solution of (8) which, when we combine it with (7), gives an exact solution

of (1). We list the results of these stages for each of the equations (13)–(15).

The ansatz for v from (13) is

v = (kt + y2 ) exp ?y2 /4t ?(t, y1 ),

2

where ?(t, y1 ) satisfies

3

?t = ?y1 y1 ? ?

2

and consequently we find that v is given by

v = (kt + y2 ) exp ?y2 /4t ? 3t/2 ?(t, y1 ),

2

where ?(t, y1 ) satisfies the (1 + 1)-dimensional heat equation.

The ansatz for v from (14) is

2

l exp ?(y2 ? 2a)2 /4t + exp ?(y2 + 2a)2 /4t

v = ea /t

?(t, y1 ),

where ?(t, y1 ) satisfies

a2

1

?2

?t + ? = ?y1 y1

2t t

and using this we eventually find that v is given by

1

v = v leay2 /t + e?ay2 /t exp ? 4a2 + y2 /4t ?(t, y1 ),

2

(17)

t

where ?(t, y1 ) satisfies the (1 + 1)-dimensional heat equation.

The ansatz for v from (15) is

ay2

exp ?y2 /4t ?(t, y1 ),

v = cos a2 + 2

t

where ?(t, y1 ) satisfies

a2

1

?2

?t + ? = ?y1 y1 ,

2t t

so that we obtain

1 ay2

v = v cos a2 + exp ? 4a2 + y2 /4t ?(t, y1 ),

2

(18)

t

t

where ?(t, y1 ) satisfies the (1 + 1)-dimensional heat equation.

324 P. Basarab-Horwath, L. Barannyk, W.I. Fushchych

We can now combine equations (17)–(19) with equation (7) to obtain new solutions

of (1):

(?x)2

( x) 3(? x)

? ?

u = [k(? x) + (?x)] exp ?((? x), (?x)),

2 4(? x) 2

( x) (4a2 +(?x)2 )

1

lea(?x)/(? x) + e?a(?x)/(? x) exp ?

u= ?((? x),(?x)),

2 4(? x)

(? x)

( x) (4a2 + (?x)2 )

1 a(?x)

?

cos a2 +

u= exp ?((? x), (?x)),

(? x) 2 4(? x)

(? x)

where ?(t, x) is any solution of the (1 + 1)-dimensional heat equation.

One can, in principle, perform the same procedure for the other conditional

symmetry operators defined in theorem 3; however, it is first necessary to obtain

some exact solutions of the systems. These latter are quite nonlinear and require

further treatment, and we leave this to a future publication.

5. Conclusion

We have been able to give a new reduction of the linear wave equation in 1 + 3

timespace dimensions to a linear heat equation in 1 + 2 timespace dimensions, that

is, a reduction of a hyperbolic equation to a parabolic one. The further reductions of

this heat equation by two-dimensional subalgebras (inequivalent under the action of

G2 (1, 2)) to ordinary differential equations leads to exact solutions in terms of special

functions. These are of interest in their own right. Conditional symmetries can also

be used to obtain new exact solutions. Using these solutions of the heat equation, one

can construct new solutions of the linear wave equation. In concluding, we remark

that the complex nonlinear wave equation

2? + F (|?|, ?µ |?|? µ |?|)? = 0,

where F is an arbitrary smooth function of its arguments and ? is a complex function,

can be reduced by the same ansatz as (7) (but with k imaginary) to a nonlinear

Schr?dinger equation with the same nonlinearity. Some of these equations admit

o

soliton solutions. We report on these results in [3].

Acknowledgments

We would like to thank the referees for valuable comments on an earlier version of

this article and for their eagle-eyed observation of mistakes. W.I. Fushchych thanks

the Swedish Institute and the Swedish Natural Sciences Research Council (NFR) for

financial support, and the Mathematics Department of Link?ping University for its

o

hospitality. P Basarab-Horwath thanks the Wallenberg Fund of Link?ping University

o

and the Tornby Fund for travel grants, and the Mathematics Institute of the Ukrainian

Academy of Sciences in Kiev for its hospitality.

1. Fushchych W.I., Serov N.I., Symmetry and exact solutions of the nonlinear multi-dimensional

Liouville, d’Alembert and eikonal equations, J. Phys. A: Math. Gen., 1983, 16, 3645–3658.

2. Fushchych W.I., Shtelen W.M., Serov N.I., Symmetry analysis and exact solutions of equations of

nonlinear mathematical physics, Dordrecht, Kluwer, 1993.

3. Basarab-Horwath P., Fushchych W.I., Barannyk L.F., Exact solutions of the nonlinear wave equation

by reduction to the nonlinear Schr?dinger equation, Preprint, Link?ping University, 1994.

o o

New solutions of the wave equation 325

4. Fushchych W.I., Barannyk L.F., Barannyk A.F., Subgroup analysis of the Galilei and Poincar?

e

groups and reduction of nonlinear equations, Kiev, Nauka Dumka, 1991 (in Russian).

5. Fushchych W.I., Nikitin A.G., Symmetries of equations of quantum mechanics, New York, Allerton

Press Inc., 1994.

6. Bluman G.W., Cole J.D., The general similarity solution of the heat equation, J. Math. Mech., 1969,

18, 1025–1042.

7. Bluman G.W., Kumei S., Symmetries and differential equations, New York, Springer, 1989.

8. Jacobson N., Lie algebras, New York, Interscience, 1962.

9. Naimark M., Stem A., Theory of group representations, Berlin, Springer, 1982.

10. Patera J., Winternitz P., Zassenhaus H., Continuous subgroups of the fundamental groups of phy-

sics. I, J. Math. Phys., 1975, 16, 1597–1614.

11. Erd?lyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions (Bateman

e

manuscript project), New York, McGraw-Hill, 1953.

12. Borhardt A.A., Karpenko D.Ja., The characteristic problem for the wave equation with mass, Dif-

ferential Equations, 1984, 20, 239–245.

W.I. Fushchych, Scientific Works 2003, Vol. 5, 326–334.

Amplitude-phase representation for solutions

of nonlinear d’Alembert equations

P. BASARAB-HORWATH, N. EULER, M. EULER, W.I. FUSHCHYCH

We consider the nonlinear complex d’Alembert equation 2? = F (|?|)? with ?

represented in terms of amplitude and phase, in (1 + n)-dimensional Minkowski space.

We exploit a compatible d’Alembert–Hamilton system to construct new types of exact

solutions for some nonlinearities.

1. Introduction

Let us consider the general nonlinear complex d’Alembert equation in (1 + n)-

dimensional Minkowski space

2? = F (|?|)?, (1)

where F is a smooth, real function of its argument, ? is a complex function of 1 + n

real variables, and

?2 ?2 ?2

2= ? 2 ? ··· ? 2 .

?x2 ?x1 ?xn

0

Equation (1) plays a fundamental role in classical and quantum field theories, and in

superfluidity and liquid crystal theory. Many exact solutions have been found using

Lie symmetry methods [6, 11, 12, 13, 8, 7], as well as with conditional symmetries [7].

In this paper we use the representation ? = ueiv , where u is the amplitude and

v is the phase (both real functions). On substituting this in (1), we find the following

system:

2u ? u(vµ vµ ) = uF (u), (2)

(3)

u2v + 2uµ vµ = 0.

We use the notation

?u ?v ?u ?v ?u ?v

? ? ··· ?

u µ vµ = .

?x0 ?x0 ?x1 ?x1 ?xn ?xn

The system (2), (3) is obviously equivalent to the starting equation (1). However,

equations (2), (3) has the advantage that it gives us the possibility of making functio-

nal and differential connections between the amplitude and phase, which substantially

simplifies the problem of integrating equation (1). Moreover, in assuming the simplest

possible relations between the amplitude and phase, we are able to construct exact

solutions of (2), (3), and hence of (1).

We now seek solutions of (2), (3). We consider two cases: (i) the amplitude

as a function of the phase, u = g(v); (ii) the phase as a function of the amplitude,

v = g(u). This is reminiscent of the polar description of plane curves in geometry. The

J. Phys. A: Math. Gen., 1995, 28, P. 6193–6201.

Solutions of nonlinear d’Alembert equations 327

system (2), (3) then yields a pair of equations for the phase v in the first case and for

the amplitude u in the second case. There then arises the question of the compatibility

of the two equations obtained, and we solve it by exploiting the compatible system

?N

(4)

w= , wµ wµ = ?,

w

where ? = ?1, 0, 1 and N = 0, 1, . . . , n. Exact solutions for the system (4) are given

in table 1 in section 2.

The system (4) is a particular case of the d’Alembert–Hamilton system

2w = F1 (w), (5)

wµ wµ = F2 (w),

The system (5) was studied by Smirnov and Sobolev in 1932, with w = w(x0 , x1 , x2 )

and F1 = F2 = 0. Collins [2–4] studied (5) with w a function of three complex

variables, and obtained compatibility conditions for the functions F1 (w), F2 (w). For

(1 + 3) and higher dimensional Minkowski space, (5) was studied by Fushchych and

co-workers [9, 10]: they obtained compatibility conditions for F1 (w), F2 (w) and some

exact solutions.

Here, we exploit the results of Fushchych et. al [10], applying them to the

system (4). Moreover, the compatibility of (4) dictates the type of nonlinearity F (u)

which can appear in (1). This is the novelty of our approach to finding some exact

solutions of (1).

2. Solutions

2.1. u = g(?). We now assume that the amplitude is a function of the phase:

u = g(v). Inserting this assumption in (2), (3), we obtain

?2g gF (g)

?

2v = (6)

= F1 (v),

g? ? 2g 2 ? g 2

g ?

g 2 F (g)

(7)

vµ vµ = = F2 (v)

g? ? 2g 2 ? g 2

g ?

with g = dg/dv.

?

We now deal with (6), (7) in two ways: (i) assume forms for F1 , F2 so as to make

equations (6), (7) compatible; (ii) transform equation (4) locally so as to agree with

(6), (7).

First, let us make the assumption

?N

F1 (v) = , F2 (v) = ?

v

with N, ? = 0. Then equations (6), (7) become a compatible system [10], and we also

find that g and F must satisfy

?2g

g2 ?

g? ? 2g ? g ? F (g) = 0,

2 2

(8)

g ? = N.

? g

From (8) it now follows that

N N

g(v) = ?v ?N/2 , ? ?4/N v 4/N ,

F (v) = ?? + ? 1?

2 2

where ? = 0 is an arbitrary real constant.

328 P. Basarab-Horwath, N. Euler, M. Euler, W.I. Fushchych

With this, we have obtained the following:

Result 1. An exact solution of (1) with nonlinearity

N N

F (|?|) = ?? + ? ?4/N ? 1? |?|4/N

2 2

is given by

?(x) = ?v(x)?N/2 eiv(x) ,

where v(x) is a solution of the compatible system (4) for N, ? = 0.

Our next step is to perform a local transformation of (4). We do this by setting

w = f (v) in (4) (with ? = 0), with f a real, smooth function such that f? = 0. With

this substitution, we obtain the system:

?

?N ?f (v)

2v = ? (9)

,

f (v)f?(v) f?3 (v)

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