ñòð. 16 |

uµ uµ = 1

goes over into the isotropic eikonal one in a space with the metric (1, ?1, . . . , ?1)

n+1

?µ ? µ = ? 2 .

u

The space-like eikonal equation

uµ uµ = ?1

70 P. Basarab-Horwath, W.I. Fushchych

goes over into the isotropic eikonal one in a space with the metric (1, 1, ?1, . . . , ?1)

n

?µ ?µ = ??2

u

whereas

uµ uµ = 0

goes over into

?µ ?µ = 0.

Thus, we see that, from solutions of the isotropic eikonal equation, we can construct

solutions of time- and space-like eikonal ones in a space of one dimension less. We

also see the importance of studying equations in higher dimensions, in particular in

spaces with the relativity groups SO(1, 4) and SO(2, 3).

It is also possible to use the isotropic eikonal to construct solutions of the Hamil-

ton–Jacobi equation in 1 + n dimensions

ut + (?u)2 = 0

which goes over into

?u ?t = (??)2

and this equation can be written as

2 2

?u ? ?t

?u + ?t

? = (??)2

2 2

which, in turn, can be written as

g AB ?A ?B = 0

with A, B = 0, 1, . . . , n + 1, g AB = diag (1, ?1, . . . , ?1) and

?u ? ?t

?u + ?t

?0 = , ?n+1 = .

2 2

It is known that the isotropic eikonal and the Hamilton–Jacobi equations have the

conformal algebra as a symmetry algebra (see [15]), and here we see the reason why

this is so. It is not di?cult to see that we can recover the Hamilton–Jacobi equation

from the isotropic eikonal equation on reversing this procedure.

This procedure of reversal is extremely useful for hyperbolic equations of second

order. As an elementary example, let us take the free wave equation for one real

function u in 3 + 1 space-time:

2 2 2 2

?0 u = ?1 u + ? 2 u + ? 3 u

and write it now as

(?0 + ?3 )(?0 ? ?3 )u = ?1 u + ?2 u

2 2

Implicit and parabolic ansatzes: some new ansatzes for old equations 71

or

2 2

?? ?? u = ?1 u + ?2 u,

x0 ?x3 x0 +x3

Now assume u = e? ?(?, x1 , x2 ). With this assumption,

where ? = , .

?=

2 2

we ?nd

2 2

?? ? = ? 1 + ? 2 ?

which is the heat equation. Thus, we can obtain a class of solutions of the free wave

equation from solutions of the free heat equation. This was shown in [1]. The ansatz

taken here seems quite arbitrary, but we were able to construct it using Lie point

symmetries of the free wave equation. A similar ansatz gives a reduction of the free

complex wave equation to the free Schr?dinger equation. We have not found a way

o

of reversing this procedure, to obtain the free wave equation from the free heat or

Schr?dinger equations. The following section gives a brief description of this work.

o

3 Parabolic ansatzes for hyperbolic equations:

light-cone coordinates and reduction to the heat

and Schr?dinger equations

o

Although it is possible to proceed directly with the ansatz just made to give a

reduction of the wave equation to the Schr?dinger equation, it is useful to put it

o

into perspective using symmetries: this will show that the ansatz can be constructed

by the use of in?nitesimal symmetry operators. To this end, we quote two results:

Theorem 1. The maximal Lie point symmetry algebra of the equation

2u = m2 u,

where u is a real function, has the basis

Jµ? = xµ ?? ? x? ?µ

P µ = ?µ , I = u?u ,

when m = 0, and

Pµ = ?µ , I = u?u , Jµ? = xµ ?? ? x? ?µ ,

D = xµ ?µ , Kµ = 2xµ D ? x2 ?µ ? 2xµ u?u

when m = 0, where

? ?

, xµ = gµ? x? ,

?u = , ?µ = µ

?u ?x

= diag (1, ?1, . . . , ?1), µ, ? = 0, 1, 2, . . . , n.

gµ?

We notice that in both cases (m = 0, m = 0), the equation is invariant under the

operator I, and is consequently invariant under ?µ ?µ + kI for all real constants k

and real, constant four-vectors ?. We choose a hybrid tetradic basis of the Minkowski

space: ?: ?µ ?µ = 0; : µ µ = 0; ?: ? µ ?µ = ?1; ?: ? µ ?µ = ?1; and ?µ µ = 1,

1

?µ ?µ = ?µ ?µ = µ ?µ = µ ?µ = 0. We could take, for instance, ? = v2 (1, 0, 0, 1),

72 P. Basarab-Horwath, W.I. Fushchych

= v2 (1, 0, 0, ?1), ? = (0, 1, 0, 0), ? = (0, 0, 1, 0). Then the invariance condition (the

1

so-called invariant-surface condition),

(?µ ?µ + kI)u = 0,

gives the Lagrangian system

dxµ du

=

?µ ku

which can be written as

d(?x) d(?x) d(?x) d( x) du

= = = = .

0 0 0 1 ku

Integrating this gives us the general integral of motion of this system

u ? ek( x)

?(?x, ?x, ?x)

and, on setting this equal to zero, this gives us the ansatz

u = ek( x)

?(?x, ?x, ?x).

Denoting ? = ?x, y1 = ?x, y2 = ?x, we obtain, on substituting into the equation

2u = m2 u,

? + m2 ?,

2k?? ? =

?2 ?2

where . This is just the heat equation (we can gauge away the linear

= +

?y 2 ?y 2

1 2

m2 ?

term by setting ? = e 2k ?). The solutions of the wave equation we obtain in this

way are given in [1].

The second result is the following:

Theorem 2. The Lie point symmetry algebra of the equation

2? + ? F (|?|)? = 0

has basis vector ?elds as follows:

(i) when F (|?|) = const |?|2 :

Jµ? = xµ ?? ? x? ?µ , Kµ = 2xµ x? ?? ? x2 ?µ ? 2xµ ??? + ??? ,

?µ ,

D = x? ?? ? ??? + ??? , M = i ??? ? ??? ,

where x2 = xµ xµ .

(ii) when F (|?|) = const |?|k , k = 0, 2:

2

Jµ? = xµ ?? ? x? ?µ , D(k) = x? ?? ?

?µ , ??? + ??? ,

k

M = i ??? ? ??? .

?

(iii) when F (|?|) = const |?|k for any k, but F = 0:

Jµ? = xµ ?? ? x? ?µ , M = i ??? ? ??? .

?µ ,

Implicit and parabolic ansatzes: some new ansatzes for old equations 73

(iv) when F (|?|) = const = 0:

Jµ? = xµ ?? ? x? ?µ , M = i ??? ? ??? ,

?µ , L = ??? + ??? ,

L1 = i ??? ? ??? , L2 = ??? + ??? , B?? ,

where B is an arbitrary solution of 2? = F ?.

(v) when F (|?|) = 0:

Jµ? = xµ ?? ? x? ?µ , Kµ = 2xµ x? ?? ? x2 ?µ ? 2xµ ??? + ??? ,

?µ ,

M = i ??? ? ??? ,

D = xµ ?µ , L = ??? + ??? ,

L1 = i ??? ? ??? , L2 = ??? + ??? , B?? ,

where B is an arbitrary solution of 2? = 0.

In this result, we see that in all cases we have M = i ??? ? ??? as a symmetry

operator. We can obtain the ansatz

? = eik( x)

?(?x, ?x, ?x)

in the same way as for the real wave equation, using M in place of I. However, now

we have an improvement in that our complex wave equation may have a nonlinear

term which is invariant under M (this is not the case for I). Putting the ansatz into

the equation gives us a nonlinear Schr?dinger equation:

o

i?? ? = ? ? + ?F (|?|)?

when k = ?1/2. Solutions of the hyperbolic equation which this nonlinear Schr?dinger

o

equation gives is described in [2] (but it does not give solutions of the free Schr?dinger

o

equation).

The above two results show that one can obtain ansatzes (using symmetries) to

reduce some hyperbolic equations to the heat or Schr?dinger equations. The more

o

interesting case is that of complex wave functions, as this allows some nonlinearities.

There is a useful way of characterizing those complex wave equations which admit

the symmetry M : if we use the amplitude-phase representation ? = Rei? for the

wave function, then our operator M becomes ?? , and we can then see that it is those

equations which, written in terms of R and ?, do not contain any pure ? terms (they

are present as derivatives of ?). To see this, we only need consider the nonlinear wave

equation again, in this representation:

2R ? R?µ ?µ + ?F (R)R = 0,

R2? + 2Rµ ?µ = 0

when ? and F are real functions. The second equation is easily recognized as the

continuity equation:

?µ (R2 ?µ ) = 0

(it is also a type of conservation of angular momentum). Clearly, the above system

does not contain ? other than in terms of its derivatives, and therefore it must admit

?? as a symmetry operator.

Writing an equation in this form has another advantage: one sees that the impor-

tant part of the system is the continuity equation, and this allows us to consider other

74 P. Basarab-Horwath, W.I. Fushchych

systems of equations which include the continuity equation, but have a di?erent ?rst

equation. It is a form which can make calculating easier.

Having found the above reduction procedure and an operator which gives us the

reducing ansatz, it is then natural to ask if there are other hyperbolic equations which

are reduced down to the Schr?dinger or di?usion equation. Thus, one may look at

o

hyperbolic equations of the form

2? = H(?, ?? )

which admit the operator M . An elementary calculation gives us that H = F (|?|)?.

The next step is to allow H to depend upon derivatives:

2? = F (?, ?? , ?µ , ?? )?

µ

and we make the assumption that F is real. Now, it is convenient to do the calculations

in the amplitude-phase representation, so our functions will depend on R, ?, Rµ , ?µ .

However, if we want the operator M to be a symmetry operator, the functions may

not depend on ? although they may depend on its derivatives, so that F must be a

function of |?|, the amplitude. This leaves us with a large class of equations, which

in the amplitude-phase form are

2R = F (R, Rµ , ?µ )R, (1)

(2)

R2? + 2Rµ ?µ = 0

and we easily ?nd the solution

F = F (R, Rµ Rµ , ?µ ?µ , Rµ ?µ )

when we also require the invariance under the Poincar? algebra (we need translations

e

for the ansatz and Lorentz transformations for the invariance of the wave operator).

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