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ln |t| ? ln |x2 ? 4 2t| +

R= ln + C2 .

v

1

2 4 2 2 ? 4 2t + |x |

x1 1

A21 = J12 , J13 , J23 , J04 ? D + M + aN

Ansatz:

2a ? 3 x2

1

? = f (?) + v ln |t|, ln x2 ,

R = g(?) + ?= .

4 t

22

Solution:

v

|x|

1

? = v ln |t| + |x| + ? x2 ? 4 2t +

8t

22

v

x2 ? 4 2t ? |x|

?

+ v ln + C1 , ? = ±1,

v

x2 ? 4 2t + |x|

22

v

4a ? 1 x2 ? 4 2t ? |x|

2a + 1 ?a

ln |t| ? ?

ln x2 + v

R= ln

2 4 2 x2 ? 4 2t + |x|

v

1

? ln |x2 ? 4 2t| + C2 .

4

5 Structure of the solutions

Most of the solutions we have obtained can be put into six classes, as follows.

Class 1: The phase and amplitude depend linearly on t and have the following

structure:

? = ?11 t + ?12 (x), R = R11 t + R12 (x)

with ?11 and R11 being constants.

Class 2: The phase and amplitude have the structure

?21

+ ?22 (x), R = R11 ln |t| + R22 (x)

?=

t

with ?21 and R21 being constants.

Class 3: The phase and amplitude depend logarithmically on t:

x2

?31

R = R31 ln |t| + R32 (w, x)

?= + ?32 (w, x), w = ,

t t

with ?31 and R31 being constants.

Some exact solutions of a conformally invariant nonlinear Schr?dinger equation

o 65

Class 4: The phase depends on t inversely, and the amplitude depends on t

inversely and logarithmically

?41 (x) ?42 (x)

?= + + ?43 (x),

t t+a

R41 (x) R42 (x)

+ R43 (x) ln |t| + R44 (x) ln |t| + a.

R= +

t t+a

Class 5: The amplitude is an implicit function of the phase:

w = (t ? 2?)2 + 2x2 ,

? = ?51 t + ?52 (w), R = R51 (w)

with ?51 a constant.

Class 6: The amplitude and phase depend on two invariants w and x2 :

?61 (w) 1

w = ? ? t,

?= , R = R61 (w).

t 2

Since equation (2) is invariant under the conformal group C(1, 4), with the in?-

nitesimal operators of the conformal algebra given in Theorem 1, we can act on the

solutions we have obtained with group elements (see [8] for the formulas giving this

action explicitly) and obtain families of solutions of equation (2). These families of

solutions, or orbits of the group passing through a given exact solution, are what

Petiau called guided waves [16].

We leave open the question of the physical interpretation of equation (2) and

its solutions. However, we note that, in as much as the system (5), (6) does not

contain Planck’s constant , the nonlinear Schr?dinger equation (2) does not describe

o

a quantal system in the standard sense of this term. The system (5), (6) is also

obtained when ? = Aei?/ is substituted into (...).

1. de BroglieL., Non-linear wave mechanics, Elsevier Publishing Company, 1960.

2. Guerra F., Pusterla M., A nonlinear Schr?dinger equation and its relativistic generalization from

o

basic principles, Lett. Nuovo Cimento, 1982, 34, 351–356.

3. Gueret Ph., Vigier J.-P., Nonlinear Klein–Gordon equation carrying a nondispersive solitonlike

singularity, Lett. Nuovo Cimento, 1982, 35, 256–259.

4. Smolin L., Quantum ?uctuations and inertia, Phys. Lett. A, 1986, 113, 408–412.

5. Doebner H.-D., Goldin G.A., On a general nonlinear Schr?dinger equation admitting di?usion

o

currents, Phys. Lett. A, 1992, 162, 397–401.

6. Fushchych W., Cherniha R., Galilei invariant nonlinear equations of Schr?dinger type and their

o

exact solutions, Ukr. Math. J., 1989, 41, 1161–1167, 1456–1463.

7. Fushchych W., Chopyk V., Symmetry and non-Lie reduction of the nonlinear Schr?dinger equati-

o

on, Ukr. Math. J., 1993, 45, 539–551.

8. Fushchych W., Shtelen W., Serov M., Symmetry analysis and exact solutions of equations of

nonlinear mathematical physics, Dordrecht, Kluwer, 1993.

9. Basarab-Horwath P., Fushchych W., Barannyk L., Exact solutions of the wave equation by

reduction to the heat equation, J. Phys. A: Math. Gen., 1995, 28, 5291–5304.

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

e

groups and reduction of nonlinear equations, Kiev, Naukova Dumka, 1991.

11. Bluman G.W., Kumei S., Symmetries and di?erential equations, New York, Springer, 1989.

12. Ovsiannikov L.V., Group analysis of di?erential equations, New York, Academic Press, 1982.

13. Olver P.J., Applications of Lie groups to di?erential equations, New York, Springer, 1993.

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

14. Fushchych W., Nikitin A., Symmetries of the equations of quantum mechanics, New York,

Allerton Press, 1994.

15. Patera J., Winternitz P., Zassenhaus H., Continuous subgroups of the fundamental groups of

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

16. Petiau G., Sur une g?n?ralisation non-lin?aire de la m?chanique ondulatoire et les propri?t?s

ee e e ee

des fonctions d’ondes correspondantes, Nuovo Cimento. Supplemento, 1958, 3IX, 542–568.

W.I. Fushchych, Scienti?c Works 2004, Vol. 6, 67–80.

Implicit and parabolic ansatzes:

some new ansatzes for old equations

P. BASARAB-HORWATH, W.I. FUSHCHYCH

We give a survey of some results on new types of solutions for partial di?erential

equations. First, we describe the method of implicit ansatzes, which gives equations

for functions which de?ne implicitly solutions of some partial di?erential equations.

In particular, we ?nd that the family of eikonal equations (in di?erent geometries) has

the special property that the equations for implicit ansatzes are also eikonal equations.

We also ?nd that the eikonal equation de?nes implicitly solutions of the Hamilton–

Jacobi equation. Parabolic ansatzes are ansatzes which reduce hyperbolic equations

to parabolic ones (or to a Schr?dinger equation). Their uses in obtaining new types

o

of solutions for equations invariant under AO(p, q) are described. We also give some

results on conformally invariant nonlinear wave equations and describe some exact

solutions of a conformally invariant nonlinear Schr?dinger equation.

o

1 Introduction

In this talk, I would like to present some results obtained during the past few years in

my collaboration with Willy Fushchych and some of his students. The basic themes

here are ansatz and symmetry algebras for partial di?erential equations.

I wrote this talk after Wilhelm Fushchych’ untimely death, but the results I give

here were obtained jointly or as a direct result of our collaboration, so it is only right

that he appears as an author.

In 1993/1994 during his visits to Link?ping and my visits to Kyiv, we managed,

o

amongst other things, to do two things: use light-cone variables to construct new

solutions of some hyperbolic equations in terms of solutions of the Schr?dinger or

o

heat equations; and to develop the germ of new variation on ?nding ansatzes. This

last piece is an indication of work in progress and it is published here for the ?rst

time. I shall begin this talk with this topic ?rst.

2 The method of implicit ansatzes

2.1 The wave and heat equations

Given an equation for one unknown real function (the dependent variable), u, say,

and several independent (“geometric”) variables, the usual approach, even in terms

of symmetries, is to attempt to ?nd ansatzes for u explicitly. What we asked was

the following: why not try and give u implicitly? This means the following: look for

some function ?(x, u) so that ?(x, u) = C de?nes u implicitly, where x represents the

geometric variables and C is a constant. This is evidently natural, especially if you

Proceedings of the Second International Conference “Symmetry in Nonlinear Mathematical

Physics. Memorial Prof. W. Fushchych Conference” (July 7–13, 1997, Kyiv), Editors M. Shkil, A. Ni-

kitin and V. Boyko, Kyiv, Institute Mathematics, 1997, 1, P. 34–47.

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

are used to calculating symmetry groups, because one then has to treat u on the same

footing as x. If we assume, at least locally, that ?u (x, u) = 0, where ?u = ??/?u,

then the implicit function theorem tells us that ?(x, u) = C de?nes u implicitly

as a function of x, for some neighbourhood of (x, u) with ?u (x, u) = 0, and that

?

uµ = ? ?µ , where ?µ = ?xµ . Higher derivatives of u are then obtained by applying

??

u

the correct amount of total derivatives.

The wave equation 2u = F (u) becomes

?2 2? = 2?u ?µ ?µu ? ?µ ?µ ?uu ? ?3 F (u)

u u

or

?µ ?µ

2? = ?u ? ?u F (u).

?u

This is quite a nonlinear equation. It has exactly the same symmetry algebra as the

equation 2u = F (u), except that the parameters are now arbitrary functions of ?.

Finding exact solutions of this equation will give u implicitly. Of course, one is entitled

to ask what advantages are of this way of thinking. Certainly, it has the disadvantage

of making linear equations into very nonlinear ones. The symmetry is not improved in

any dramatic way that is exploitable (such as giving a conformally-invariant equation

starting from a merely Poincar? invariant one). It can be advantageous when it comes

e

to adding certain conditions. For instance, if one investigates the system

2u = 0, uµ uµ = 0,

we ?nd that uµ uµ = 0 goes over into ?µ ?µ = 0 and the system then becomes

2? = 0, ?µ ?µ = 0.

In terms of ordinary Lie ansatzes, this is not an improvement. However, it is not

di?cult to see that we can make certain non-Lie ansatzes of the anti-reduction type:

allow ? to be a polynomial in the variable u with coe?cients being functions of x.

For instance, assume ? is a quintic in u: ? = Au5 + Bu + C. Then we will have the

coupled system

2A = 0, 2B = 0, 2C = 0,

Aµ Aµ = Bµ Bµ = Cµ Cµ = Aµ Bµ = Aµ Cµ = Bµ Cµ = 0.

Solutions of this system can be obtained using Lie symmetries. The exact solutions

of

2u = 0, uµ u µ = 0

are then obtained in an implicit form which is unobtainable by Lie symmetry analysis

alone.

Similarly, we have the system

2u = 0, uµ u µ = 1

which is transformed into

2? = ?uu , ?µ ? µ = ? 2

u

Implicit and parabolic ansatzes: some new ansatzes for old equations 69

or

25 ? = 0, ?A ?A = 0,

where 25 = 2 ? ?u and A is summed from 0 to 4.

2

It is evident, however, that the extension of this method to a system of equations

is complicated to say the least, and I only say that we have not contemplated going

beyond the present case of just one unknown function.

We can treat the heat equation ut = u in the same way: the equation for the

surface ? is

?? · ??

?

??

?t = .

?u ?u

If we now add the condition ?u = ?? · ??, then we obtain the system

?u = ?? · ??

?t = ?,

so that ? is a solution to both the heat equation and the Hamilton–Jacobi equation,

but with di?erent propagation parameters.

If we, instead, add the condition ?2 = ?? · ??, we obtain the system

u

? ? ?uu , ?2 = ?? · ??.

?t = u

The ?rst of these is a new type of equation: it is a relativistic heat equation with

a very large symmetry algebra which contains the Lorentz group as well as Galilei

type boosts; the second equation is just the eikonal equation. The system is evidently

invariant under the Lorentz group acting in the space parametrized by (x1 , . . . , xn , u),

and this is a great improvement in symmetry on the original heat equation.

It follows from this that we can obtain solutions to the heat equation using Lorentz-

invariant ansatzes, albeit through a modi?ed equation.

2.2 Eikonal equations

Another use of this approach is seen in the following. First, let us note that there are

three types of the eikonal equation

uµ uµ = ?,

namely the time-like eikonal equation when ? = 1, the space-like eikonal one when

? = ?1, and the isotropic eikonal one when ? = 0. Representing these implicitly, we

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