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m is a parameter, u = u(x0 , . . . , xn ) is a complex function.

– The nonlinear Dirac equation

?

i? µ ?µ + F4 (??) ? = 0, (0.5)

? ?

? µ are 4 ? 4 Dirac matrices, ?, ? spinors, F4 smooth and depending on ??. The

special case

?

(??µ ?)? µ

µ

(0.6)

i? ?µ + ? ? ?=0

?

[(??? ?)?? ? ?]1/3

Proceedings of the XV International Conference on “Differential Geometric Methods in Theoretical

Physics”, Eds. H.D. Doebner, J.D. Hennig, World Scientific, Singapore – New Jersey – Hong Kong, 1987,

P. 544–549.

176 W.I. Fushchych

can be considered as a conformally invariant analog of the Dirac–Heisenberg equation

for a spinor field.

If we require these equations to be invariant with respect to a group larger than the

Poincar? or the Galilei group, a special form is imposed on F and we can construct

e

a whole family of solutions from known ones from such a symmetry.

?

We denote by P (1, n) the extended Poincar? group, i.e. the Poincar? group P (1, n)

e e

?

and scale transformations, and by AP (1, n) its Lie algebra.

§ 1. The symmetry

To construct solutions of (0.1)–(0.6) we need to know their symmetry properties.

?

Theorem 1. The wave equation (0.1) is invariant under P (1, n) iff

F1 (u) = ?1 ur , (1.1)

r = 1,

(1.2)

F1 (u) = ?2 exp(u),

?1 , ?2 , r constants.

n+3

The proof is given in [3]. (0.1) is in the case (1.1) and for r = n?1 invariant under

the conformal group C(1, n) ? P (1, n).

Theorem 2. For F2 = 0 equation (0.2) is invariant under IGL(1, n+1), the group of

linear inhomogeneous transformations of R(1, n + 1), and C(1, n + 1), the conformal

group of R(1, n + 1). The basis elements of the corresponding Lie algebra have the

form

PA = ig AB ?/?xB , (1.3)

LAB = xA PB , A, B = 0, . . . , n + 1,

xn+1 ? u, (1.4)

KA = xA D, D = ixA Pa ,

g AB is the metric tensor in R(1, n + 1).

The Monge–Ampere equation is invariant, in particular under linear transforma-

tions which preserve the quadratic form

s2 = x2 ? x2 ? · · · ? x2 ? u2

0 1 n

containing independing variables x and the depending variable u equally.

?

Theorem 3. Equation (0.2) with F2 = 0 is invariant under P (1, n + 1) iff

n?1

F2 (u) = ?(1 ? u? u? ) (1.5)

.

2

Theorem 4. The maximal, in the sense of Lie, invariance group of the equation

?

(0.3) is P (1, n + 1).

?

Theorem 5. (0.4) is invariant under the extended Galilei group G(1, n) which in-

cludes G(1, n), scale and projective transformations, iff

F3 = ?|u|4/n . (1.6)

where n is the number of spatial variables.

Theorem 6. The nonlinear Dirac equation (0.5) is invariant under the conformal

group C(1, n) iff

? ?

F4 (??) = ?(??)4/n . (1.7)

Symmetry and exact solutions of some multidimensional nonlinear equations 177

All theorems listed above can be proved by Lie’s method. The proofs are as a rule

cumbersome, so we omit them.

§ 2. Solutions of nonlinear equations

To construct exact solutions of (0.1)–(0.6) we use the symmetry properties of the

equations. The solutions in question are multiparametrical and due to their symmetry

we use the following ansatz

(2.1)

u(x) = f (x)?(?) + g(x),

where ?(?) an unknown function depending on new variables (m = n ? 1)

?(x) = {?1 , ?2 , . . . , ?m }

chosen from the invariants of the symmetry group of the equation. More precisely ?

and f , g are determined from the equations

dx0 dx1 dxn du

= ··· = (2.2)

= = ,

A0 A1 An B

where Aµ , B are functions defining infinitesimal transformations of the invariance

group

xµ = xµ + ?Aµ , u = u + ?B,

Aµ = cµ? x? + dµ , B = au + b,

where cµ? , dµ , a, b are group parameters. Variables ? are just the first integrals of

(2.2).

In the special case that ? depends on one variable, the partial differential equation

for u reduces to an ordinary differential equation for ?. Solutions of this ODE give

through (2.1) solutions of the original PDE.

Below we list some simple solutions of (0.1)–(0.6).

1. The nonlinear wave equation

2u + ?ur = 0, (2.3)

r = 1,

1

1?k

?

? 1 ? k2 (?? y ? )2 + y ? y?

u(x) = ,

(2.4)

2

? ? ?? = ?1, y? = x? + a? ;

1

1?k

?

1 ? k 2 ?? y ? ? ? y?

u(x) = ,

(2.5)

2

?? ?? = ?1;

?? ?? = ? ? ?? = 0,

? (1 ? k)2

2

?? ? = ?

? ? ?

1?k

(2.6)

u(x) = [F (?? x ) + ? x? ] , = 0,

2 1+k

F arbitrary smooth, a? , ?? , ?? are constants satisfying the above conditions. (2.4)–

(2.6) give a family of solutions of equation (2.3). As it is seen from (2.4)–(2.6), for

178 W.I. Fushchych

k > 1 the solutions have a singularity at ? = 0 and cannot be obtained by a standard

perturbation method.

2. Solutions of the Monge–Ampere equation

For det(uµ? ) = 0 arbitrary smooth functions

?k = ?? x? ,

k

(2.7)

u = ?(?1 , ?2 , . . . , ?n?1 ),

?k = (?0 , . . . , ?n ) arbitrary constant vectors, are solutions. Additional solutions in

k k

explicit and in implicit form are

u = (?µ xµ )2 ? ?2 x2 , ?2 ? ?0 ? ?1 ? · · · ? ?n ;

2 2 2

(2.8)

u = x2 /(? · x), ? · x ? ?? x? ; (2.9)

?? x? ? ?n+1 u = ?2 (?? x? ? ?n+1 u), (2.10)

?2 is smooth, ?? are parameters;

u = ? ?1 (x, u) (? · x)2 ? ?2 x2 , ?(x, u) = 1 + bµ xµ ? bn+1 u. (2.11)

3. Solutions of the generalized Euler–Lagrange equation

For 2u(1 ? u? u? ) + uµ? uµ u? = 0 the function

u = ?(?? x? ) + ?? x? (2.12)

is a solution, where ? is smooth, and the parameters satisfy the following conditions

?? ? ? + ?? ?? (1 ? ?? ? ? ) = 0.

A solution in implicit form is

?? x? ? ?n+1 u = ?(?? x? ? ?n+1 u),

(2.13)

?? ?? ? ?n+1 ?? ? ? ? ?n+1 ? (?µ ? µ ? ?n+1 ?n+1 )2 = 0.

2 2

4. Solutions of the nonlinear Dirac equation

?

i? µ ?µ + ?(??)k = 0. (2.14)

Consider the case k = 1/3, then eq. (2.14) is invariant under C(1, 3). To reduce eq.

(2.14) to the system of ODE we use the ansatz

(2.15)

?(x) = A(x)?(?),

where A(x) is a 4 ? 4 matrix, ?(?) is a four-component function, depending on one

invariant variable ?. More specifically

A(x) = (? ? x? )(xµ xµ )?2 , (2.16)

? = ? ? x? (xµ xµ )?1 , ? ? ?? > 0. (2.17)

(2.15)–(2.17) reduces (2.14) to the following system of ODE

d?

= i?(? ? ?? )?1 (??)1/3 (? · ?)?. (2.18)

?

d?

Symmetry and exact solutions of some multidimensional nonlinear equations 179

Solving eq. (2.18), we get the following solution of (2.14)

?(x) = (? · x)(x? x? )?2 exp{i??(? · ?)?)? =

?·? (2.19)

= (? · x)(x? x? )?2 cos(????) + i sin(????) ?,

?

where ? is a constant spinor, ? = (? ? ?? )1/2 , ? = (??)1/3 (? ? ?? )?1 . (2.19) is confor-

?

mally invariant.

In the same way solutions of nonlinear Schr?dinger, Navier–Stokes, Liouville equa-

o

tions have been constructed [1, 2, 3, 6]. We can even solve PDE’s which are nonin-

variant with respect to P (1, 3), e.g.

2 2 2 2 2 2 2 2

?0 ?u ?1 ?u ?2 ?u ?3 ?u

2u = + + + ,(2.20)

x0 ?x0 x1 ?x1 x2 ?x2 x3 ?x3

where ?0 , ?1 , ?2 , ?3 are parameters, xµ = 0. This reduces with the Lorentz-invariant

ansatz u = ?(x2 ) to an ODE

2

d2 ? d? d?

? = x2 ? x? x? .

= ?2

? 2 +2 ,

d? d? d?

1. Fushchych W.I., The symmetry of mathematical physics problems, in Algebraic-Theoretical Studies

in Mathematical Physics, Kiev, Institute of Mathematics, 1981, 6–28.

2. Fushchych W.I., On symmetry and particular solutions of some multidimensional equations of

mathematical physics, in Algebraic-Theoretical Methods in Mathematical Physics Problems, Kiev,

Institute of Mathematics, 1983, 4–23.

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

Liouville, d’Alembert and eikonal equations, J. Phys. A: Math. Gen., 1983, 16, ¹ 15, 3645–3656.

4. Fushchych W.I., Serov N.I., The symmetry and some exact solutions of the multidimensional

Monge–Ampere equation, Dokl. Acad. Nauk. USSR, 1983, 273, ¹ 3, 543–546; 1984, 278, ¹ 4,

847–851.

5. Fushchych W.I., Shtelen W.M., On some exact solutions of the nonlinear Dirac equation, J. Phys.

A: Math. Gen., 1983, 16, ¹ 2, 271–277.

6. Fushchych W.I., Shtelen W.M., On some exact solutions of the nonlinear equations of quantum

electrodynamics, Phys. Lett. B, 1983, 128, ¹ 3–4, 215–217.

7. Fushchych W.I., Shtelen W.M., Zhdanov R.Z.. On the new conformally invariant equations for

spinor fields and their exact solutions, Phys. Lett. B, 1985, 159, ¹ 2–3,189–191.

8. Fushchych W.I., On Poincar?-, Galilei-invariant nonlinear equations and methods of their solution,

e

in Group-Theoretical Studies of Equations of Mathematical Physics, Kiev, Institute of Mathematics,

1985, 4–20.

W.I. Fushchych, Scientific Works 2001, Vol. 3, 180–189.

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