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2m
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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