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? ? (1.5)
j = 3, 4.
Exact solutions of nonlinear Dirac’s and Schr?dinger’s equations
o 309

Note 1. The conformally-invariant Dirac–G?rsey equation (1956)
u
?
{?µ pµ + D(??)1/3 }? = 0 (1.6)

belongs to the class (1), (1.4).
Note 2. The equation of the type

{?µ pµ + D?µ (?? µ ?) · [(??? ?)(?? ? ?)]?1/3 }? = 0
? ? ? (1.7)

is invariant under the conformal group C(1, 3).
?
2. The Ans?tzes for P (1, 3)-invariant equation. To be specific let us consider
a
the nonlinear spinor equation of the type
?
{?µ pµ + ?(??)1/2k }? = 0, (2.1)

where ?, k are arbitrary constants, k = 0.
We look for solutions of (2.1) in the form Fushchych (1981)

(2.2)
? = A(x)?(?),

where A(x) is 4 ? 4 matrix, ?(?) is 4-component column-function depending on three
new variables ? = {?1 , ?2 ?3 }.
For the Ansatz (2.2) to work effectively it is necessary to find A(x), ? in a form
which after a substitution of (2.2) into (1) would yield an equation for ?(?) depending
only on new variables ?.
This requirement is met if the following equalities are satisfied:
?
QA(x) ? ?µ (2.3)
+ ? A(x) = 0,
?xµ

?? l
µ
(2.4)
? = 0, l = 1, 2, 3,
?xµ

where ? µ (x), ?(x) are the coefficients of the infinitesimal operators Q = {Q1 , Q2 , . . .}
?
of the group P (1, 3). In our case the generators of the P (1, 3) have the form

Jµ? = xµ p? ? x? pµ + Sµ? . (2.5)
P µ = pµ ,

Thus the problem of describing Ans?tze of the form (2.2) reduces to the construc-
a
tion of the general solution to the system of equations (2.3), (2.4) with the given ? µ ,
?.
As an example let us consider the case where in (2.3), (2.4) the operators Q have
the simple form

Q = {Q1 = J03 , Q2 = P1 , Q3 = P2 }.

Then the system (2.3), (2.4) has the form
i
x0 p3 ? x3 p0 + ?0 ?3 A(x) = 0, (2.6)
p1 A(x) = 0, p2 A(x) = 0,
2

(x0 p3 ? x3 p0 )? = 0, (2.7)
p1 ? = 0, p2 ? = 0.
310 W.I. Fushchych

It follows from (2.7) that ? = ?(x0 , x3 ) = x2 ? x2 . We look for the solutions to
0 3
(2.6) in the form
(2.8)
A(x) = exp{?0 ?3 g(x)}.
After a substitution of (2.8) into (2.6), we obtain
?g ?g 1
? = 0. (2.9)
x0 + x3
?x3 ?x0 2
The particular solution of eq. (2.9) is given by the expression
1
g(x) = ln(x0 + x3 ).
2
Thus we have
1
(2.10)
A(x) = exp ?0 ?3 ln(x0 + x3 ) .
2
Without going into the technical details on solving the system (2.3), (2.4) we give
some expressions for the matrix A(x) and ?.
Example 2.1.
1
A(x) = (x0 ? x2 )?k exp ?1 (?2 ? ?0 ) ln(x0 ? x2 ) , (2.11)
2a

?1 = x2 ? x2 ? x2 x?2 , ?2 = (x0 ? x2 )x?2 ,
0 1 2 3 3
(2.12)
?1
?3 = ax1 (x0 ? x2 ) ? ln(x0 ? x2 ), a = 0.

If the parameter a = 0, then
x1
?1 (?2 ? ?0 ) . (2.13)
A(x) = exp
2(x0 ? x3 )
Example 2.2.
A(x) = 2(x0 + 2x1 + ?)?k/2 ?
(2.14)
1 1 x2
?0 ?2 ln(2x0 + 2x1 + ?) ? ?2 ?3 tg?1
? exp , ? = 0,
4 2 x3

?1 = (2x0 + 2x1 + ?) exp{2(x1 ? x0 )? ?1 },
x2 (2.15)
?1
?3 = b ln x2 + x2 + 2 tg?1
?2 = (2x0 + 2x1 + ?) x2 + x2 , .
2 3 2 3
x3
3. 3.1. Reduced equations. The Ansatz (2.2) with the matrices (2.10), (2.11),
(2.14) and new variables ? gives the following reduced equations
k(?2 ? ?0 )? + [(?0 ? ?2 )(?1 + a?2 ?2 ?3 ) + (?0 + ?2 )?2 ?
22 2

?? 2 ??
? 2a?1 ?1 ?3 ?2 ? 2?3 ?1 ?2 ] + [(?0 ? ?2 )?2 ? ?3 ?2 ]
2
+ (3.1)
??1 ??2
?? ?
+ [a?1 + (?2 ? ?0 )(?3 + 1)] = i?(??)1/2k ?,
??3
Exact solutions of nonlinear Dirac’s and Schr?dinger’s equations
o 311

?? ?? ?? ?
= i?(??)1/2k ?, (3.2)
(?0 + ?1 ) + ?2 + ?3
??1 ??2 ??3

1 ?? ?
(1 ? 2k)?3 ? + 2(?3 + a?2 ) = i?(??)1/2k ?. (3.3)
2 ??2

The Ansatz (2.2) with the matrix A(x) = 1, ?1 = x0 + x3 , ?2 = x1 , ?3 = x2 leads
to the equation
?? ?? ?? ?
= i?(??)1/2k ?. (3.4)
(?0 + ?3 ) + ?1 + ?2
??1 ??2 ??3
It turns out that some of the reduced equations possess substantially more sym-
metries than the initial equation (2.1). For example the equation (3.4) is invariant
under infinite-dimensional Lie algebras. More exactly, the following statement is true:
Theorem 4. The system (3.4) is invariant under the infinitely dimensional algebra
whose basis elements have the form
For the case k = 1
? ? 1
Q1 = ?1 (?1 ) + ?2 (?2 ) + [?1 ?1 + ?2 ?2 ](?0 + ?3 ),
??2 ??3 2
? ?
Q2 = ??2 + ?3 ,
??3 ??2
? ? ?
?
Q3 = ?0 (?1 ) + ?0 (?1 ) ?2 + ?3 +
??1 ??3 ??2
1?
?
+ ?0 + ?0 (?1 )(?1 ?2 + ?2 ?1 )(?0 + ?3 ),
2
Q4 = ?3 (?1 )?4 (?0 + ?3 ).

For the case k = 1
? ? ? 1
Q2 = ??2
Q1 = , + ?3 + ?2 ?3 .
??1 ??3 ??2 2
? ? 1?
Q3 = ?1 (?1 ) + ?2 (?1 ) + [?1 (?2 )?1 + ?2 (?1 )?2 ](?2 + ?3 ),
??2 ??3 2
? ? ?
Q4 = ?1 + ?2 + ?2 + k,
??1 ??2 ??3
Q5 = ?3 (?1 )?4 (?0 + ?3 ),

where ?0 (?1 ), ?1 (?1 ), ?2 (?1 ), ?3 (?1 ) are arbitrary smooth functions, a dot desig-
nates differentiation with respect to ?1 .
3.2. The reduction of the nonlinear spinor equation to a system of ODE. Here
we give the explicit form of some ODE’s to which the Dirac equation is reduced:
? ?
i?2 ?(?) = ?(??)1/2k ?, (3.5)

i ? ?
(?0 + ?3 )? + i[?(?0 ? ?3 )]? = ?(??)1/2k ?, (3.6)
2
312 W.I. Fushchych

i ?1/2 ? ?
?2 ? + 2i? 1/2 ?2 ? = ?(??)1/2k ?, (3.7)
?
2
i
[?(? + 1)]?1 (2? + 1)[?0 + ?3 )? + i(?0 + ?3 )? = ?(??)1/2k ?,
? ? (3.8)
2
1
i(?0 + ?3 )? + i (?0 + ?3 )? ?1 + (?0 ? ?3 )?4 ? = ?(??)1/2k ?,
? ? (3.9)
4
d?
i? ?1 (?0 + ?3 )? + i(?0 + ?3 )? = ?(??)1/2k ?,
? ? ?
?? (3.10)
.
d?
The full list of the systems of ordinary differential equations is given in Fushchych
and Zhdanov (1988).
4. The explicit solutions of the Dirac equation. Some of the ODE’s can be
?
I. Case k = 1/2, with the nonlinearity ?(??), ? = x2 + x2 ,
2 3

1 x2
?(x) = (x2 + x2 )?1/4 exp ? ?2 ?3 tg?1 ?
2 3
2 x3
(4.1)
??
? x2
(?3 + a?2 ) ln(x2 + x2 ) + 2 tg?1
? exp ?i? ?,
2 3
2(1 + a2 ) x3
? is constant spinor, a is a real parameter.
?
II. Case k = 1/2, nonlinearity ?(??)1/2k , ? = x2 + x2 ,
2 3

1 x2 2i?k 2k?1
?(x) = (x2 + x2 ) exp ? ?2 ?3 tg?1 (x2 + x2 ) 4k ?3 ?. (4.2)
exp
1 ? 2k
2 3 2 3
2 x3
?
III. Case k = 1/2, nonlinearity ?(??)1/2k , ? = x0 + x3 ,
1? ?
? (?1 ?1 + ?2 ?2 ) + ?3 ?4 (?0 + ?3 ) ?
?(x) = exp
2 (4.3)
? exp i?(??)1/2k ?1 (x1 + ?1 ) ?,
?

where ?1 , ?2 , ?3 arbitrary smooth functions of ?.
?
IV. Case k = 1, nonlinearity ?(??)1/2 ,
?0 x0 ? ?1 x1 ? ?2 x2 ?0 x0
exp i?(??)1/2 2 (4.4)
?(x) = ? ?.
x0 ? x2 ? x2
2 ? x2 ? x2 )3/2
(x0 1 2
1 2
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