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4 Reduction and antireduction
Under the term “reduction–antireduction”, we understand a decreasing of dimension
of an equation with respect to independent variables and increasing (antireduction)
by the number of dependent variables. That is we have simultaneously the process of
reduction (by the number of independent variables) and antireduction (increasing the
number of reduced systems with respect to the original equation) [25].
In the classical Lie approach as a rule the number of components of dependent
variables for reduced systems does not increase.
Example 1. Let us consider the nonlinear acoustics equation (Khokhlov–Zabolotskaja
equation)
u01 ? (u1 u)1 ? u22 ? u33 = 0, (4.1)

u = u(x1 , x2 , x3 ).

The ansatz (Fushchych and Myronyuk, 1991 [26])
1 1
x1 ?(1) (?0 , ?2 , ?3 ) + x2 ?(2) (?0 , ?2 , ?3 ) + ?(3) (?0 , ?2 , ?3 ), (4.2)
u=
61
3
?0 = x0 , ?2 = x2 , ? = x3

antireduces four-dimensional equation (4.1) to the system of coupled three-dimensional
equations for functions ?(1) , ?(2) , ?(3)

? 2 ?(1) ? 2 ?(2) (2) 2
2 + ?? 2 = (? ),
??2 3
? 2 ?(1) ? 2 ?(1) ??(1)
? ?(1) ?(2) = 0, (4.3)
+ +
2 2
??2 ??3 ??0
2 (3) 2 (3)
1 (2) (3) 1 ??(1)
?? ?? 1
2 + ?? 2 ? 3 ? ?? + (?(1) )2 = 0.
??2 3 ??0 9
3

The formula (4.2) gives a non-Lie ansatz for equation (4.1).
348 W.I. Fushchych

Example 2. Let us consider the equation for short waves in gas dynamics

2u01 ? 2(2x1 + u1 )u11 + u22 + 2?u1 = 0,
(4.4)
u = u(x0 = t, x1 , x2 ).

The ansatz (Fushchych and Repeta 1991, [27])
3/2
u = x1 ?(1) (?0 , ?2 ) + x2 ?(2) (?0 , ?2 ) + x1 ?(3) + ?(4) ,
1
(4.5)
?0 = x0 , ?2 = x2

antireduces one three-dimensional scalar equation (4.4) to a system of two-dimensio-
nal equations for four functions

? 2 ?(1) ? 2 ?(2)
(3)
? = 0, 2 = 0, 2 = 0,
??2 ??2
(4.6)
? 2 ?(4) ??(1)
9 (1) 2 1
= ?(1) 3?(2) + ? ? .
=? ,
2
??2 4 ??0 2

4.1 Antireduction and ansatzes for the nonlinear heat equation
Let us consider the nonlinear one-dimensional heat equation
?u ? ?u
(4.7)
= a(u) + F (u),
?t ?x ?x

?2u
?u
(4.8)
= + F (u).
?x2
?t
We consider an implicit ansatz

h t, x, u, ?(1) (?), ?(2) (?) . . . , ?(N ) (?) = 0, (4.9)

which reduces the two–dimensional equation (4.7) to the system of ODE for functi-
ons ?(1) , . . . , ?(N ) . We have constructed a quite long list of ansatzes which reduce
equation (4.7) to the system of ODE (Zhdanov R. and Fushchych W. 1994, [33]).
Example 3. If in (4.7)

a(u) = ?u?3/2 , F (u) = ?1 u + ?2 u5/2 , (4.10)

then the ansatz [33] is as follows

u?3/2 = ?(1) (t) + ?(2) (t)x + ?(3) (t)x2 + ?(4) (t)x3 , (4.11)

2 3 3
?(1) = 2??(1) ?(3) ? ?(?(2) )2 ? ?1 ?(1) ? ?2 ,
?
3 2 2
2 (2) (3) 3
?(2) = ? ?? ? + 6??(1) ?(4) ? ?1 ?(2) ,
?
3 2 (4.12)
2 3
?(3) = ? ?(?(3) )2 + 2??(2) ?(4) ? ?1 ?(3) ,
? ?
3 2
3
?(4) = ? ?1 ?(4) .
?
2
Ansatz ’95 349

Having solved the system of ODE (4.12), by formula (4.11) we construct exact soluti-
ons of the equation (4.7).
Example 4. If in (4.8)

F (u) = ? + ? ln u ? ? 2 (ln u)2 u, (4.13)

then the ansatz

ln u = ?(1) (t) + e?x ?(2) (t) (4.14)

reduces (4.8) to the system of ODE

?(1) = 2 + ??(1) ? ? 2 (?(1) )2 ,
?
(4.15)
?(2) = ? + ? 2 ? 2? 2 ?(1) ?(2) .
?

It is possible to construct solutions of system (4.15) in the explicit form. Depending
on the sign of the quantity d = ? 2 + 4?? 2 we get the following solutions of the
nonlinear equation (4.8), (4.13).

Case 4.1 d > 0
?2
d1/2 t d1/2 t
1
? ? d1/2 tg
exp ?x + ? 2 t + (4.16)
u = c cos .
2? 2
2 2

Case 4.2 d < 0
?2
|d|1/2 t |d|1/2 t
1
? + |d|1/2 th
exp ?x + ? 2 t + (4.17)
u = c ch .
2? 2
2 2

Case 4.3 d = 0
1
u = ct?2 exp ?x + ? 2 t + (4.18)
(?t + 2).
2? 2 t

Example 5. If in (4.7)

a(u) = ?uk , F (u) = ?1 u + ?2 u1?k , (4.19)

then the ansatz

uk = ?(1) (t) + ?(2) (t)x + ?(3) (t)x2 (4.20)

antireduces (4.7) to the system of ODE

?(1) = 2??(1) ?(3) + ?k ?1 (?(2) )2 + k?2 ,
?
?(2) = 2?(1 + 2k ?1 )?(2) ?(3) + k?1 ?(2) , (4.21)
?
?(3) = 2?(1 + 2k ?1 )(?(3) )2 + k?1 ?(3) .
?
350 W.I. Fushchych

5 Non-Lie symmetry, new relativity principles
5.1 Non-Lie symmetry Schr?dinger equation
o
Let us consider the Schr?dinger equation
o
p2
?
?a (5.1)
i u(x0 , x) = 0.
?x0 2n
It is well known that the maximal (in the Lie sense) invariance algebra (5.1) is
the full Galilei algebra AG2 (1, 3) = P0 , Pa , Jab , Ga , D, A
? ?
, Pa = ?i
P0 = i , a = 1, 2, 3,
?x0 ?x0
= xa pb ? xb pa , Ga = x0 pa ? mxa , (5.2)
Jab
1
D = 2x0 P0 ? xk Pk , A = x0 D ? x2 P0 + mx2 .
0 a
2
Operators Ga generate the standard Galilei transformations:
t > t = exp {iGa va } t exp {?iGa va } = t, (5.3)

xa > xa = exp {iGb vb } xa exp {?iGc vc } = xa + va t. (5.4)

Let us put the following question: do symmetries which are not reduced for the
algebra (5.2) exhaust for equation (5.1)?
o
non-Lie, nonlocal) which are not reduced to the Galilei algebra AG2 (1, 3) [29].
One of results in this direction is the following:
Theorem 6. (Fushchych and Seheda 1977 [28]). The Schr?dinger equation (5.1) is
o
invariant with respect to the Lorentz algebra AL(1, 3)
Jab = xa pb ? xb pa , (5.5)
1
p = (p2 + p2 + p2 )1/2 = (??)1/2 . (5.6)
J0a = (pGa + Ga p), 1 2 3
2m
It is not difficult to check that the operators Jab , J0c ? AL(1, 3) satisfy the
commutation relations
[Jab , J0c ] = i(gac Jb0 ? gbc Ja0 ), [J0a , J0b ] = ?iJab .
It is important to point out that J0a are integral-differential symmetry operators and
generate nonlocal transformations
xa > xa = exp {iJob Vb } xa exp {?iJ0c Vc } = Galilei transform. (5.4), (5.7)

t > t = exp {iJ0a Va } t exp {?iJ0b Vb } = t. (5.8)

Hence the operators J0a (5.6) generate new transformations which do not coincide
with the known Galilei and Lorentz transformation. Thus we have new relativity
principle. It is defined by formulae (5.7), (5.8).
Ansatz ’95 351

5.2 Time is absolute in relativistic physics
The four-component Dirac equation lies in the foundation of the modern quantum
mechanics
?µ pµ ? = m?(x0 , x1 , x2 , x3 ). (5.9)
Here ?µ are 4 ? 4 Dirac matrices.
Since the time of discovery of this equation it is known that (5.9) is invariant with
respect to the Poincar? algebra AP (1, 3) = Pµ , Jµ? with the basis elements
e
? i
Jµ? = xµ p? ? x? pµ + Sµ? , (?µ ?? ? ?? ?µ ).
(1)
(5.10)
Pµ = i , Sµ? =
?xµ 4
(1)
Operators Jµ? generate the standard Lorentz transformations
(1)
t > t = exp iJ0a va t exp {?iJ0b vb } , (5.11)

(1)
xa > xa = exp iJ0b vb xa exp {?iJ0c vc } . (5.12)

Hence, the fundamental statement follows that time t ? T (1) and space x ? R(3) are
the single pseudo-Euclidean space-time with the metric
s2 = x2 ? x2 ? x2 ? x2 . (5.13)
0 1 2 3

Let us put another question: Do there exist symmetries in equation (5.10) which
cannot be reduced to the algebra AP(1,3) (5.11)?