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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)?

Answer: The Schr?dinger equation (5.1) has additional symmetries (supersymmetries,

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)?

Answer: The Dirac equation (5.9) has a wide additional symmetry (supersymmetry,

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