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?|x|2
U ?|x|U
wa wa = Ua Ua + ?xa Ua + exp .
t 2t 2t

This theorem is proved in the same way as the first one. The only difference is
that one should substitute into the invariance condition (2.9) the coefficients A and B
from (2.1) and (2.2) and the values of ? µ , ? from (1.4b).
It should be noted that equations of the form (2.19) are obtained as a particular
case of (2.20), i.e. when the function B in (1.7) is independent on the derivatives U .
I
Invariance under PGT automatically implies invariance under the rotation group.
The further restriction of the class of equations (2.19) is achieved by the require-
ment for the equations to be invariant under the projective operator ? (1.4c) and the
operator of scale transformations D (1.4d). The two following theorems are proved in
quite a similar way to the ones above.
Theorem 3. Among equations (2.19) only equations
U
g tn/2 w , (2.21)
?U = ?Ut + 2
t
where g is an arbitrary differentiable function, admit the operator ? (1.4c).
Theorem 4. Among equations (2.19) only equations
?
U U U
? constant,
tn/2 w =
?U = ?U1 + ?1 2 ,
t ?(t, x) ? (2.22)
?1 = constant, ? = constant,
where
1/2 n
?|x|2
1 ?
exp ? (2.23)
?(t, x) =
2 ?t 4t
The Galilean relativistic principle and nonlinear partial differential equation 491

is a fundamental solution of (1.1), admit the operator ? (1.4c) and the operator

D = 2t?t + xa ?xa + (2/? ? n)U ?U . (2.24)

Note 1. If one implies ? = 0 in (2.22), the obtained equation has the form

?U = ?Ut + ?1 U/t2 (2.25)

which may be reduced to (1.1) by means of the local substitution

U = W (t, x) exp(?1 /?t) ? = 0.

Note 2. The coefficients of all classes of equations constructed above contain (expli-
citly or implicitly) the fundamental solution ?(t, x) of (1.1). This is apparently due to
the fact that ?(t, x) (with an approximation to an arbitrary constant) is the complete
solution of the system
? = ?U0 ,
(2.26)
Ga (U ) ? tUa + 1 ?xa U = 0, a = 1, n.
2

Note 3. The above theorems may be generalised for the systems of equations of the
form
?U (k) = A(k) t, x, U (1) , . . . , U (m) +
(2.27)
+B (k) t, x, U (1) , . . . , U (m) , k = 1, 2, . . . , m.

In particular, amongst the equations (2.27) only equations
(k)
?U (k) = ?U0 + U (k) g (k) t, w(1) , . . . , w(m) , k = 1, 2, . . . , m,

where w(k) = U (k) exp ?|x|2 /4t , g (k) are arbitrary differentiable functions, are inva-
riant under the Galilean transformations with the infinitesimal operators
? 1 ? ?
? ?xa U (1) + · · · + U (m)
Ga = t , a = 1, n.
?U (1) ?U (m)
?xa 2

3. The second-order equations,
invariant under the Galilean transformations
In this section we shall construct all the equations of the form

(3.1)
Ut = C(t, x, U )?U + K(t, x, U, U ),
I

where C(t, x, U ), K(t, x, U, U ) are arbitrary differentiable functions, invariant under
I
the operators Ga (1.5), generating the GT (1.6). Also we shall distinguish all the
second-order equations of the form (1.8) which admit the following operators:

(3.2)
Ga = tPa , Pa = ? a , P0 = ? t , a = 1, n.

These operators satisfy the commutational relations

[Ga , P0 ] = ?Pa . (3.3)
[Ga , Pb ] = 0, [Pµ , P? ] = 0,
492 W.I. Fushchych, R.M. Cherniha

It turns out that the class of such equations is rather broad. In particular, it contains
the many-dimensional Monge–Amp` re equation (see Fushchych and Serov [8]) and
e
the non-relativistic analogue of the latter. All these equations are considerably nonli-
near, and as a rule they cannot be reduced to the form containing a linear plus a
nonlinear term.
The following statement gives the solution of the first problem, which was posed
at the beginning of this section.
Theorem 5. (3.1) is invariant under the GT (1.6) if and only if

(3.4)
C(t, x, U ) = f (t, U ),

K(t, x, U, U ) = g(t, U, U ) ? xa Ua /t, (3.5)
I I

where f , g are arbitrary differentiable functions.
To prove this theorem one should repeat the same procedures used in proving
theorem 1, with the only obvious difference that the coefficient functions of the Ga
operator, i.e.

? 0 = 0, ? a = ga t, a = 1, n, ?=0

should be substituted into (2.9).
Now let us formulate several more statements, giving the complete description of
the equations of class (3.1), invariant under Ga , Jab and the operators

? = t2 ?t + txa ?xa , (3.6)

(3.7)
D = 2t?t + xa ?xa .

Theorem 6. Among the set of equations (3.1) only the equations given by
Ut = f (t, U )?U + g(t, U, wn+1 ) ? xa Ua /t,
(3.8)
wn+1 = Ua Ua , Ua = ?U/?xa

are invariant under the operators Ga and Jab , a, b = 1, n.
Theorem 7. (3.8) is invariant under the projective transformations generated by the
operator (3.6) if and only if

g(t, U, wn+1 ) = t?2 g(U, t2 wn+1 ), (3.9)
f (t, U ) = f (U ),

where f , g are arbitrary differentiable functions.
Theorem 8. Amongst equations of the form (3.8) only equations

Ut = f (U )?U + Ua Ua g(U ) ? xa Ua /t (3.10)

are invariant under the projective and scale transformations generated by the
operators (3.6) and (3.7).
Theorem 9. The maximal IA of the simplest linear equation from the class (3.10):

Ut = ??U ? xa Ua /t, ? = constant (3.11)
The Galilean relativistic principle and nonlinear partial differential equation 493

is an algebra SLi(1, n) with basic operators:

Jab = xa ?b ? xb ?a , ? = t2 ?t + txa ?xa ,
Ga = t?a , I = U ?U ,
|x|2
xa n
?
D = 2t?t + xa ?xa , Pa = ?xa + I, P t = ?t + I.
2t 4?t2
2?t
Note 4. (3.11), by means of the local substitution

?|x|2
U = W (t, x)tn/2 exp
4t
or, in the equivalent notation,
1/2 n
?|x|2
W (t, x) 1 ?
exp ?
U= , ?(t, x) =
?(t, x) 2 ?t 4t

may be reduced to (1.1) for the function W (t, x).
Note 5. The classes of equations given in theorems 5 and 6 can be obtained from the
equations given in theorems 1 and 2. For this purpose it would be enough to apply
the above substitution from note 4.
Note 6. Equations invariant under GT (1.6) (see theorem 5) can be transformed by
means of the substitution of the independent variables

t = ?(t ),
xa = ?(t )xa + ?(a) (t ), a = 1, n,

where ?(t ) = constant, ?(a) , a = 1, n being arbitrary differentiable functions, to the
equations given by

Ut = f (t , U )?U + g (t , U , U ),
I

where
U (t , x ) = U (t, x),
d?
(?(t ))?2 f (?(t ), U ),
f (t , U ) =
dt
d?
g(?(t ), U , U (?(t ))?1 )+
g (t, , U , U ) =
dt
I I

d?(a) (t ) d? (a)
(?(t ))?1 ? ? (t )(?(t ))?2 Ua .
+
dt dt
In particular if

?(a) (t ) = 0,
?(t ) = t , a = 1, n

one obtains the equations

Ut = t ?2 f (t , U )?U + g(t , U , U t?1 ).
I
494 W.I. Fushchych, R.M. Cherniha

Consequence 2. It follows from the theorems given in §§ 2 and 3 that the nonlinear
diffusion equation (1.9) is invariant neither under PGT (1.2) and (1.3), nor under
GT (1.6). It means that the Galilean principle of invariance is not satisfied by (1.9).
Nonlinear equations, invariant under PGT and x and t translations, are obtained by
Fushchych [5].
Now let us proceed in solving the second problem: to describe all the second-order
equations

(3.12)
F (x0 , x1 , U, U0 , U1 , U00 , U01 , U11 ) = 0

in the two-dimensional space (x0 , x1 ), which are invariant under GT and translations
generated by operators (3.2).
Theorem 10. Amongst the set of equations (3.12) only the equations given by

F1 (w(I) , w(II) , U, U1 , U11 ) = 0 (3.13)

are invariant under ot (1.6) and translations.
(3.13) contains the following notation:

U0 U1 U00 U01
w(I) = det w(II) = det (3.14)
,
U01 U11 U10 U11

of the determinant of matrices, the elements of which are the first- and second-order
derivatives of the function U . Here F1 is an arbitrary differentiable function.
Proof. The invariance of (3.12) under translations, i.e. operators P0 , P1 , is equivalent
to the requirement
?F ?F
(3.15)
= = 0.
?x0 ?x1
Taking into account (3.15) we obtain the following expression for the action of the
2
twice prolonged operator X on the manifold (3.12) (see (2.6))

?F ?F ?F
+ ?µ + ? µ? (3.16)
? = 0, µ, ? = 0, 1.
?U ?Uµ ?Uµ?
F =0

The coefficient functions of operators {Ga } are given by

? 0 = ? = 0, ? 1 = t. (3.17)

The coefficient functions {?µ } = {?0 , ?1 }, {? µ? } = {? 00 , ? 01 , ? 10 , ? 11 } are determined
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