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? = 0, 6,
F is an arbitrary smooth function.
Such well-known equations are contained in the class (5):
1. u2 ? u2 ? 1 = 0 — the eikonal equation; (6)
0 1

ii , 1993, 10, . 52–58.
106 W.I. Fushchych, V.A. Tychynin

2. u11 ? u00 [u2 u11 ? 2u0 u1 u10 + u2 u00 ] = 0 — the Born–Infeld equation; (7)
0 1

3. u00 u11 ? u2 = 0 — the Monge–Amper? equation;
e (8)
10

4. u0 = f (u1 )u11 , f (u1 ) = f (u?1 )u?2 — the nonlinear heat equation [5]. (9)
1 1

Particularly, such equation as
u0 ? u?1 u11 = 0 (10)
1

is contained in the last class (9).
(1)
Let u (x0 , x1 ) be a known solution of Eq. (5). To construct a new solution
(2)
u (x0 , x1 ) let us write the first solution replacing in it an argument x1 for parameter
(1)
? : u (x0 , ? ) and substitute it to the hodograph transformation formula (1). So, we
obtain the solutions generating formula for Eq. (5).
(2) (1)
u (x0 , x1 ) = ?, x1 = u (x0 , ? ). (11)
Let us now describe some class of (1)-linearizable equations. Making use of for-
mulae (1) to transform general linear second order PDE
bµ? (y)vµ? + bµ (y)vµ + b(y)v + c(y) = 0, (12)
y = (y0 , y1 ), µ, ? = 0, 1,
we obtain
b00 (x0 , u)u?3 (u2 u11 ? 2u0 u1 u10 + u2 u00 ) ?
0 1
1
? 2b10 (x0 , u)u?3 (u1 u10 ? u0 u11 ) + b11 (x0 , u)u?3 u11 + (13)
1 1
+ b0 (x0 , u)u?1 u0 + b1 (x0 , u)u?1 ? b(x0 , u)x1 ? c(x0 , u) = 0.
1 1

bµ? , bµ , c are arbitrary smooth functions, b10 = b01 . Summation over repeated indices
is understood in the space R(1, 1) with the metric gµ? = diag (1, ?1). The repeated
use of this transformation to Eq. (12) turn us again to the Eq. (11).
For any equation of the class (12) the principle of nonlinear superposition is sati-
sfied
(3) (1) (1) (2)
u (x0 , x1 ) = u (x0 , x1 ? ? ),
u (x0 , x1 ) = u (x0 , ? ), (14)
(k) (3)
Here u (x0 , x1 ), k = 1, 2 are known solutions of Eq. (12), u (x0 , x1 ) is a new solution
of this equation. Parameter ? must be eliminated due to second equality of the sys-
tem (13). For example, such equations important for applications are contained in this
class (12):
u0 ? u?2 u11 = 0, u0 u11 ? u1 u10 = 0,
1
u0 u11 ? 2u0 u1 u10 + u2 u00 = 0, u0 ? c(x0 , u)u1 = 0.
2
1

Let us consider now an example of constructing new solutions from two known ones
by means of solutions superposition formula (13).
Example 1. A nonlinear heat equation
u0 ? u?2 u11 = 0
1
Hodograph transformations and generating of solutions 107

is reduced to the linear equation
v0 ? v11 = 0 (15)
Therefore, the formula (13) is true for (14). The functions
v
(1) (2)
x1 ? 2x0
u = x1 , u= (16)
(3)
are both partial solutions of Eq. (14). We construct a new solution u of this Eq. (14)
(1) (2)
via u and u . It has the form
1 1
(3)
u (x0 , x1 ) = ? ± + x1 ? 2x0 , (17)
2 4
2. Hodograph-invariant and -linearizable equations in R(1, 3). The hodo-
graph transformation of a scalar function u(x) of four independent variables x =
(x0 , x1 , x2 , x3 ) has the form
(18)
v(x) = y1 , x1 = v(y), x? = y? , ? = 0, 2, 3.
Prolongation formulae for (18) are obtained via calculations [6, 7]:
?1 ?1 ?3
u? = ?v1 v? , u11 = ?v1 v11 ,
u 1 = v1 ,
?3 ?3 2
u1? = ?v1 (v1 v1? ? v? v11 ), v?? = ?v1 (v1 v?? ? 2v? v1 v1? + v? v11 ),
2
(19)
?3
u?? = ?v1 [v1 (v1 v?? ? v? v1? ) ? v? (v1 v1? ? v? v11 )].
Here ?, ? = 0, 2, 3, ? = ?. Making use of involutivity of the transformation (18) we
list for it a such set of absolute differential invariant expressions of order ? 2:
f 2 (u1 , u?1 ), f 3 (u? , ?u?1 u? ),
f 0 (x0 , x2 , x3 ), f 1 (x1 , u), 1 1
f 4 (u11 , ?u?3 u11 ), f 5 (u1? , ?u?3 (u1 u1? ? u? u11 )),
1 1
(20)
f 6 (u?? , ?u?3 (u2 u?? ? 2u1 u? u1? + u2 u11 )).
1
1 ?
f 7 (u?? , ?u?3 [u1 (u1 u?? ? u? u1? ) ? u? (u1 u1? ? u? u11 )]).
1

There is no summation over ? here, as before, f 0 is an arbitrary smooth function, f j ,
j = 1, 7 are arbitrary symmetric.
An equation invariant under transformation (18) has the form
F ({f ? }) = 0 (21)
(? = 0, 7).
The solutions generating formula has the same form as (10)
(2) (1)
u (x0 , x1 , x2 , x3 ) = ?, x1 = u (x0 , ?, x2 , x3 ). (22)
(1) (2)
Here u (x) is a known solution of Eq. (21), u (x) is its new solution. The following
well-known equations are contained in this class (21):
u2 ? ua ua ? 1 = 0, a = 1, 3, the eikonal equation;
1. 0
(1 ? u? u? )2u ? uµ u? uµ? = 0, µ, ? = 0, 3, the Born–Infeld equation [8];
2.
det(uµ? ) = 0 the Monge–Amper? equation.
e
3.
108 W.I. Fushchych, V.A. Tychynin

Here summation over repeated indices is understood in the space R(1, 3) with the
metric gµ? = diag (1, ?1, ?1, ?1).
2u = ?µ ? µ u = u00 ? u11 ? u22 ? u33
is the d’Alembert operator,
ua ua = u2 + u2 + u2 + u2 = (?u)2 .
1 2 2 3

The class of hodograph-linearizable equations in R(1, 3) is constructed analogously
as above. Making use of transformation (18) for linear equation (11), written in
R(1, 3), we get

b11 (x? , u)u?3 u11 + b?? (x? , u)u?3 (u2 u?? ? 2u1 u? u10 + u2 u11 ) +
1
1 1 ?
+ b?? (x? , u)u?3 [u1 (u1 u?? ? u? u10 ) ? u? (u1 u1? ? u? u11 )] + (23)
1
+ b1 (x? , u)u?1 u? ? b(x? , u)x1 ? c(x? , u) = 0, x? = (x0 , x2 , x3 ).
1

Here ?, ? = 0, 2, 3 and summation over ? is understood in the space R(1, 2) with
metric g?? = diag (1, ?1, ?1).
?
Note, that multidimensional nonlinear heat equation
u0 ? u?2 (1 + u2 + u2 )u11 ? u22 ? u33 + 2u?1 (u2 u12 + u3 u13 ) = 0 (24)
2 3
1 1

reduces due to transformation (18) to linear equation v0 = ?(3) v, where ?(3) ?
2 2 2
?1 + ?2 + ?3 is the Laplace operator.
So, the solutions superposition formula for the equations (23) and (24) is
(3) (1)
u (x0 , x1 , x2 , x3 ) = u (x0 , ?, x2 , x3 ), (25)
(1) (2)
u (x0 , ?, x2 , x3 ) = u (x0 , x1 ? ?, x2 , x3 ). (26)

Example 2. Let partial solutions of Eq. (24)
1
x ? c2 2
92
(1) (2)
u = x0 ? x2 ? x3 ? ln 1 c3 (x1 ? c4 )2 ? x2 ? x2
u=
, 2 3
c1 4
(3)
be initial for generating a new solution u . Then this new solution of Eq. (24) is
determined via (25), (26) by the equality
(3) (3) 2
u 2 (x) + x2 + x2 = c3 x1 ? c2 ? c1 exp{x0 ? x2 ? x3 ? u (x)} ,
2 3
(27)
9
c3 = c2 , c2 = c4 + c2 .
43
Thus, the formula (27) gives us a new solution of Eq. (24) in the implicite form.
3. Hodograph-invariant and -linearizable systems of PDE in R(1, 1). Let
us consider two functions uµ (x0 , x1 ), µ = 0, 1 of independent variables x0 , x1 . The
hodograph transformation in this case, as is known [2], has the form
u0 (x0 , x1 ) = y0 , u1 (x0 , x1 ) = y1 , x0 = v 0 (y0 , y1 ), x1 = v 1 (y0 , y1 ),
(28)
? = u1 u0 ? u1 u0 = 0, ? ? = v1 v0 ? v0 v1 = 0.
10 10
10 01
Hodograph transformations and generating of solutions 109

The first and second order derevatives are changing as

u1 = ? ??1 v0 , u1 = ?? ??1 v0 , u0 = ?? ??1 v1 , u0 = ? ??1 v1 ,
0 1 0 1
(29)
1 0 1 0


u1 = ?? ??3 · [(v0 )2 (v0 v11 ? v0 v11 ) + (v1 )2 (v0 v00 ? v0 v00 ) ?
0 10 01 0 10 01
11
? 2v1 v0 (u1 v10 ? v0 v10 )],
00 0 01
0
u1 = ?? ??3 · [(v0 )2 (v0 v11 ? v0 v11 + (v1 )2 (v0 v00 ? v0 v00 ) ?
1 10 01 1 10 01
00
? 2v0 v1 (v0 v10 ? v0 v10 )],
11 10 01

u1 = ? ??3 · [v0 v0 (v0 v11 ? v0 v11 ) + v1 v1 (v0 v00 ? v0 v00 ) ?
01 10 01 01 10 01
10
? (v0 v10 ? v1 v10 )(v1 v0 + v0 v1 )],
10 01 10 10
(30)
u0 = ?? ??3 [(v0 )2 (v1 v11 ? v1 v11 ) + (v1 )2 (v1 v00 ? v1 v00 ) ?
0 01 10 0 01 10
11
? 2v1 v0 (v1 v10 ? v1 v10 )],
00 01 10

u0 = ?? ??3 [(v0 )2 (v1 v11 ? v1 v11 ) + (v1 )2 (v1 v00 ? v1 v00 ) ?
1 01 10 1 01 10
00
? 2v1 v0 (v1 v10 ? v1 v10 )],
11 01 10

u0 = ?? ??3 [v0 v0 (v1 v11 ? v1 v11 ) + v1 v1 (v1 v00 ? v1 v00 ) ?
01 01 10 01 01 10
10
? (v1 v10 ? v1 v10 )(v1 v0 + v0 v1 )].
01 10 10 10


Let us now construct the absolute differential invariants with respect to (28)–(30) of
order ? 2. Making use of involutivity of this transformation we get

f 1 (xµ , uµ ), f 2 (uµ , ?u? ),
µ = 0, 1, µ = ?, µ, ? = 0, 1,
µ ?

there is no summation over repeated indices here,

f 3 (uµ , ?? ?1 uµ ), µ = ?, µ, ? = 0, 1;
? ?
?3
f (u11 , ?? [(u0 )2 (u1 v11 ? u0 u1 ) + (u0 )2 (u1 u0 ? u0 u1 ) ?
41 0
0 0 0 11 1 0 00 0 00
? 2u1 u0 (u0 u10 ? u0 u10 )]),
00 10 01

f 5 (u1 , ?? ?3 · [(u1 )2 (u1 u0 ? u0 u1 ) + (u1 )2 (u1 u0 ? u0 u1 ) ?
00 0 0 11 0 11 1 0 00 0 00
? 2u0 u1 (u0 v10 ? u0 u10 )]),
11 10 01

f 6 (u1 , ?? ?3 · [u0 u1 (u1 u0 ? u0 u1 ) + u0 u1 (u1 u0 ? u0 u1 ?

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. 26
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