ñòð. 126 |

? ?

534 W.I. Fushchych, R.Z Zhdanov

The general solution of the system (47) has the form [5]

? = exp{i??1 (??)1/2k ?}?,

?

where ? is an arbitrary constant 4-component column. Substituting the resulting

expression for ? = ?(?) in (46) gives us the class of exact solutions of the nonlinear

Dirac equation containing three arbitrary functions.

Nonlinear equations of mathematical and theoretical physics that admit nontrivial

conditional symmetry have been analyzed in [14].

3. Reduction of number of independent and number of dependent variab-

les of PDE. Suppose (3) is an involutive set of operators that satisfy the condition

R2 ? R1 = ? > 0. In this case we have to modify somewhat the above technique

of reducing PDE by means of ans?tzes invariant with respect to the involutive set

a

(3). Note that the case in which (3) are basis operators of a subalgebra of the Lie

invariance algebra of a given equation satisfying the condition R1 < R2 leads to

“partially invariant” solutions [18].

We wish to solve the initial system of PDE in implicit form:

? = 0, m ? 1,

? ? (x, u) = 0, (48)

where ? ? are smooth functions satisfying the condition

m?1

det ?? ? /?u? (49)

= 0.

?,?=0

As a result, (1) and (2) assume the form

HA (x, u, ?, ? , . . . , ? ) = 0, (50)

A = 1, M ,

r

1

? ? ?

(51)

?aµ (x, u)?xµ + ?a (x, u)?u? = 0, a = 1, N ,

where ? = {? s ?/?xµ1 · · · ?xµp ?u?1 · · · ?u?q , p + q = s}.

s

It is clear that, as they are defined in the space of the variables x, u, ?(x, u), the

operators (3) satisfy the condition R1 = R2 (since the coefficients of ??? are all zero).

By means of the same reasoning as in the proof of Theorem 1, we may establish the

following result. There exists a change of variables (17) that reduces the system (51)

to the form

µ = 0, R1 ? 1, ? = 0, ? ? 1.

? ?

(52)

?xµ = 0, ?u ? = 0,

If the system (48), (50) is conditionally invariant with respect to the set of opera-

tors (3) and if condition (52) holds, it may be rewritten as follows:

? a (x , u ) = 0, ? = 0, m ? 1,

(53)

HA (xR1 , . . . , xn?1 , u ? , . . . , u m?1 , ?, ? , . . . , ? ) = 0,

r

1

where the symbol ? denotes the collection of partial derivatives of the function ? of

s

order s with respect to the variables xR1 , . . . , xn?1 , u ? , . . . , u m?1 .

Integrating (52) yields the ansatz of the field w? :

? = 0, m ? 1,

? ? = F ? (xR1 , . . . , xn?1 , u ? , . . . , u m?1 ), (54)

Conditional symmetry and reduction of partial differential equations 535

where F ? are arbitrary smooth functions. But the ansatz of the field u ? (x ) cannot

be obtained by substituting (54) in the relations ? ? (x , u (x )) = 0, ? = 0, . . . , m ? 1,

since the inequality R2 ? R1 = ? > 0 violates the condition (49) (if ? > 0, the matrix

?? ? /?ui? m?1 has null columns).

?,?=0

To overcome this problem, we shall, by definition, let the expressions

? = ?, m ? 1,

F ? (xR1 , . . . , xn?1 , u ? , . . . , u m?1 ) = 0,

j = 0, ? ? 1

u j = Cj ,

be the ansatz of the field u ? = u ? (x ) invariant with respect to the set of operators

(55)

Qj = ?xj?1 , j = 1, R1 , Xi = ?u i?1 , i = 1, ?.

The latter ansatz may be rewritten in the form

? = 0, ? ? 1,

u ? = C? ,

(56)

? = 0, m ? ? ? 1,

u ?+? = ?? (xR1 , . . . , xn?1 ),

where ?? are arbitrary smooth functions and C? are arbitrary constants.

Rewriting (56) in terms of the initial variables gives us

? = 0, ? ? 1,

g ? (x, u) = C? ,

(57)

? = 0, m ? ? ? 1.

g ?+? (x, u) = ?? (fR1 (x, u), . . . , fn?1 (x, u)),

Moreover, substituting (57) in the initial system of PDE (1) or, equivalently,

substituting the expressions ? ? = g ? ? C? , ? = 0, . . . , ? ? 1, ? ? = g ?+? ? ?? ,

0 ? ? ? m ? ? ? 1 in the PDE (50) yields a system of M differential equations

for m ? ? functions. Consequently, the dimension of the system (1) decreases by R1

independent and ? dependent variables.

Let us rewrite (57) in a form more convenient in applications. For this purpose,

note that, without loss of generality, we may renumber the operators (3) satisfying

the condition R2 ? R1 = ? > 0 in such a way that the first R1 operators satisfy the

condition

R1 n?1 R1 m?1 n?1

?

rank ?aµ = rank ?aµ , ?a

a=1 µ=0 a=1 ?=0 µ=0

and the last N ? R2 operators are linear combinations of the previous R2 operators.

Let ?j (x, u), j = 1, . . . , m+n?R2 , be the complete set of functionally independent

first integrals of the system (51) and, moreover,

=m??

m?? m?1

rank ??j /?u? j=1 ?=0

and let ?j (x, u) be the solutions of the equations Q1+R1 ?(x, u) = 1 with i = 1, 2, . . . , ?.

Then (57) may be expressed in the following equivalent form:

?i (x, u) = Ci , i = 1, ?,

(58)

j = 1, m ? ?.

?j (x, u) = ?j (?R1 (x, u), . . . , ?n?1 (x, u)),

Definition 4. Expressions (58) are called the ansatz of the field u? = u? (x) invari-

ant with respect to the involutive set of operators (3) provided R2 ? R1 ? ? > 0.

The above reasoning may be summarized in the form of a theorem.

536 W.I. Fushchych, R.Z Zhdanov

Theorem 2. Suppose that the system of PDE (1) is conditionally invariant with

respect to the involutive system of operators (3) and, moreover, that R1 < R2 .

Then the system (1) is reduced by the ansatz invariant with respect to the set of

operators (3).

Example 1. The system of two wave equations

2u = 0, 2v = 0 (59)

is invariant with respect to a one-parameter group with infinitesimal operator Q = ?v .

Since R1 = 0 and R2 = 1, the parameter ? is equal to 1. The complete set of first

integrals of the equation ??(x, u, v)/?v = 0 is given by the functions

?µ = xµ , µ = 0, 3, ?4 = u,

whence the ansatz for the field u(x), v(x) invariant under the operator Q has the

form (58),

u = ?(?0 , ?1 , ?2 , ?3 ), v = C, C = const.

Substituting the above expressions in (59) yields

?? 0 ? 0 ? ? ? 1 ? 1 ? ? ? 2 ? 2 ? ? ? 3 ? 3 = 0

i.e., the number of dependent variables of the initial system (59) is reduced.

Example 2. Consider the system of nonlinear Thirring equations

ivx = mu + ?1 |u|2 v, iuy = mv + ?2 |v|2 u, (60)

where u, v are complex functions of x, y and ?1 , ?2 are real constants.

The above system admits a one-parameter transformation group with generator

Q = iu?u + iv?v ? iu? ?u? ? iv ? ?v? .

Following the change of variables

u(x, y) = H1 (x, y) exp{iZ1 (x, y) + iZ2 (x, y)},

v(x, y) = H2 (x, y) exp{iZ1 (x, y) ? iZ2 (x, y)},

where Hj and Zj are the new dependent variables, Q assumes the form Q = ?Z1 .

Consequently, the ansatz invariant under Q has the form

u(x, y) = H1 (x, y) exp{iC + iZ2 (x, y)},

(61)

v(x, y) = H2 (x, y) exp{iC ? iZ2 (x, y)}.

Substitution of (61) in (60) yields a system of four PDE for the three functions

H1 , H2 , and Z2 ,

H2x = mH1x sin 2Z2 , H1y = ?mH2 sin 2Z2 ,

2

H2 Z2x = mH1 cos 2Z2 + ?1 H1 H2 ,

?H1 Z2y = mH2 cos 2Z2 + ?2 H2 H1 .2

Example 3. A group analysis of the one-dimensional gas dynamics equations

ut + uux + ??1 px = 0, pt + (up)x + (? ? 1)pux = 0 (62)

?t + (u?)x = 0,

Conditional symmetry and reduction of partial differential equations 537

has been carried out by Ovsyannikov [1], who established, in particular, that the

invariance algebra of the system of PDE (62) contains the basis element

(63)

Q = p?p + ??? .

The complete set of functionally independent first integrals of the equation Qw(t,

x, u, p, ?) = 0 is: ?1 = u, ?2 = p??1 , ?3 = t, and ?4 = x. Consequently, the ansatz

invariant under Q (63) may be chosen in the form

p??1 = ?2 (t, x), ln ? + F (p??1 ) = C,

u = ?1 (t, x), (64)

where C = const and F is some smooth function.

Substituting the ansatz (64) in the system of PDE (62) yields a system of three

differential equations for the two unknown functions ?1 (t, x) and ?2 (t, x):

?

?1 + ?1 ?1 ? ?2 F (?2 )?2 = 0,

t x x

?t + ? ?x + (? ? 1)? ?x = 0,

2 12 21

(65)

?

?1 ((1 ? ?)?2 F (?2 ) ? 1) = 0,

x

Thus we have achieved a reduction of the number of dependent variables of the

gas dynamics equations.

It is of interest that if ?1 = 0, it follows from the third equation of the system (65)

x

that F = ? + (1 ? ?)?1 ln(??1 p). Substituting this expression in (62) yields p = k?? ,

k ? R1 , which is the relation that characterizes a polytropic gas.

1. Ovsyannikov L.V., Group analysis of differential equations, Moscow, Nauka, 1978 (in Russian).

2. Olver P., Applications of Lie groups to differential equations, New York, Springer, 1986.

3. Sidorov A.F., Shapeev V.P., Yanenko N.N., Method of differential constraints and its applications

in gas dynamics, Novosibirsk, Nauka, 1984 (in Russian).

4. Fushchych W.I., Shtelen V.M., Serov N.I., Symmetry analysis and exact solutions of nonlinear

equations of mathematical physics, Kiev, Naukova Dumka, 1989 (in Russian).

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Rep., 1989, 172, ¹ 4, 123–174.

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system, Phys. Lett. A, 1989, 141, ¹ 3–4, 113–115.

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1951 (Russian translation).

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(in Russian).

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differential equations, Quart. J. Math., 1952, 3, ¹ 12, 250–259.

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a

on, Ukr. Math. J., 1990, 42, ¹ 7, 958–962.

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538 W.I. Fushchych, R.Z Zhdanov

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equation, Lett. Nuovo Cim., 1982, 34, ¹ 67, 498–501.

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in Mathematical Science, New York, Academic, 1982, 1–6.

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W.I. Fushchych, Scientific Works 2002, Vol. 4, 539–556.

On the general solution of the d’Alembert

equation with nonlinear eikonal constraint

W.I. FUSHCHYCH, R.Z. ZHDANOV, I.V. REVENKO

We construct general solutions of the system of nonlinear differential equations 2u = 0,

uµ uµ = 0 in the four- and five-dimensional complex pseudo-Euclidian spaces. The obtai-

ned results are used to reduce multi-dimensional nonlinear wave equation to ordinary

differential equations.

1. Introduction. In the present paper we construct general solution of the multi-

dimensional system of partial differential equations

2n u ? 0,

(1)

uµ uµ ? u2 1 ? u2 2 ? · · · ? u2 n?1 = 0

x x x

in the four- and five-dimensional pseudo-Euclidian space. In (1) u = u(x0 , x1 , . . .,

xn?1 ) ? C 2 (Cn , C1 ). Hereafter the summation over the repeated indices in the pseudo-

Euclidian space M (1, n) with the metric tensor gµ? = diag (1, ?1, . . . , ?1) is under-

stood.

We suggest a new algorithm of construction of exact solutions of the nonlinear

d’Alembert equation

24 u = ?uk , ?, k ? R1 (2)

via solutions of the system of PDE (1).

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