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. 122
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where f1 , f2 are arbitrary smooth enough functions. Consequently, we have the
following maps transforming subsets of solutions of the linear d’Alembert equation
(79) into another subsets of its solutions:
?1 2
? ? ? ?
u > f1 (u) ?A? (u)A? (u) Aµ (u)xµ + B(u) +
(1)
?3 µ??? 2 ?1
? ? ? ?
+ ?A? (u)A? (u) ? Aµ (u)A? (u)A? (u)x? + C(u) + f2 (u),
?1
1/2
? ? ? ?
u > f1 (u) ?A? (u)A? (u) Aµ (u)xµ + B(u)
(2) + f2 (u),
2
x0 ? x3 u > f1 (x0 ? x3 ) ln x1 + C1 (x0 ? x3 )
(3) +
2 ?1/2
+ x2 + C2 (x0 ? x3 ) + f2 (x0 ? x3 ),
?3/2 µ???
? ? ? ?
u > ?A? (u)A? (u)
(4) ? Aµ (u)A? (u)A? (u)x? + C(u) ,
x0 ? x3 > f1 (x0 ? x3 )(x1 cos C1 (x0 ? x3 ) +
(5)
+ x2 sin C1 (x0 ? x3 ) + C2 (x0 ? x3 ).

Note that in the cases 4, 5 function f2 is absorbed by arbitrary functions C, C2 .
And one more remark seems to be noteworthy. If one takes as a particular solution
of the system (2) the function u(x) = (x0 x1 ± x2 x2 + x2 ? x2 )/(x2 + x2 ) and
1 2
1 2 0
General solution of the d’Alembert equation with a nonlinear eikonal constraint 515

substitutes it into the first, second and fourth Ans?tze from (70), then the following
a
Ans?tze are obtained:
a
x0 x1 ± x2 x2 + x2 ? x2
? 1 2 0
x2 x2 x2 x2 ,
(1) w(x) = ? + + ,
1 2 3 0
x2 + x2
1 2

x0 x1 ± x2 x2 + x2 ? x2
w(x) = ? x2 + x2 ? x2 , 1 2 0
(2) ,
1 2 0 2 + x2
x1 2

x0 x1 ± x2 x2 + x2 ? x2
1 2 0
(4) w(x) = ? x3 , .
2 + x2
x1 2

Provided the above Ans?tze do not depend on the second argument, the usual Lie
a
Ans?tze are obtained which are invariant under some subgroups of the Poincar? group
a e
P (1, 3) [19]. Consequently, if we imagine invariant solutions as dots in a solution
space of the nonlinear d’Alembert equation, then through some of them one can
draw curves which are conditionally-invariant solutions. In this respect a number of
interesting questions arise, let us mention two of these:

(1) Is any invariant solution of the nonlinear d’Alembert equation (33) a particular
case of some more general conditionally-invariant solution?
(2) Does there exist such conditionally-invariant solution of Eq. (33) that all invari-
ant solutions of Eq. (33) are its particular cases? (saying about invariant soluti-
ons we mean solutions invariant under some subgroup of the symmetry group
of Eq. (33)).

An answer to the first question seems to be positive. A positive answer to the
second one would provide us with a concept of a “general invariant solution”. But so
far this problem is completely open and needs further investigation.
Acknowledgments. One of the authors (R. Zhdanov) is supported by the Alexan-
der von Humboldt Foundation.

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Contents
O. Bedrij, W.I. Fushchych, Fundamental constants of nucleon-meson dynamics . . . . 1
A.F. Barannyk, W.I. Fushchych, On maximal subalgebras of the rank n ? 1
of the conformal algebra AC(1, n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
W.I. Fushchych, Conditional symmetries of the equations of mathematical physics . 9
W.I. Fushchych, L.F. Barannik, V.I. Lagno, Invariants of one-parameter
subgroups of the conformal group C(1, n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
.I. , .. i, ii i i
, iii i i . . . . . 21
.I. , .I. , i i
i ii i i . . . . . . . . . . . . . . . . . . . . . . . . . 31
.I. , .I. , i i i ii
i i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
W.I. Fushchych, V.I. Chopyk, Symmetry analysis and ansatzes
for the Schr?dinger equations with the logarithmic nonlinearity . . . . . . . . . . . . . . 56
o
W.I. Fushchych, V.I. Chopyk, V.P. Cherkasenko, Symmetry and exact
solutions of multidimensional nonlinear Fokker–Planck equation . . . . . . . . . . . . . . 67
W.I. Fushchych, V.I. Lagno, R.Z. Zhdanov, On nonlinear representation
of the conformal algebra AC(2, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
.. , .. , .. ,
. . . . . . . . . . . . . . . . . . . . . . . . . . 80
W.I. Fushchych, N.I. Serov, L.A. Tulupova, The conditional invariance
and exact solutions of the nonlinear diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . 89
W.I. Fushchych, W. Shtelen, P. Basarab-Horwath, A new conformal-invariant
non-linear spinor equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
W.I. Fushchych, V.A. Tychynin, Generation of solutions for nonlinear
equations via the Euler–Amper? transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
e
W.I. Fushchych, V.A. Tychynin, Hodograph transformations and generating
of solutions for nonlinear differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
W.I. Fushchych, I.A. Yegorchenko, New conditionally invariant solutions
for non-linear d’Alembert equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112
W.I. Fushchych, R.Z. Zhdanov, Anti-reduction of the nonlinear wave equation . . 116
W.I. Fushchych, R.Z. Zhdanov, I.V. Revenko, On the new approach
to variable separation in the wave equation with potential . . . . . . . . . . . . . . . . . . . 120
R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych, Orthogonal and non-orthogonal
separation of variables in the wave equation utt ? uxx + V (x)u = 0 . . . . . . . . . . 126
.. , .. , .. ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . 139
N. Euler, A. K?hler, W.I. Fushchych, Q-symmetry generators and exact
o
solutions for nonlinear heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
.I. , .. i, i-iii ii
i i–i i-i . . . . . . . . . . . . . . . . . . . . . . . . . 165
.I. , .. , .. , i i
i i i ’ i – . . . . . . . . . . . . . . . . . . . . . . . . . . .173
W.I. Fushchych, R.O. Popovych, Symmetry reduction and exact solutions
of the Navier–Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
W.I. Fushchych, R.O. Popovych, G.V. Popovych, Ans?tze of codimension one a
for the Navier–Stokes field and reduction of the Navier–Stokes equation . . . . 240
W.I. Fushchych, R.Z. Zhdanov, Antireduction and exact solutions
of nonlinear heat equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
W.I. Fushchych, R.Z. Zhdanov, Conditional symmetry and anti-reduction
of nonlinear heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
W.I. Fushchych, R.Z. Zhdanov, V.I. Lahno, On linear and non-linear
representations of the generalized Poincar? groups in the class of Liee
vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
W.I. Fushchych, I.M. Tsyfra, Nonlocal ansatzes for nonlinear wave equation . . . . 267
W.I. Fushchych, I.M. Tsyfra, V.M. Boyko, Nonlinear representations
for Poincar? and Galilei algebras and nonlinear equations
e
for electromagnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych, On the new approach
to variable separation in the two-dimensional Schr?dinger equation . . . . . . . . . . 283
o
R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych, Separation of variables
in the two-dimensional wave equation with potential . . . . . . . . . . . . . . . . . . . . . . . . . 291
P. Basarab-Horwath, L. Barannyk, W.I. Fushchych, New solutions
of the wave equation by reduction to the heat equation . . . . . . . . . . . . . . . . . . . . . . .311
P. Basarab-Horwath, N. Euler, M. Euler, W.I. Fushchych, Amplitude-phase
representation for solutions of nonlinear d’Alembert equations . . . . . . . . . . . . . . . 326
O. Bedrij, W.I. Fushchych, Planck’s constant is not constant in different
quantum phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
W.I. Fushchych, Ansatz ’95 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
W.I. Fushchych, Galilei invariant nonlinear Schr?dinger type equations o
and their exact solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
W.I. Fushchych, A.F. Barannyk, Yu.D. Moskalenko, On new exact solutions
of the multidimensional nonlinear d’Alembert equation . . . . . . . . . . . . . . . . . . . . . . 365
W.I. Fushchych, V.M. Boyko, Symmetry classification of the one-dimensional
second order equation of hydrodynamical type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
W.I. Fushchych, R.M. Cherniha, Galilei-invariant nonlinear systems
of evolution equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
W.I. Fushchych, V. Chopyk, P. Nattermann, W. Scherer, Symmetries
and reductions of nonlinear Schr?dinger equations of Doebner–Goldin type . . 393
o
W.I. Fushchych, O.V. Roman, R.Z. Zhdanov, Symmetry reduction and exact
solutions of nonlinear biwave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
W.I. Fushchych, O.V. Roman, R.Z. Zhdanov, Symmetry and some exact
solutions of non-linear polywave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
W.I. Fushchych, R.Z. Zhdanov, O.V. Roman, Symmetry properties,
reduction and exact solutions of biwave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
V.I. Lahno, R.Z. Zhdanov, W.I. Fushchych, Symmetry reduction and exact
solutions of the Yang–Mills equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
R.Z. Zhdanov, W.I. Fushchych, Conditional symmetry and new classical
solutions of the Yang–Mills equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
R.Z. Zhdanov, W.I. Fushchych, On non-Lie ansatzes and new exact solutions
of the classical Yang–Mills equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
R.Z. Zhdanov and W.I. Fushchych, Symmetry and reduction of nonlinear
Dirac equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
R.Z. Zhdanov, V.I. Lahno, W.I. Fushchych, Reduction of the self-dual
Yang–Mills equations. I. The Poincar? group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
e
R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych, On the new approach
to variable separation in the time-dependent Schr?dinger equation o
with two space dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
R.Z. Zhdanov, I.V. Revenko, W.I. Fushchych, On the general solution
of the d’Alembert equation with a nonlinear eikonal constraint
and its applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

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