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Ansatz ’95
W.I. FUSHCHYCH

In this talk I am going to present a brief review of some key ideas and methods which
were given start and were developed in Kyiv, at the Institute of Mathematics of National
Academy of Sciences of Ukraine during recent years.

Plan of the talk

The simplest classification of equations.
What is ansatz? The problem of PDE reduction without symmetry.
Conditional symmetry. How can we expand symmetry of PDE?
Conditional symmmetry of Maxwell and Schr?dinger systems.
o
Q-conditional symmetry of the nonlinear wave equation, which is not invariant with
respect to the Lorentz group.
Conditional symmetry of the Poincar?–invariant d’Alembert equation.
e
Conditional symmetry of the nonlinear heat equation.
Reduction and Antireduction.
Antireduction of the nonlinear acoustics equation.
Antireduction of the equation for short waves in gas dynamics.
Antireduction of nonlinear heat equation.
Nonlocal symmetry, new relativity principles.
Non–Lie symmetry of the Schr?dinger equation.
o
Time is absolute in relativistic physics.
New equations of motions.
High–order parabolic equation in Quantum Mechanics.
Nonlinear generalization of the Maxwell equations.
Equations for fields with the spin 1/2.
How to extend symmetry of on equation with arbitrary coefficients?



1 Classification of equations
Every field of science must begin from some classification. We have today a lot of
classifications of differential equations: parabolic, hyperbolic, elliptic, ultrahyperbolic

J. Nonlinear Math. Phys., 1995, V.2, N 3–4, P. 216–235.
Ansatz ’95 339

etc. I believe that it is most appropriate for our Conference to divide all equations of
?
mathematics into two classes: B and B

Newton
Eikonal
B
Hamilton
Euler
' $
Heat
B?Beauty
' $ Monge-Ampere
Born-Infeld
Kernel d’Alembert
Maxwell
&%
Navier-Stokes
Korteweg-de Vries
& %
Boussinesq
Schr?dinger
o
Dirac
Yang-Mills
...
It is seen from the adduced picture that all fundamental equations of mathemati-
cal physics are united into one class B. From the point of view of existing now
classifications they belong to essentially different classes. Equations from the class B
have wide symmetry, and by this feature they are substantially different from other
equations of mathematics.
It is important to point out that there are close relations among these different
equations, which have not been investigated yet till now. For example, if we know
solutions of the heat equation, we can construct solutions for the wave (d’Alembert)
equation. By means of solutions of the Dirac equation, solutions of the Maxwell, heat,
Yang–Mills, and other equations [18] can be obtained.


2 Ansatz reduction of PDE without using symmetry
Let us consider a PDE

L(x, u, u(1) , u(2) , . . . u(n) ) = 0,
?u
u = u(x), x = (x0 , x1 , . . . , xn ), u(1) = (u0 , u1 , u2 , . . . , un ), uµ = ,
(2.1)
?xµ
?2u
u(2) = (u00 , u01 , . . . , unn ), uµ? = .
?xµ ?x?
340 W.I. Fushchych

?
Depending on the explicit form of L, equation (2.1) can belong to B or B. In
mathematical physics we often come across equations of the following type:

Lu ? 2u ? F (x, u, u(1) ) = 0. (2.2)

What can we say today about solutions of equations (2.1), (2.2)? The answer is
trivial: Nothing.
If equation (2.2) belongs to the class B and is invariant with respect to the Poincar?
e
group P (1, n), that is, a nonlinear function F (x, u, u(1) ) has the special form

?u ?u
(2.3)
F (x, u, u(1) ) = F u,
?xµ ?xµ
then for equation (2.2) we can construct some classes of exact solutions, study Pain-
lev? properties, construct approximate solutions, study asymptotic properties, etc.
e
Definition 1. (W. Fushchych, 1981, 1983 [1, 2, 3]) We shall call a formula

(2.4)
u = f (x)?(?) + g(x),

an ansatz for equation (2.2) if after substitution of (2.4) we get an equation for the
function ?(?) which depends only on new variables ? = (?1 , ?2 , . . . , ?n?1 ), where
f (x), g(x) are given functions.
If (2.4) is an ansatz for (2.2), then the latter is reduced (the number of independent
variables decreases by one) to an equation for the function ?(?).
Thus the problem of reduction of an equation reduces to description of three
functions f (x), g(x), ? which leads to an equation for ?(?) with less number of
variables.
We can display schematically the process of reduction for an 4–dimensional equati-
on in the following way:

Basic Equations
4 Dim 4 Dim
 Z
 Z
 Z
 Z
E3 E3
 Z

= ˜
Z
3 Dim 3 Dim
S 
S 
S 
E2 E
S 
?2
? S 
2 Dim 2 Dim
S 
S 
Z 
Z 
S 
Z 
S 
Z 
E1 E1
S 
Z 
w
S ?
/
˜
Z =
1 Dim 1 Dim
ODE
Ansatz ’95 341

E3 is a set of three-dimensional equations, E2 is a set of two-dimensional equations,
E1 is a set of one-dimensional equations with the following inclusion E3 ? E2 ? E1 .
That is, from one principal equation we obtain the whole set of ODE. Having
solved the ODE, we find exact solutions of a multidimensional equation.
Description of ansatzes of the form (2.4) for the nonlinear wave equation is an
extremely difficult nonlinear problem. In the simplest case, when we put f (x) = 1,
g(x) = 0 for the nonlinear Poincar?–invariant d’Alembert equation
e

2u = F (u), (2.5)

the problem of reduction of (2.5) to ODE reduces to construction of solutions for the
following overdetermined system for ? (Fushchych W., Serov M. 1983 [3])

2? = F2 (?),
2 2 2 2
(2.6)
?? ?? ?? ?? ?? ??
? ? ? ··· ?
= = F2 (?).
?xµ ?xµ ?x0 ?x1 ?x2 ?xn
If ? is a solution of the system (2.6), then the multidimensional equation (2.5) reduces
to ODE with variable coefficients

(2.7)
a2 (?)?(?) + a1 (?)?(?) + a0 (?)?(?) F (?) = 0
? ?

A solution of equation (2.5) has the form

(2.8)
u(x0 , . . . xn ) = ?(?), ? = ?(x0 , x1 , . . . , xn ),

? is a solution of equation (2.7).
Compatibility and general solutions of system (2.6) are described in detail in
papers of Zhdanov, Revenko, Yehorchenko, Fushchych (1987–1993, [4–6]). As we
see, without using explicitly the symmetry of equation (2.5), we can reduce a multi-
dimensional wave equation to ODE. It is obvious that all ansatzes and solutions, which
are constructed on the basis of the classical method by Sophus Lie, can be obtained
within the framework of our approach. The subgroup analysis of the Poincar? group
e
P (1, n) (Patera J., Winternitz P., Zassenhaus H., 1975–1983, [7, 8] Fedorchuk V.,
Barannyk A., Barannyk L., Fushchych W., 1985–1991 [9–11]) gives only a part of
possible ansatzes.
Note 1. P. Clarkson and M. Kruskal (1989 [12]) implemented the approach suggested
by us in 1981–1983 [1, 2, 3] for the one-dimensional Boussinesq equation and const-
ructed in explicit form ansatzes and solutions which cannot be obtained within the
framework of the classical S. Lie method. In the literature, this approach is often
called the “direct method of reduction”. I believe that it would be more consistent and
correct to call this method of construction of PDE solutions a method of ansatzes.


3 Conditional symmetry
The Lie symmetry, as known, is a local symmetry of the whole set of solutions.
The Lie algorithm enables us to define the invariance algebra for an arbitrary given
equation and to construct ansatzes.
342 W.I. Fushchych

The term and the concept “conditional symmetry” was introduced and developed
in our papers (1983–1993, [2, 3, 13–18]). This extremely simple concept has appeared
to be efficient and enabled us to discover a nature of many ansatzes which could not
be obtained within the framework of the Lie method.
Conditional symmetry is the symmetry of subsets of equation’s solutions. Knowing
conditional symmetry of an equation, we can construct non–Lie ansatzes and soluti-
ons. It is more difficult to study conditional symmetry of a given equation than to
study its classical Lie symmetry. The difficulty is related to the fact that to find
conditional symmetry of an equation, it is necessary to solve nonlinear determining
equations.
During recent years, there are intensive studies in this promising direction, and
today we can make following general conclusion:
Corollary 1. Principal nonlinear equations of mathematical physics have conditional
symmetry.
Let us denote by the symbol
(3.1)
Q = Q1 , Q2 , . . . , Qr
some set of operators which does not belong to the invariance algebra (IA) of equation
(2.1).
Definition 2. (Fushchych W., Nikitin A., Shtelen W. and Serov M., 1987 [13, 14, 18],
Fushchych W. and Tsyfra I. (1987 [15])). Equation (2.1) is said to be conditionally
invariant under the operators Q from (3.1), if there exists a supplementary condition
on the solutions of (3.1) of the form
(3.2)
L1 (x, u, u(1) , . . . , u(n) ) = 0
such that (3.1) together with (3.2) is invariant under Q.
Thus, one has the following criterion of conditional invariance [13, 15, 18]
(3.3)
Qs L = ?0 L + ?1 L1 ,

(3.4)
Qs L1 = ?2 L + ?3 L1 ,

where ?0 , ?1 , ?2 , ?3 are some differential expressions, Qs is the s-th prolongation by
Lie.
Definition 3. We shall say that an equation is Q-conditionally invariant if the additi-
onal equation L1 = 0 is a quasilinear equation of the first order
L1 (x, u, u(1) ) ? Qu = 0, (3.5)

? ?
(3.6)
Q = ?µ (x, u) + ?(x, u) ,
µ

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