ñòð. 82 |

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

ñòð. 82 |