On classification and invariants of second order non-parabolic linear partial differential equations in two variables

0.2 MB·

by  U. Bekbaev

In the nameof Allah, Most gracious, Most Merciful.
7
ON CLASSIFICATION AND INVARIANTS OF SECOND ORDER
1
0 NON-PARABOLIC LINEAR PARTIAL DIFFERENTIAL
2
EQUATIONS IN TWO VARIABLES

n
a
J U.BEKBAEV

6

Abstract. The paper deals with second order abstract linear partial differ-
]
P ential equations (LPDE) over a partial differential field with two commuting
A differential operators. In terms of usual differential equations the main con-
tent can be presented as follows. The classification and invariants problems
.
h forsecondorderLPDEswithrespecttotransformations
t x=x(ξ,η), y=y(ξ,η), u=h(x,y)v(ξ,η),
a
m where x,y are independent and u is the dependent variable of the LPDE,
are considered. Solutions to these problems are given for different subclasses
[
of non-parabolic LPDE which appear naturally in the equivalence problem

1 of LPDE. A criterion for reducibility of such LPDE to LPDE with constant
v coefficients isofferedaswell.
4
1
5
1
1. Introduction
0
1. Let
0
A(x,y)u +B(x,y)u +C(x,y)u +a(x,y)u +b(x,y)u +c(x,y)u=0
7 xx xy yy x y
1 be a second order linear partial differential equation (LPDE) over the ring
v: C∞(R2, ∂ , ∂ ). Under the transformation of dependent variable v = h(x,y)−1u
∂x ∂y
i and independent variables ξ = ξ(x,y),η = η(x,y) it transforms to an analogous
X
(equivalent) one
r
a
A (ξ,η)v +B (ξ,η)v +C (ξ,η)v +a (ξ,η)v +b (ξ,η)v +c (ξ,η)v =0,
1 ξξ 1 ξη 1 ηη 1 ξ 1 η 1
and equality δ = g−1∂- the chain rule is valid, where ∂ is column vector with
”coordinates” ∂1 =∂/∂x, ∂2 =∂/∂y, δ is column vector with ”coordinates” δ1 =
∂/∂ξ, δ2 =∂/∂η, g is matrix g =(gi) with g1 =ξ ,g1 =η ,g2 =ξ ,g2 =η
j i,j=1,2 1 x 2 x 1 y 2 y
and u(x,y)=h(x,y)v(ξ(x,y),η(x,y)).
A function f∂(x) of x ,x ,...,x and a finite number of their partial derivatives
1 2 6
is said to be invariant (relative invariant) if
fδ(V )=f∂(V) (respectively, fδ(V )=h(x,y)k∆(x,y)lf∂(V))
1 1
1991 Mathematics Subject Classification. Primary35A99,Secondary35J15, 35L10,12H20.
Key words and phrases. linear partial differential equation; change of variables; equivalence;
invariant;fiberbundledspace.
ThisresearchissupportedbytheMOHEMalaysiaundergrantFRGS14-153-0394.
1
2 U.BEKBAEV
for any V = (A(x,y),B(x,y),C(x,y),a(x,y),b(x,y),c(x,y)) from the domain of
f∂(x), non-vanishing h(x,y) and change of independent variables ξ = ξ(x,y),η =
η(x,y), where
V =(A (ξ,η),B (ξ,η),C (ξ,η),a (ξ,η),b (ξ,η),c (ξ,η)),
1 1 1 1 1 1 1
k,l are fixed integers and ∆(x,y) = ξ η −ξ η . Such invariants are important
x y y x
in the classification theory of differential equations. For example usage of a single
relative invariant DhVi = B2 −4AC, namely the discriminant, gives us a rough
classification of the second order real LPDE (elliptic, hyperbolic and parabolic
differential equations). In general every relative invariant f∂(x) separates its own
domain into two parts:
{V :f∂(V)6=0} and {V :f∂(V)=0}
and never an element of the first set and an element of the second one can be
equivalent to the same LPDE.
Wewillconsideranabstractalgebraicanalogoftheabovesituation. Forthatwe
use some notations, terminology and results from the theory of differential algebra
which one can find in [9].
Let(F,∂1,∂2)be anydifferentialfieldwiththecommutingdifferentialoperators
∂1,∂2. Consider
GL∂(2,F)={g =(gi) ∈GL(m,F):∂kgi =∂igk at i,j,k =1,2}
j i,j=1,2 j j
and any second order linear partial differential equation
(1.1) A(∂1)2u+B∂1∂2u+C(∂2)2u+a∂1u+b∂2u+cu=0,
where A,B,C,a,b,c are given elements of F and u is an unknown element of F.
By transformations v = h−1u, δ = g−1∂, where g ∈ GL∂(2,F), h ∈ F∗ =
F \{0}, ∂ is column vector with ”coordinates”∂1,∂2 and δ is column vector with
”coordinates” δ1,δ2, it is transformed to an analogous one
(1.2) A (δ1)2v+B δ1δ2v+C (δ2)2v+a δ1v+b δ2v+c v =0.
1 1 1 1 1 1
Moreoverthe coefficients of these equations are related in the following way.
A =h(A(g1)2+Bg1g2+C(g2)2),
1 1 1 1 1
B =h(2Ag1g1+B(g1g2+g1g2)+2Cg2g2),
 1 1 2 1 2 2 1 1 2
C =h(A(g1)2+Bg1g2+C(g2)2),
(1.3)  bBa11(=∂=1hhh(g(AA12∂+∂11g∂g21121h++g11BB)∂∂+112gg2212C2++∂2CCh∂∂g2122g,g2212++aagg111++bbgg212))++22AA∂∂11hhgg111++
1 2 2 2 2 2 2
B(∂1hg2+∂2hg1)+2C∂2hg2,
Definitionc11=.1A. (E2∂q1u)2ahti+onB2s∂(11∂.12)ha+ndC((2∂1.22))2har+eas∂a1idh+tobb∂e2hG+=ch.F∗ ×GL∂(2,F)
-equivalent if there exist h∈F∗, g ∈GL∂(2,F) such that δ =g−1∂ and system of
equalities (1.3) is valid. In such case we write V =τ∂hh,g;Vi, where
1
V =(A ,B ,C ,a ,b ,c ), V =(A,B,C,a,b,c).
1 1 1 1 1 1 1
For the convenience even in the abstract differential field case we use
u (u , u , u ) to denote ∂1u (respectively, ∂2u, δ1u, δ2u).
x y ξ η
Let C = {a ∈ F : ∂1a = ∂2a = 0} be the constant field for (F,∂1,∂2) and x
standfor the variablevector (x ,x ,...,x ). The set ofalllinear partialdifferential
1 2 6
ON CLASSIFICATION AND INVARIANTS 3
equations (1.1) over F of order ≤ 2 is nothing than F6. Whenever we speak
about topology we mean the ∂- differential Zariski topology [9]. If W ⊂ F6 is
e
an irreducible closed subset we use ChW ;∂i to denote the field of ∂-differential
e
rationalfunctionsonW withconstantcoefficients. WeuseC{x;∂}todenotethe∂-
e
differentialringof∂-differentialpolynomialsinindependentvariables(x ,x ,...,x )
1 2 6
over C and Chx;∂i for the field of ∂-differential rational functions in x over C.
Definition 1.2. A differential rational function f∂hxi ∈ ChW ;∂i is called to be
e
a G - invariant (relative invariant) if the equality
fg−1∂hτ∂hh,g;xii=f∂hxi (respectively, fg−1∂hτ∂hh,g;xii=hk∆lf∂hxi)
is valid for any (h,g) ∈ G whenever f∂ is defined at x ∈ W , where k,l are fixed
e
integers and ∆=detg.
Letus denotethe fieldofF∗×GL∂(2,F)-invariant∂-differentialrationalfunc-
tions over W by ChW ;∂iG.
e e
The main aim of the present paper is the classification of second order non-
parabolic LPDE with respect to the above considered action of F∗ ×GL∂(2,F)
and description of the corresponding field of F∗×GL∂(2,F)-invariant differential
rational functions over C. In this paper it is done for some G -invariantsubsets of
second order non-parabolic LPDE which appear naturally in their G-equivalence
problem. A criterion for reducibility of such equations to LPDE with constant
coefficients is offered as well.
Forthe secondordergenericLPDEthe G-equivalenceandinvariantsareconsid-
eredin[1]withoutclassificationandreducibilitytoLPDEwithconstantcoefficients
problems. In particular in this paper all results of [1], except for Theorem 3.1, are
reobtained by the use of more general method than of [1] (see case a) considered
in Section 3). For the equivalence and invariants problems of such equations with
respect only to the change of independent variables one can see [2, 3].
We would like to note that in spite of the importance of second order LPDE
their equivalence and invariants problems have not been explored much. As to the
similarproblemsfortheordinaryLDEtheyareconsideredbymanyauthors,seefor
example [7, 10, 11, 12, 13, 15]. For an algebraic approach, by means of differential
algebra, to the equivalence and invariance problems of ordinary LDE one can see
[4, 5, 6].
The construction of the paper is as follows. Section 2 is about the key results
which are presented in an appropriate form to investigate the classification and
invariantsproblems for anyorderLPDE.Section3 is aboutapplicationsofSection
2 results to different subsets of non-parabolic second order LPDE.
2. Preliminaries
In this section we use l,m,n for any fixed natural numbers and prove some
commonresultstousetheminthenextsectiontosolveclassificationandinvariants
problems for the second order LPDE.
Let (F,∂) stand for a field F with fixed commuting system of differential oper-
ators ∂1,∂2,...,∂m and C be its constant field i.e.
C ={a∈F :∂1a=∂2a=...=∂ma=0},
4 U.BEKBAEV
where ∂ stands for the column with ”coordinates” ∂1,∂2,...,∂m. Elements of Fm
are assumed to be written as row vectors and
GL∂(m,F)={g ∈GL(m,F):∂igj =∂jgi for i,j,k=1,2,...,m}.
k k
Proposition 2.1. If the system of differential operators ∂1,...,∂m is linear in-
dependent over F then the differential operators δ1,...,δm, where δ = g−1∂,g ∈
GL(m,F), commute with each other if and only if g ∈GL∂(m,F).
Proof. Let g ∈ GL(m,F), δ = g−1∂. It is clear that linear independence of
∂1,...,∂m implies linear independence of δ1,...,δm. For any i,j = 1,2,...,m we
have ∂j = m gjδk,
k=1 k
m m m m
∂iP∂j = (∂i(gj)δk+gj∂iδk)= ∂i(gj)δk+ gjgiδsδk.
k k k k s
k=1 k=1 k=1s=1
X X XX
Therefore due to ∂i∂j =∂j∂i one has
m ∂i(gj)δk+ m m gjgiδsδk =
(2.1) k=1 k k=1 s=1 k s
m ∂j(gi)δk+ m m gjgiδkδs
P k=1 k P k=1P s=1 k s
If δkδs =δsδk forPany k,s=1,2,..P.,m thPen due (2.1) one has
m m
∂i(gj)δk = ∂j(gi)δk
k k
k=1 k=1
X X
i.e. ∂i(gj) = ∂j(gi) for any i,j,k = 1,2,...,m because of linear independence of
k k
δ1,...,δm. Thus in this case g ∈GL∂(m,F).
Vice versa, if ∂i(gj) = ∂j(gi) for any i,j,k = 1,2,...,m then due (2.1) at any
k k
a∈F one has
m m m m
gjgiδsδka= gjgiδkδsa for any i,j =1,2,...,m.
k s k s
k=1s=1 k=1s=1
XX XX
These equalities can be written in the following matrix form
g(δ1δa,...,δmδa)gt =g(δ1δa,...,δmδa)tgt,
where t means the transpose. Therefore (δ1δa,...,δmδa) = (δ1δa,...,δmδa)t i.e.
δkδsa = δsδka for any k,s = 1,2,...,m, which completes the proof of Proposition
2.1. (cid:3)
Furtheritisassumedthatthecommutingsystemofdifferentialoperators∂1,...,∂m
is linear independent over F. Note that for any g ∈GL∂(m,F) and δ = g−1∂ the
constant field of (F;δ) is the same C and equality
(2.2) GLδ(m,F)=g−1GL∂(m,F)
holds true.
Furthereachelement(h,g)∈GL(n,F)×GL∂(m,F)isassumedtobepresented
as a block-diagonalmatrix with the blocks ”h” and ”g” on the main diagonal.
We consider the following groupoid G (generated by GL∂(m,F) and GL(n,F))
as a fiber bundled space with base
B={δ :there exists g ∈GL∂(m,F) such that δ =g−1∂},
andfiberGL(n,F)×GLδ(m,F)overδ ∈Bandmapss:G→B,t:G→B,where
by definition s(δ,(h,g)) = δ and t(δ,(h,g)) = g−1δ. Whenever s(δ ,(h ,g )) =
1 1 1
ON CLASSIFICATION AND INVARIANTS 5
t(δ,(h,g)) the product (δ,(h,g))(δ ,(h ,g ))=((g )−1δ,(hh ,gg )) is defined. All
1 1 1 1 1 1
needed axioms of groupoid [8, 14] are checked easily.
WeconsideralsofiberbundledspaceDEwiththesamebaseBandfiberFl =Fl
δ
over δ ∈ B, where F means the field F considered as δ-differential field. The
δ
algebra of polynomials C{x;∂}, where x = (x ,x ,...,x ), can be extended to the
1 2 l
C{DE}-the algebra of differential polynomials on DE in the following natural
way: f∂{x} ∈ C{x;∂} generates polynomial map (fδ{x}) ∈ C{DE}, where
δ∈B
fδ{x} stands for the polynomial obtained by substitution δ for ∂ into f∂{x}. In
other words each element of C{DE} when considered on (δ,Fl) is a δ-differential
δ
polynomial and its coefficients do not depend on δ. By ChDEi we denote the
correspondingfieldofrationalfunctions. LetususeI todenotethen-orderidentity
n
matrix and we use e for I also. We consider W = (δ,W )⊂DE a sub-fiber
m δ∈B δ
bundled space and C{W}, ChWi in the natural way.
S
Let G⋄W stand for the fiber bundled space with the same base B and fiber
GL(n,F)×GLδ(m,F)×W over δ.
δ
Byalgebraic(”anti”)representationofGonW wemeanamapT :G⋄W →W,
where T :(δ,(h,g,v))7→(g−1δ,τδhh,g;vi), such that
(2.3) τg−1δhh ,g ;τδhh,g;vii=τδhhh ,gg ;vi
1 1 1 1
whenever g ∈ GLδ(m,F), v ∈ W , h,h ∈ GL(n,F) and g ∈GLg−1δ(m,F). It is
δ 1 1
assumed that τδhI ,I ;vi=v for any (δ,v).
n m
We say that
W0 = (δ,W0δ)⊂W is invariant (G−invariant) if τδhh,g;vi∈W0g−1δ
δ∈B
[
whenever (h,g)∈GL(n,F)×GLδ(m,F) and v∈W .
0δ
Further it is assumed that each component of (τδ) is from ChWi that is a
δ∈B
differential rational function in components of h,g,v over C .
Assumption. There exists a nonempty G-invariant subset W of W and a map
0
P =(Pδ) :W →G, where P :(δ,v)7→(δ,Pδhvi), such that
δ∈B 0
(2.4) Pg−1δhτδhh,g;vii=(h−1,g−1)Pδhvi
whenever δ ∈ B, v ∈ W and (h,g) ∈ GL(n,F)×GLδ(m,F). It is also assumed
δ
that all components of (Pδhvi) are differential rational functions in v over C.
δ∈B
Interms ofcomponentsofPδhvi=(Pδhvi,Pδhvi) equality(2.4)canbe written
0 1
in the form:
Pg−1δhτδhh,g;vii=h−1Pδhvi, Pg−1δhτδhh,g;vii=g−1Pδhvi.
0 0 1 1
Note that G-stabilizer of any (δ,v)∈W is trivial. The next result is a classifi-
0
cation theorem for the elements of W with respect to G.
0
Theorem 2.2. Pairs (∂,u),(δ,v) ∈ W are G-equivalent, that is v = τ∂hh,g;ui
0
and δ =g−1∂ for some (h,g)∈GL(n,F)×GL∂(m,F), if and only if
P∂hui−1∂ =Pδhvi−1δ and τ∂hP∂hui;ui=τδhPδ(vi;vi.
1 1
Proof. If v=τ∂hh,g;ui and δ =g−1∂ then
Pδhvi−1δ =(Pg−1∂hτ∂hh,g;uii)−1g−1∂ =(g−1P∂hui)−1g−1∂ =P∂hui−1∂
1 1 1 1
6 U.BEKBAEV
and
τδhPδhvi;vi=τδhPg−1∂hτ∂hh,g;uii;vi=τδh(h−1,g−1)P∂hui;vi=
τg−1∂h(h−1,g−1)P∂hui;τ∂hh,g;uii=τ∂hP∂hui;ui
Visa versa, if P∂hui−1∂ = Pδhvi−1δ and τ∂hP∂hui;ui = τδhPδhvi;vi then for
1 1
h=P∂huiPδhvi−1,g =P∂huiPδhvi−1 due to (2.2), (2.3) and the condition of the
0 0 1 1
theorem one has
τ∂hh,g;ui=τ∂hP∂huiPδhvi−1,P∂huiPδhvi−1;ui=
0 0 1 1
τP1∂hui−1∂hPδhvi−1,Pδhvi−1;τ∂hP∂hui,P∂hui;uii=
0 1 0 1
τP1δhvi−1δhP0δhvi−1,P1δhvi−1;τδhP0δhvi,P1δhvi;vii =τδhIn,Im;vi=v.
(cid:3)
This theorem means that for the canonical representative of G-equivalent to
(∂,u) elements of W one can take
0
(P∂hui−1∂,τ∂hP∂hui;ui).
1
The next result is a criterion for reducibility(equivalence) to an element with
constant components.
Theorem 2.3. An element (∂,u)∈W is G-equivalent to an element (δ,v)∈W ,
0 0
whereallcomponentsofvareconstants,thatisv∈Cl,ifandonlyifallcomponents
of
τ∂hP∂hui;ui
are constants.
Proof. The ”If” part. If v = τ∂hh,g;ui ∈ Cl for some (h,g) ∈ GL(n,F) ×
GL∂(m,F) then all entries of matrix Pg−1∂hτ∂hh,g;uii = (h−1,g−1)P∂hui are
in C as well. That is at any i=1,2,...,m one has
0=∂i((h−1,g−1)P∂hui)=−(h−1∂ihh−1,g−1∂igg−1)P∂hui+(h−1,g−1)∂iP∂hui
which implies equality
(∂ihh−1,∂igg−1)=∂iP∂huiP∂hui−1.
Thereforeonehas(h,g)=P∂hui(h ,g )forsomeh ∈GL(n,C)andg ∈GL(m,C)⊂
0 0 0 0
GL∂(m,F) as far as the solution of the system
∂iXX−1 =∂iP∂huiP∂hui−1, i=1,2,...,m
is unique up to such product. The equality
v=τ∂hh,g;ui=τg0−1∂hh0,g0;τ∂hP∂hui;uii
indicates that τ∂hP∂hui;uii=τ∂hh−1,g−1;vi∈Cl as well.
0 0
The ”only if” part is evident. (cid:3)
Further in this section (F,∂) is considered to be a differential-algebraic closed
field of characteristic zero, W is assumed to be a closed, irreducible (in the differ-
δ
ential Zariski topology) subset of Fl and W is dense in W .
δ 0δ δ
The following set ChWiG ={f =(fδhxi) ∈ChWi:
δ
fg−1δhτδhh,g;vii=fδhviwhenever(h,g,v)∈GL(n,F)×GLδ(m,F)×W ,δ ∈B}
δ
ON CLASSIFICATION AND INVARIANTS 7
is said to be the field of G-invariant rational functions on W. From the very
definition it is clear that any f = (fδhxi) ∈ ChWiG is fully defined by its fiber
δ
f∂hxi. Thereforeforthissetonecanusemoreinformativeandtransparentnotation
ChW ;∂iGL(n,F)×GLδ(m,F) orChW ;∂iG-thefieldofG-invariantrationalfunctions
e e
on W .
e
Theorem 2.4. The field of G-invariant rational functions ChW ;∂iG is invariant
e
with respect to the commuting system of differential operators δ1,δ2,...,δm, where
δ = P∂hxi−1∂, and as such differential field it is generated over C by the system
1
of components of τ∂hP∂hxi;xi that is
ChW ;∂iG =Chτ∂hP∂hxi;xi;δi.
e
Proof. It is evident that all components of τ∂hP∂hxi;xi are in ChW ;∂iG. If
e
f∂hxi = fg−1∂hτ∂hh,g;xii for all (h,g) ∈ GL(n,F)×GL∂(m,F) then, in par-
ticular,
f∂hxi=fP1∂hui−1∂hτ∂hP∂hui,P∂hui;xii
0 1
whenever u ∈ W . It implies, as far as W is dense in W , that for the variable
0e 0e e
vector y∈W the equality
0
f∂hxi=fP1∂hyi−1∂hτ∂hP∂hyi,P∂hyi;xii
0 1
holds true. In y = x case one gets that f∂hxi = fP1∂hxi−1∂hτ∂hP∂hxi,P∂hxi;xii.
0 1
(cid:3)
Corollary2.5. ThefieldChW ;∂iisgeneratedoverChW ;∂iG asaδ =P∂hxi−1∂-
e e 1
differential field by the system of components of P∂hxi that is
ChW ;∂i=ChW ;∂iGhP∂hxi;δi.
e e
Proof. Due to Theorem 2.4 one has equality
ChW ;∂iGhP∂hxi;P∂hxi−1∂i=Chτ∂hP∂hxi;xi,P∂hxi;P∂hxi−1∂i.
e 1 1
Thereforetoprovethecorollaryitisenoughtoshowthat∂,aswellasx,canbeex-
pressedbyδ =P∂hxi−1∂andthesystemofcomponentsofP∂hxiandτ∂hP∂hxi;xi.
1
Indeed ∂ =P∂hxiδ and
1
τP1∂hxi−1∂hP∂hxi−1;τ∂hP∂hxi;xii=x.
(cid:3)
Theorem 2.6. The equality δ−tr.deg.ChW ;∂iG/C =dim(W )−(n2+m) holds
e e
true.
Proof. We will show that the system
P∂hxi,P∂ hxi,P∂ hxi,...,P∂ hxi
0 1,1 1,2 1,m
is δ-algebraic independent over ChW ;∂iG, where P∂hxi stands for all its compo-
e 0
nents and P∂ hxi,P∂ hxi,...,P∂ hxi stands for the first row of the matrix P∂hxi.
1,1 1,2 1,m 1
Note that the system can be presented in the form e P∂hxi, where e stands for
1 1
(I ,1,0,0,...,0).
n
If pδ[Z ,z ,...,z ] is any δ-differential polynomial over ChW ;∂iG for which
0 1 m e
pδ[e P∂hxi]=0thenitshouldremainbetruewithrespecttothechanges∂ 7→g−1∂
1
8 U.BEKBAEV
and x7→τ∂hh,g;xi, where h∈GL(n,F) and g ∈GL∂(m,F). As far as δ and the
coefficients of pδ[Z ,z ,...,z ] are invariant with respect to such changes one has
0 1 m
0=pδ[e P∂hxi]=pδ[e Pg−1∂hτ∂hh,g;xii]=pδ[e (h−1,g−1)P∂hxi]
1 1 1
thatis pδvv[e1(h−1,g−1)P∂hvi]=0 for any v ∈W0e,where pδvv[Z0,z1,...,zm] stands
for the polynomial obtained from pδ[Z ,z ,...,z ] by substitution v for x. But for
0 1 m
anyh0 ∈GL(n,F)andg ∈GLδv(m,F)the systemh0 =h−1P0∂hvi, g =g−1P1∂hvi
has solution h = P∂hvih−1 ∈ GL(n,F), g = P∂hvig−1 ∈ GL∂(m,F). Therefore
0 0 1
pδvv[e1(h0,g)] = 0 for any h0 ∈ GL(n,F) and g ∈ GLδv(m,F). In particular it
implies that for any g =δvu∈GLδv(m,F) the equality
(2.5) pδv[h ,δ1u ,δ1u ,...,δ1u ]=0
v 0 v 1 v 2 v m
holdstrue,whereg =δ u=δ (u ,u ,...,u )standsforthesquarematrix(δiu )
v v 1 2 m v j i,j=1,2,...,m
. Notethatifpδvv[Z0,z1,...,zm]isnonzerothenthesameistrueforpδvv[Z0,δv1z1,...,δv1zm]
. Equality (2.5) shows that the δ -differential polynomial
v
detZ det(δ (z ,z ,...,z ))pδv[Z ,δ1z ,δ1z ,...,δ1z ]
0 v 1 2 m v 0 v 1 v 2 v m
vanishes at any (Z ,z ,z ,...,z ) ∈ Fn2+m. As far as detZ det(δ (z ,z ,...,z ))
0 1 2 m 0 v 1 2 m
isanonzeropolynomialonecanconcludethatpδvv[Z0,δv1z1,δv1z2,...,δv1zm]hastobe
zero polynomial at any v ∈ W . Therefore pδ[Z ,δ1z ,δ1z ,...,δ1z ] = 0 that is
0e 0 1 2 m
pδ[Z ,z ,...,z ] itself is zero polynomial. Due to the equality
0 1 m
ChW ;∂iGhP∂hxi;δi=ChW ;∂iGhP∂hxi−1;δi and P∂hxi−1 ∈GLδhm,Chx;∂ii
e e 1
the δ-algebraic transcendence degree of the system of components of P∂hxi over
ChW ;∂iG equals exactly to n2+m. Now taking into account the equalities
e
dimW =∂−tr.deg.ChW ;∂i/C =δ−tr.deg.ChW ;∂i/C
e e e
one can conclude that the equality δ−tr.deg.ChW ;∂iG/C =dim(W )−(n2+m)
e e
is true. (cid:3)
3. Classification and invariants of the second order non-parabolic
LPDE
Inthissectionweconsiderapplicationsoftheprevioussectionresultstothecor-
respondingproblemsforthesecondorderLPDEintwovariables,whichcorresponds
to n=1,m=2,l=6 case of that section. We get the needed applications via the
construction the corresponding map P∂ satisfying conditions of the Assumption.
In general neither W nor P∂ is unique.Therefore our construction of P∂ should
0e
be considered as one of the possible constructions. We refrain from reformulation
allresultsofSection2foreachparticularcaseasfarasthoughthe reformulationis
not difficult but space consuming process. Further we use the following functions
ON CLASSIFICATION AND INVARIANTS 9
a∂hVi=AC −A C+BC −B C−bB+2aC,
x x y y
b∂hVi=A B−AB +A C−AC −aB+2bA,
x x y y
c∂hVi=((a∂hVi) −(b∂hVi) )2DhVi,
DhVi y DhVi x
c∂hVi=−(c∂hVi)x,
1 2c∂hVi
c∂hVi=−(c∂hVi)y,
2 2c∂hVi
γ∂hVi=A((c∂hVi) +c∂hVi2)+B((c∂hVi) +c∂hVic∂hVi)+
1 x 1 1 y 1 2
C((c∂hVi) +c∂hVi2)+ac∂hVi+bc∂hVi+c.
2 y 2 1 2
γ∂hVi=A((a∂hVi) + a∂hVi2)+B((a∂hVi) + a∂hVib∂hVi)+
0 DhVi x DhVi2 DhVi y DhVi DhVi
C((b∂hVi) + b∂hVi2)+aa∂hVi +bb∂hVi +c, where DhVi=B2−4AB,
DhVi y DhVi2 DhVi DhVi
to investigate classificationand invariants problems for the secondorder LPDE.
These functions are introduced in [1].
Thefollowingtheoremfrom[1]reducessystemofequalities(1.3)toamoresimple
equivalence under the condition DhVi6=0.
Theorem 3.1. If DhVi=6 0 and c∂hVi=6 0 (c∂hVi= 0) then system of equalities
(1.3) is equivalent to the following system of equalities.
A B /2 A B/2
1 1 =hgt g,
B /2 C B/2 C
1 1
(3.1)  (cid:18) abDδδhhhVVV111iii !=g−(cid:19)1 baD∂∂hhhVV(cid:18)Viii−−hh−−11∂∂21hh !(cid:19),
DhV1i DhVi
γδhV i=hγ∂hVi (respectively, γδhV i=hγ∂hVi.)
, where gt stands fo1r the transpose of g. 0 1 0
One of the interesting questions in theory of LPDE is the equivalence problem
of LPDE to LPDE with constant coefficients. For a solution of this question for
any order ordinary LDE one can see [4, 5, 6]. Here we would like to show that the
above result can be used to treat this problem even without construction of the
correspondingmapP∂hxi(seeTheorem2.3). Notethatfornon-parabolicequation
(1.1) the condition V ∈C6 is equivalent to
a∂hVi b∂hVi
(A,B,C, , ,γ∂hVi)∈C6,
DhVi DhVi 0
whichcanbeseendirectlyfromthedefinitionofthefunctionsa∂hVi,b∂hVi,γ∂hVi.
0
It implies that for (1.1) to be G-equivalent to LPDE with constant coefficients at
least c∂hVi should be zero that is the equality
a∂hVi b∂hVi
∂2 =∂1
DhVi DhVi
is necessary.
Corollary 3.2. Non-parabolic equation (1.1), for which ∂2a∂hVi = ∂1b∂hVi and
DhVi DhVi
γ∂hVi=6 0, is G-equivalent to a LPDE with constant coefficients if and only if the
0
system
(∂igg−1)tM∂hVi+M∂hVi∂igg−1+∂iM∂hVi=0,i=1,2
∂igg−1m∂hVi=∂im∂hVi,i=1,2
10 U.BEKBAEV
A/γ∂hVi B/(2γ∂hVi)
has a solution in GL∂(2,F), where M∂hVi= 0 0 ,
B/(2γ∂hVi) C/γ∂hVi
(cid:18) 0 0 (cid:19)
a∂hVi +γ∂hVi−1∂1γ∂hVi
m∂hVi= DhVi 0 0 .
b∂hVi +γ∂hVi−1∂2γ∂hVi !
DhVi 0 0
Proof. The relation V =τ∂hh,g;Vi∈C6 is equivalent to
1
aδhV i bδhV i
(A ,B ,C , 1 , 1 ,γδhV i)∈C6.
1 1 1 DhV i DhV i 0 1
1 1
Therefore due to Theorem 3.1 it is equivalent to h−1∂h=−γ∂hVi−1∂γ∂hVi and
0 0
∂i(gtM∂hVig)=0,i=1,2
∂i(g−1m∂hVi)=0,i=1,2
(cid:26)
which can easily be presented in the following form
(∂igg−1)tM∂hVi+M∂hVi∂igg−1+∂iM∂hVi=0,i=1,2
.
∂igg−1m∂hVi=∂im∂hVi,i=1,2.
(cid:26)
(cid:3)
It should be noted that matrices ∂1gg−1,∂2gg−1 are defined uniquely from the
linear system
(3.2) (∂igg−1)tM∂hVi+M∂hVi∂igg−1+∂iM∂hVi=0,i=1,2
ξ η
due to DhVi 6= 0. In terms of the ordinary notation x x for g it means
ξ η
y y
(cid:18) (cid:19)
that expressions
ξ η ξ η ξ η
1| xx xx |, 1| x x |, 1| xy xy |,
∆ ξ η ∆ ξ η ∆ ξ η
(3.3) y y xx xx y y
ξ η ξ η ξ η
1| x x |, 1| yy yy |, 1| x x |,
∆ ξ η ∆ ξ η ∆ ξ η
xy xy y y yy yy
which representallentries of the matrices ∂1gg−1,∂2gg−1, are defined uniquely by
the entries of M∂hVi.
Therefore system (3.2) can be written in the form
∂1gg−1 =N∂hVi
(3.4) { 1 .
∂2gg−1 =N∂hVi
2
In particular it implies that equalities ∂igg−1m∂hVi=∂im∂hVi,i=1,2 are noth-
ing than some relations between entries of M∂hVi and m∂hVi needed for the re-
ducibility of (1.1) to LPDE with constant coefficients. So the problem is reduced
to the solvability of (3.4) in GL∂(2,F).
Remark 3.3. One can ask about the admissible values of six expressions given
in (3.3) . That is if one attaches some values to these six expressions how to
knowwhethertheobtainedsystemoffirstordernonlineardifferentialequationshas
solution in GL∂(2,F) or not? In other words one can ask about the integrability
conditions of the obtained system of six equations. It can be answered in the
following way. Construct matrices N∂hVi,N∂hVi using the attached values and
1 2
equalities (3.4). The integrability condition for the system is given by the next
Proposition.