New Examples of Irreducible Local Diffusion of Hyperbolic PDE's
Local diffusion of strictly hyperbolic higher-order PDE's with constant coefficients at all simple singularities of corresponding wavefronts can be explained and recognized by only two local geometrical features of these wavefronts. We radically disprove the obvious conjecture extending this fa...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України
2020
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| Цитувати: | New Examples of Irreducible Local Diffusion of Hyperbolic PDE's. Victor A. Vassiliev. SIGMA 16 (2020), 009, 21 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859568570615201792 |
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| author | Vassiliev, Victor A. |
| author_facet | Vassiliev, Victor A. |
| citation_txt | New Examples of Irreducible Local Diffusion of Hyperbolic PDE's. Victor A. Vassiliev. SIGMA 16 (2020), 009, 21 pages |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Local diffusion of strictly hyperbolic higher-order PDE's with constant coefficients at all simple singularities of corresponding wavefronts can be explained and recognized by only two local geometrical features of these wavefronts. We radically disprove the obvious conjecture extending this fact to arbitrary singularities: namely, we present examples of diffusion at all non-simple singularity classes of generic wavefronts in odd-dimensional spaces, which are not reducible to diffusion at simple singular points.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 009, 21 pages
New Examples of Irreducible Local Diffusion
of Hyperbolic PDE’s
Victor A. VASSILIEV †‡
† Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
E-mail: vva@mi-ras.ru
URL: http://www.mi-ras.ru/~vva/
‡ National Research University Higher School of Economics, Moscow, Russia
Received September 29, 2019, in final form February 18, 2020; Published online February 24, 2020
https://doi.org/10.3842/SIGMA.2020.009
Abstract. Local diffusion of strictly hyperbolic higher-order PDE’s with constant coeffi-
cients at all simple singularities of corresponding wavefronts can be explained and recognized
by only two local geometrical features of these wavefronts. We radically disprove the obvi-
ous conjecture extending this fact to arbitrary singularities: namely, we present examples
of diffusion at all non-simple singularity classes of generic wavefronts in odd-dimensional
spaces, which are not reducible to diffusion at simple singular points.
Key words: wavefront; discriminant; critical point; morsification; vanishing cycle; hyperbolic
PDE; fundamental solution; lacuna; sharp front; diffusion; Petrovskii condition
2020 Mathematics Subject Classification: 35L67; 58K05; 32-04
1 Introduction
This is a work in lacuna theory of hyperbolic PDE’s initiated by I.G. Petrovskii [17] and expanded
significantly by J. Leray [13], M. Atiyah, R. Bott and L. G̊arding [5, 6, 11], and others. However,
the methods of the work belong mainly to singularity theory of differentiable functions.
There are two known sources of local diffusion (i.e., the local irregularity of continuations)
of solutions of strictly hyperbolic partial differential equations with constant coefficients: non-
singular points of wavefronts, at which the Davydova signature condition fails, and cuspidal
edges at which the investigated local connected component of the complement of a wavefront is
the “bigger” one. All cases of local diffusion of waves at simple (i.e., of classes Ak, Dk, E6, E7
or E8) singular points of wavefronts of generic hyperbolic PDE’s can be reduced to these two:
the diffusion arises in such a component in a neighbourhood of a simple singularity if and only
if the boundary of this component contains points of one of these two basic types. In particular,
this is true for all generic hyperbolic PDE’s in the ≤ 7-dimensional spaces.
We show that for non-simple singular points of wavefronts in RN , N odd, this is not more the
case. Namely, we indicate local components of complements of generic wavefronts at their points
of all “parabolic” singularity classes P8, X9, and J10, such that solutions of the corresponding
PDE’s have diffusion at the most singular points of their boundaries only (i.e., at the points of
these parabolic types), but are sharp at all simpler points. The singularities of these three classes
occur close to all other non-simple singularities of generic wavefronts, therefore such additional
examples of diffusion also occur at all of them.
The proofs use a program counting topological types of morsifications of critical points of
real analytic functions.
This paper is a contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor
of Dmitry Fuchs. The full collection is available at https://www.emis.de/journals/SIGMA/Fuchs.html
mailto:vva@mi-ras.ru
http://www.mi-ras.ru/~vva/
https://doi.org/10.3842/SIGMA.2020.009
https://www.emis.de/journals/SIGMA/Fuchs.html
2 V.A. Vassiliev
1.1 Hyperbolic PDE’s of higher orders (after [5, 6], see also [10, 20])
Let F be a linear partial differential operator with constant coefficients: F has the form of
a polynomial (generally with complex coefficients) in variables ∂/∂xj where xj are the coor-
dinates in RN . It is convenient to identify these variables ∂/∂xj with coordinates ξj in dual
space ŘN of our space RN , with pairing operation
〈(x1, . . . , xN ), (ξ1, . . . , ξN )〉 =
N∑
j=1
xjξj .
The projectivization of this dual space is denoted by ŘPN−1; its k-dimensional subspaces cor-
respond via the projective duality to (N − k − 2)-dimensional subspaces in RPN−1.
Let d be the degree of this polynomial F . The following Cauchy problems in the half-space
RN+ = {x |x1 > 0} for this operator are considered. Given any regular function ϕ in this half-
space, and a collection of d regular functions ψk, k = 0, 1, . . . , d − 1, defined on its boundary
hyperplane RN−11 ≡ {x1 = 0} of this half-space, the task is to find a function u in RN+ , such that
F (u) ≡ ϕ in RN+ , and ∂k
∂xk1
u ≡ ψk on hyperplane RN−11 for any k = 0, . . . , d− 1.
A fundamental solution of our operator F is any distribution in RN such that this operator
applied to it is equal to the delta function at 0. Operator F is called hyperbolic (or sometimes
hyperbolic in the sense of Petrovskii) if it admits a fundamental solution whose support belongs
to a proper cone S(F ) in the half-space RN+ with the vertex 0 (i.e., this vertex is the unique
intersection point of this cone with RN−11 ). Such a fundamental solution (if it exists) is uniquely
determined by F ; it is called principal fundamental solution of F and is denoted by E(F ). It
may be considered as a (generalized) wave caused by an instant pointwise perturbation and
propagating in the future (x1 > 0) part of our space-time only.
If F is hyperbolic, then any Cauchy problem of the described form has a unique solution,
which is regular and depends continuously on the data {ϕ;ψ0, . . . , ψd−1} of the problem; more-
over, the value of this solution at any point x ∈ RN+ depends only on the values of these data in
the capsized cone x − S(F ) with the vertex at point x. Indeed, such a solution is provided by
the convolution of E(F ) with the data of our problem.
The hyperbolicity property implies strong conditions on the principal symbol of F (i.e., the
homogeneous part Fd of highest degree of polynomial F ). First of all, if the operator F is
hyperbolic then this principal part is necessarily a real (up to a constant factor) polynomial;
therefore studying solutions of the corresponding equation we can and will assume that it is just
real. Further, the cone A(F ) ⊂ ŘN of zeros of this principal part Fd should have exactly d real
intersection points (maybe counted with multiplicities) with any line in ŘN parallel to the ξ1
axis (i.e., to the line orthogonal to the hyperplane RN−11 ).
If this cone A(F ) is a non-singular manifold outside the origin in ŘN , then this condition is
also sufficient for hyperbolicity of entire operator F . Otherwise some additional conditions on
the lower terms of F should be satisfied.
Hyperbolic operators with such smooth cones A(F ) are called strictly hyperbolic. They form
a dense subset in the space of all hyperbolic operators. We consider in this work only strictly
hyperbolic operators.
The principal fundamental solution of any hyperbolic operator is regular (i.e., locally coincides
with appropriate non-singular analytic functions) everywhere outside the wavefront W (F ), which
is a conic (i.e., invariant under positive dilations) closed semi-algebraic subvariety of positive
codimension in RN+ ; moreover the support of this principal fundamental solution lies in the
convex hull of the wavefront.
Namely, the wavefront of a strictly hyperbolic operator consists of points x ∈ RN+ such
that their orthogonal hyperplanes in the space ŘN are tangent to the cone A(F ). For general
New Examples of Irreducible Local Diffusion of Hyperbolic PDE’s 3
Figure 1. Projectivized zero set and wavefront of a cubic operator in R3.
hyperbolic operators also some points corresponding to the hyperplanes “not in general position”
with the singular set of A(F ) should be added.
Given a point x ∈ W (F ), the principal fundamental solution E(F ) is called holomorphically
sharp in a local (close to x) connected component C of the regularity domain RN \ W (F )
if there is a smooth analytic function in a neighbourhood of x in RN coinciding with E(F )
in the intersection of this neighbourhood and this component C. E(F ) is C∞-sharp in such
a component if it can be extended to a C∞-smooth function on the closure of this component.
Example 1.1. The wavefront of the standard wave operator
∂2
∂x21
− c2
(
∂2
∂x22
+
∂2
∂x23
)
in R3 is the cone
c2x21 = x22 + x23, x1 ≥ 0;
the principal fundamental solution of this operator is sharp at all its points in the “exterior”
component of the complement of W (F ), and is not in the interior one (growing there asymp-
totically as (distance from the wavefront)−1/2 close to the regular points of this cone). On the
contrary, principal fundamental solutions of wave operators in space-time of even dimension
greater than 2 are sharp from both sides of the cone.
A local component of RN \ W (F ), in which the fundamental solution is sharp, is called
a (holomorphic or C∞-) local lacuna. The negation of sharpness is called a diffusion of waves.
Remark 1.2 (on terminology). Roughly, elementary wave E(F ) is sharp at a point of its front,
if its singularity behaves like a squall, not predictable by the behaviour of the regular part of
the wave before meeting with it. Diffusion is an opposite situation: in this case the shock front
spreads some signs of its occurrence around it. The word “lacuna” was originally applied by
Petrovskii to the domains where the fundamental solution is just equal to zero; so it indicated
also the areas in RN+ and RN−11 such that the values of the data of Cauchy problem in them
are not important for the behavior of the solution of the problem in a given point x. The local
lacunas were introduced in [5, 6].
Wavefronts can be singular along the rays in RN+ , which correspond by projective duality to
parabolic points of the projectivization A∗(F ) ⊂ ŘPN−1 of the cone A(F ) ⊂ ŘN , i.e., the points
at which the second fundamental form of this hypersurface in ŘPN−1 is degenerate.
4 V.A. Vassiliev
b
A2 A2
A3
A2
3
1
2 -
Figure 2. Cuspidal edge and swallowtail.
Example 1.3. The projectivization of the zero set of a non-singular homogeneous polynomial
in Ř3 is a cubic curve in the projective plane ŘP 2, which can consist either of two components
as in Fig. 1 (left) or only the right-hand one of these components. The one-component case
never is hyperbolic, and the two-component one is hyperbolic if the ξ1-axis is directed inside
the domain bounded by its left-hand component. The wavefront of the latter operator looks as
a cone in R3
+ whose image in RP 2 is shown in Fig. 1 (right): its three singular points come from
three inflection points of the cubic curve (one of which is in our picture placed at infinity).
The properties of sharpness and diffusion at the points of a wavefront are invariant under
dilations of RN . Therefore we will consider projectivizations W ∗(F ) ⊂ RPN−1 of wavefronts and
speak on the diffusion or sharpness in a local connected component of RPN−1\W ∗(F ) if it holds
in the preimage of this component under the obvious projection RN+ \W (F )→ RPN−1 \W ∗(F ).
These properties are related with the local geometry of the projectivized wavefront W ∗(F ),
which by construction is the variety projective dual to the projectivization A∗(F ) ⊂ ŘPN−1
of the cone A(F ). Two basic types of points of wavefronts spreading the diffusion in such
components are described in the following two subsections.
A classification of singular points of projectivized wavefronts of strictly hyperbolic operators
(and, more generally, of hypersurfaces projective dual to smooth ones) coincides with a classi-
fication of parabolic points of smooth hypersurfaces, and hence with a classification of critical
points of smooth functions, see, e.g., [3]. The relation between them is provided by generating
functions of these fronts, see Section 1.3 below. In particular, non-singular points of wave-
fronts correspond to non-parabolic points of A∗(F ) and to Morse critical points of generating
functions, while the simplest standard singularities, semicubic cuspidal edges and swallowtails,
shown correspondingly in the left and the right parts of Fig. 2, are related with critical points
of types A2 and A3. For the surfaces in RPN−1 with arbitrary N ≥ 4, these pictures should
be multiplied by a piece of RN−4. The singularities shown in Fig. 1 (right) for case N = 3
have type A2: the cubic inflection points of curve A∗(F ) with local equation ξ0 = ξ31 + O
(
ξ41
)
in ŘP 2 result in cusps of projective dual curve, which are locally ambient diffeomorphic to the
semicubic parabola
{
x3 + y2 = 0
}
; they appear as transversal slices of the picture of Fig. 2.
1.2 Davydova condition at non-singular points of a wavefront
A non-singular piece of a projectivized wavefront locally divides RPN−1 into two parts. Choose
a normal direction to this wavefront W ∗(F ) at a point of such a piece and consider the second
fundamental form of the wavefront with respect to this direction. This quadratic form in N − 2
variables is non-degenerate as the operator is strictly hyperbolic.
New Examples of Irreducible Local Diffusion of Hyperbolic PDE’s 5
Definition 1.4. The Davydova condition [8] is satisfied if the positive inertia index of this
quadratic form is even.
If this condition is not satisfied, then we surely have local diffusion in the component of the
complement of the wavefront, indicated by the chosen normal direction (see [8]). Moreover,
the analytic continuation of the fundamental solution from this component to a neighbourhood
of such a point of the wavefront in CN has a two-fold ramification along the (complexified)
wavefront. Conversely, if the Davydova condition holds, then the wavefront is sharp in this
local component: this was proved in [7] by analytic estimates, and explained in [6] in terms of
monodromy and removable singularities.
1.3 Cuspidal edges and generating functions
The simplest singularities of a wavefront in RPN−1 are its cuspidal edges, at whose points it is
locally ambient diffeomorphic to the product of the semicubical parabola and RN−3, see Figs. 1
(right) and 2 (left). Space RPN−1 is locally divided by such a hypersurface into two parts, one
of which is “bigger”, and the other one “smaller”. According to [11], the principal fundamental
solution is never sharp in the bigger component, but can be sharp in the smaller one in some
additional conditions. To describe these conditions, we need the following notion of generating
functions of wavefronts.
Let X ∈ RPN−1 be a point of a projectivized wavefront. The corresponding hyperplane
in ŘPN−1 (i.e., projectivization of the hyperplane in ŘN orthogonal to the line {X} ⊂ RN )
is tangent to projectivized zero set A∗(F ) of the principal symbol of F at a point ξ. Choose
affine coordinates (ξ0, . . . , ξN−2) in ŘPN−1 with the origin in this point in such a way that this
tangent plane is defined by equation ξ0 = 0. Hypersurface A∗(F ) can be then specified close
to ξ by an equation of the form
ξ0 = f(ξ1, . . . , ξN−2), (1.1)
where f(0) = 0, df(0) = 0. Variables ξ1, . . . , ξN−2 form a system of local coordinates on A∗(F ).
The projective duality map A∗(F ) → RPN−1, and also the germ of its image at X, are then
determined by our function f : for any point of the variety A∗(F ) we draw the tangent hyperplane
at this point, and mark the corresponding point in RPN−1.
Definition 1.5. The function germ f defined in this way is called a projective generating function
of the wavefront at point X ∈ RPN−1 corresponding to the hyperplane {ξ0 = 0} ⊂ ŘPN−1.
It follows easily from this definition that any two generating functions of the same germ of
the wavefront, defined by different choices of affine coordinates ξi, can be obtained one from the
other by a diffeomorphism of space of arguments and a dilation of the target line by a non-zero
(maybe negative) coefficient.
Remark 1.6. If occasionally the hyperplane corresponding to X is tangent to hypersurfa-
ce A∗(F ) at several its points, then such a map is defined in a neighbourhood of any of these
points; the germ of the wavefront at point X will then consist of several locally irreducible
branches corresponding to all these tangency points and described by their projective generating
functions.
If generating function f is Morse, then the corresponding piece of the wavefront is smooth
(and was considered in the previous subsection). A cuspidal edge of the wavefront occurs when f
has the simplest non-Morse singularity of type A2, see, e.g., [2, 3]. According to L. G̊arding [11],
a wavefront is sharp at its cuspidal edge if and only if dimension N is odd, the inertia indices
of the quadratic part of the function germ f are even, and the investigated component of the
6 V.A. Vassiliev
complement of the wavefront is the “smaller” one. (The rank of the quadratic part of a function
of this class depending on N − 2 variables is equal to N − 3, hence in the case of odd N both
inertia indices are of the same parity.) For example, these conditions are satisfied for the interior
component in Fig. 1 (right).
Remark 1.7. The part “only if” of this G̊arding’s result claims that diffusion at cuspidal edges
appears in some seven cases, depending on the parities of N and inertia indices of the quadratic
part of f , and on the choice of a component. In some six out of these seven cases, diffusion
follows already from the Davydova’s obstruction: we can indicate a smooth piece of wavefront
approaching the cuspidal edge, such that sharpness in our component fails at the points of
this piece. The only exceptional case is that of odd N and odd inertia indices of f : in this
case the wavefront is sharp for the “bigger” component at all neighboring smooth points of
its surface, however regularity of the fundamental solution fails when we approach its singular
stratum.
1.4 Other simple singularities
The most common equivalence relation of singularity classes of germs of functions
(
Rn, 0
)
→ R
is provided by the group Diff0 of local diffeomorphisms
(
Rn, 0
)
→
(
Rn, 0
)
acting on functions
by composition with these diffeomorphisms. The space of function germs splits into the orbits
of this action. This splitting is not discrete: some singularity classes can depend on continuous
parameters separating continuously many orbits of this action.
A natural primary segment of the classification of singularities of smooth functions (and
hence also of wavefronts) is formed by so-called simple singularities Ak, Dk, E6, E7 or E8,
see [3]. For this part of the classification the splitting into the Diff0-orbits is still discrete:
by definition these are the singularities such that a neighbourhood of any of them in the
function space is covered by representatives of only finitely many orbits of this action. The
name “simple” and canonical notation of these classes come from a deep and multiform rela-
tion of these classes with the simple Lie algebras. All local lacunas close to singular points
of wavefronts of these types were counted in [18] (except for cases A2 and A3 of cuspidal
edges and swallowtails, which were in detail described by L. G̊arding [11]). In [19] an easy
geometric characterization of these lacunas was found: it turned out that all cases of the dif-
fusion close to any simple singular points of generic wavefronts can be reduced to the two
cases described above. Namely, the following theorem was proved there (the technical notion
of projective versality used in it and satisfied for generic wavefronts will be defined in Sec-
tion 1.5).
Theorem 1.8 (see [19]). If projectivized wavefront W ∗(F ) of a strictly hyperbolic operator F
has a simple singularity at its point X and is projective versal at this point, then the principal
fundamental solution of F is sharp at point X in a certain local connected component of the
complement of W ∗(F ) if and only if
1) the Davydova condition is satisfied at all smooth point of W ∗(F ) in the boundary of this
component, and
2) this boundary does not contain the cuspidal edges of W ∗(F ), close to which our component
is the “bigger” one.
Example 1.9 (see [11]). Close to a swallowtail (see Fig. 2), only the following components of
the complement of the wavefront are local lacunas: component 2 if N is odd and the positive
inertia index i+ of the quadratic part of the generating function is even; component 3 if i+ is
odd and N is arbitrary.
New Examples of Irreducible Local Diffusion of Hyperbolic PDE’s 7
Remark 1.10.
1. If N is even, then condition 2) in this theorem is unnecessary: if it is not satisfied at an
edge point, then condition 1) also fails at some neighboring smooth points of the wavefront.
2. The restrictions from Sections 1.2 and 1.3 prove the part “only if” of this theorem (which
holds also for not necessarily projective versal wavefronts); a proof of the part “if” is based
on the properness of the so-called Lyashko–Looijenga maps of miniversal deformations of
simple singularities, see [14, 15].
A natural question arising from this theorem (and formulated explicitly by V.P. Palamodov
about 1991, see [19]) asks whether it can be extended to all singularity classes of wavefronts of
strictly hyperbolic operators. We show below that the answer is negative in the case of odd N
for all non-simple singularity classes.
1.5 Generating families of wavefronts and versality
The projectivized wavefront W ∗(F ) ⊂ RPN−1 can be considered close to any its point X as
the real discriminant variety of a deformation of the corresponding generating function (1.1).
Namely, this deformation depends on the (N −1)-dimensional parameter x = (x0, x1, . . . , xN−2)
and consists of functions fx defined by formula
fx ≡ f(ξ1, . . . , ξN−2)− x0 −
N−2∑
j=1
xjξj . (1.2)
The parameters x0, . . . , xN−2 of this deformation form a coordinate system in an affine chart
of space RPN−1 with the origin at the point X: this chart consists of hyperplanes in ŘPN−1
defined by equation
ξ0 = x0 +
N−2∑
j=1
xjξj (1.3)
in local coordinates ξi. Such a point x ∈ RPN−1 belongs to the real discriminant of deforma-
tion (1.2) (i.e., fx has a real critical point close to 0 with critical value 0) if and only if the
hyperplane in ŘPN−1 given by (1.3) is tangent to A∗(F ); so the real discriminant of (1.2) is
exactly the variety projective dual to A∗(F ).
We will consider the case when this deformation is sufficiently representative, namely is
a versal deformation of f . Let us remind this notion (for an expanded description see [3]).
Suppose we have a smooth function ϕ(y) : Rn → R with dϕ(0) = 0, and its deformation
Φ(y, λ) :
(
Rn×Rl, 0
)
→ (R, 0), i.e., a family of functions ϕλ : Rn → R depending on the parameter
λ ∈ Rl, ϕ0 ≡ ϕ. This deformation is called versal if any other deformation Ψ:
(
Rn × Rk, 0
)
→
(R, 0) of the same function ϕ can be reduced to it by an appropriate map of parameters and
a family of local diffeomorphisms of the space Rn depending on these parameters: in formulas,
there should be a smooth map θ :
(
Rk, 0
)
→
(
Rl, 0
)
and a family of local (i.e., defined in
a neighbourhood of the origin) diffeomorphisms Hκ : Rn → Rn smoothly depending on the
parameter κ ∈ Rk, H0 ≡ {the identity map}, such that
Ψ(y,κ) = Φ(Hκ(y), θ(κ)) for any y and κ sufficiently close to the origin. (1.4)
It is known that
• For any function singularity ϕ with finite Milnor number µ(ϕ), almost all its deformations
depending on ≥ µ(ϕ) parameters are versal;
8 V.A. Vassiliev
• such a singularity does not have versal deformations depending on less than µ(ϕ) parame-
ters (versal deformations depending on exactly µ(ϕ) parameters are called miniversal),
• if Φ is a versal deformation of ϕ then for any sufficiently small perturbation ϕ̃ of our
singularity ϕ there is a point λ ∈ Rl close to the origin such that the functions ϕ̃ and ϕλ
can be transformed one into the other by a local diffeomorphism in Rn;
• if Φ is a miniversal deformation of ϕ (thus depending on µ(ϕ) parameters), and Ψ an
arbitrary versal deformation of ϕ depending on k parameters, then the corresponding
map θ in (1.4) is a submersion close to the origin in Rk, sending the discriminant of
deformation Ψ to the discriminant of deformation Φ.
Similar notion and facts hold for singularities of holomorphic functions of complex variables.
Definition 1.11. Deformation (1.2) is called a (projective) generating family of the wave-
front W ∗(F ) at point X. The hypersurface A(F ) is projective versal at its point X if the
corresponding deformation (1.2) is versal, see [3].
Example 1.12. Suppose that generating function f has a singularity of type Ak at a point
Ξ ∈ ŘPN−1. Then there is a smooth parametric curve u :
(
R1, 0
)
→
(
RN−2, 0
)
such that the
function |grad f | grows as the kth degree of the parameter along it. The linear hull of the first j
derivatives of this curve at 0 is uniquely defined by this condition for any j ≤ k − 1. The
hypersurface with equation ξ0 = f(ξ1, . . . , ξN−2) is projective versal at point Ξ if and only if all
these first k − 1 derivatives are linearly independent in RN−2.
The projective versality at simple singular points is a condition of general position, however
for more complicated singularity classes, which split into continuous families of orbits of the
group Diff0 of local changes of variables
(
Rn, 0
)
→
(
Rn, 0
)
, this is generally not true. For
instance, a generic hyperbolic operator of order ≥ 3 in R8 can have singularities of type P8
(see [3] and the next Section 1.6) in some discrete set of points of the projectivized wavefront;
the generating family (1.2) is in this case transversal to the entire stratum {P8} in the space of
function singularities, but not to its particular orbits, and hence is not versal.
1.6 Parabolic singularities and main result
Parabolic singularities of function germs (see [3]) form the next natural family of singularity
classes after the simple ones. Besides some nice algebraic properties distinguishing them, it is
important for us that they are confining for the family of simple singularities (see [3]), that is,
any non-simple singularity can be turned to a parabolic one by an arbitrarily small perturbation.
Therefore, proving that some property (e.g., diffusion) holds close to any non-simple singularity,
it is enough to prove it for these classes only.
The parabolic classes are listed in Table 1. Any function germ Rn → R of this class can be
reduced by a smooth change of variables to one of the normal forms indicated in this table. The
letter Q in it denotes non-degenerate quadratic forms in the variables missing in the first parts
of these formulas, e.g., ±ξ24 ± · · · ± ξ2n for singularities P 1
8 and P 2
8 , and ±ξ23 ± · · · ± ξ2n for five
other singularity types. So, the corank of a parabolic singularity (i.e., the number of essential
variables) is equal to 3 in the case of P8 singularities, and to 2 for X9 and J10.
Proposition 1.13 (see [3]). All sufficiently small perturbations of a function germ with a para-
bolic critical point have only simple critical points or parabolic critical points of the same class
as the perturbed one.
Indeed, a small perturbation cannot increase the Milnor number, or the corank or the mul-
tiplicity of a singularity.
New Examples of Irreducible Local Diffusion of Hyperbolic PDE’s 9
Table 1. Real parabolic singularities
Notation Normal form Restrictions
P 1
8 ξ31 + αξ1ξ2ξ3 + ξ22ξ3 + ξ2ξ
2
3 +Q α > −3
P 2
8 ξ31 + αξ1ξ2ξ3 + ξ22ξ3 + ξ2ξ
2
3 +Q α < −3
±X9 ±
(
ξ41 + αξ21ξ
2
2 + ξ42 +Q
)
α > −2
X1
9 ξ1ξ2
(
ξ21 + αξ1ξ2 + ξ22
)
+Q α2 < 4
X2
9 ξ1ξ2(ξ1 + ξ2)(ξ1 + αξ2) +Q α ∈ (0, 1)
J3
10 ξ1
(
ξ1 − ξ22
)(
ξ1 − αξ22
)
+Q α ∈ (0, 1)
J1
10 ξ1
(
ξ21 + αξ1ξ
2
2 + ξ42
)
+Q α2 < 4
Theorem 1.14. Suppose that N is odd, the projectivized wavefront W ∗(F ) ⊂ RPN−1 of a strict-
ly hyperbolic operator F has a singularity of one of classes indicated in Table 1 at a point
X ∈W ∗(F ), and one of two conditions is satisfied:
a) the positive inertia index of the quadratic part Q of the corresponding generating function
is odd;
b) the singularity type of W ∗(F ) at point X is one of the following four: X1
9 , X2
9 , J3
10, or J1
10.
Then
1) the principal fundamental solution E(F ) has a holomorphic diffusion in all local compo-
nents of the complement of the wavefront W ∗(F ) at the point X;
2) if the singularity of the wavefront at point X is projective versal, then there is a local (close
to the point X) connected component of the complement of the wavefront, such that the
fundamental solution of F is sharp in this component at all points of the wavefront in the
boundary of this component, except for the points of the most singular stratum (i.e., for
parabolic singular points of the corresponding class).
Both statements of this theorem in Case (b) follow from the same statements of Case (a)
for the same singularity types, because the four classes mentioned there are invariant under the
multiplication of functions by −1, which preserves also the condition of sharpness, but changes
(in the case of odd N) the parities of inertia indices of quadratic forms of rank N −4. Therefore
we will consider in our proofs only Case (a), i.e., assume that positive inertia indicex of quadratic
form Q is odd. For a proof of Statement 1 of this theorem see [20] for all parabolic singularity
classes except for P 2
8 , and [21] for P 2
8 ; see also Remark 2.5 in p. 12 below. Statement 2 in
Case (a) will be proved in Section 4 for singularities P 1
8 and P 2
8 , and in Section 3 for all the
other parabolic singularity types.
Remark 1.15. The singular points of wavefronts of three parabolic classes not mentioned in
Case (b) of this theorem in the case of odd N and even positive inertia index of form Q do have
local lacunas in their neighborhoods, see [21].
2 Topological reformulations
2.1 Sharpness and the local Petrovskii condition
Holomorphic sharpness of wavefronts can be detected by a topological criterion, the local Petro-
vskii condition, introduced in [6] and reformulated in [18] in terms of the topology of Milnor fibers
of generating functions. This condition generalizes a global homological condition from [17].
10 V.A. Vassiliev
This condition has sense for all real holomorphic function singularities with finite Milnor
numbers. Let us remind the needed topological objects (see [3, 16]). Let f :
(
Cn,Rn, 0
)
→
(C,R, 0) be a holomorphic function germ with df(0) = 0; suppose that 0 is an isolated critical
point of f , so that its Milnor number µ(f) is finite. Let B ⊂ Cn be a sufficiently small ball
centered at point 0, and fλ a very small perturbation of function f which is non-discriminant
(i.e., the variety f−1λ (0) is non-singular in B). The manifold Vλ ≡ f−1λ (0) ∩B is then called the
Milnor fiber corresponding to the perturbation fλ. It is a smooth (2n−2)-dimensional manifold
with boundary ∂Vλ ≡ Vλ ∩ ∂B, and is homotopy equivalent to the wedge of µ(f) spheres of
dimension n− 1. In particular H̃n−1(Vλ) ' Zµ(f) and
H̃n−1(Vλ, ∂Vλ) ' Zµ(f); (2.1)
here H̃ denotes the reduced homology groups, so that H̃n−1 ≡ Hn−1 if n > 1.
If perturbation fλ is real (that is, fλ(Rn) ⊂ R), then there are two important elements of the
group (2.1): the even and the odd local Petrovskii classes.
The even Petrovskii class is realized by the fundamental class of the manifold Vλ ∩Rn of real
points of the Milnor fiber, oriented as the boundary of the domain where fλ ≤ 0. For example,
if the function f has a local minimum at its critical point, then the even Petrovskii class of its
perturbation f + ε, ε > 0, is trivial: it is represented by the empty cycle.
In this work we will deal with the odd Petrovskii class whose construction is more tricky,
see, e.g., [20, Chapter IV]. However, we will be mainly interested not in this class itself but
in the condition of its triviality, which has an easy characterization: the odd Petrovskii class
associated with the perturbation fλ(x1, . . . , xn) of function f is equal to zero if and only if the
even Petrovskii class of perturbation fλ(x1, . . . , xn) + x2n+1 − x2n+2 of function f(x1, . . . , xn) +
x2n+1 − x2n+2 is equal to zero.
Definition 2.1. The local Petrovskii class in the group (2.1) is defined as the even local Petro-
vskii class if n is even or the odd local Petrovskii class if n is odd. A non-discriminant point λ
satisfies the local Petrovskii condition if the corresponding local Petrovskii class is equal to 0.
Proposition 2.2. If f is the generating function of the projectivized wavefront W ∗(F ) of
a strictly hyperbolic operator at its point X, the corresponding critical point of f is isolated,
and fx is a small non-discriminant real perturbation of f of the form (1.2), then the principal
fundamental solution E(F ) is holomorphically sharp at X in the local (close to X) component
of RPN−1 \W ∗(F ) containing point x if and only if the corresponding local Petrovskii class is
equal to 0 in group H̃N−3(Vx, ∂Vx).
Part “if” of this statement is proved in [6], part “only if” in [18].
2.2 Functoriality of local Petrovskii classes under adjacencies of singularities
Let Φ:
(
Cn×Cl,Rn×Rl, 0
)
→ (C,R, 0) be a deformation of our function germ f , parameterized
by points λ ∈ Cl, i.e., Φ is a family of functions fλ ≡ Φ(·, λ) : Cn → C such that f0 ≡ f and
fλ
(
Rn
)
⊂ R if λ ∈ Rl ⊂ Cl.
Let Σ ⊂ Rl be the discriminant of this deformation, i.e., the set of parameters λ ∈ Rl
such that variety Vλ is singular in B. This set contains the real discriminant (see Section 1.5)
but generally can be greater than it, including also parameter values λ of functions fλ having
imaginary critical points in B with zero critical values. This set divides parameter space Rl into
several connected components in a neighbourhood of the origin.
By the construction of Petrovskii classes, the local Petrovskii condition is satisfied or is not
satisfied simultaneously for all points from any such component.
New Examples of Irreducible Local Diffusion of Hyperbolic PDE’s 11
Consider some such component C; let λ̃ ∈ Σ ∩ C̄ be a discriminant value of the parameter,
which lies in the boundary of C. By definition of the discriminant, the corresponding function fλ̃
has several (at least one) critical points aj ∈ B with critical value 0. Consider a set of very
small non-intersecting balls Bj ⊂ B around all these points.
Let λ ∈ C be a non-discriminant value of the parameter which is very close to λ̃, so that the
corresponding level set Vλ is transversal to the boundaries of all balls Bj , and varieties Vλ ∩Bj
can be considered as the Milnor fibers of the corresponding critical points aj of function fλ̃.
Proposition 2.3.
1. For any real critical point aj ∈ B of function fλ̃ with fλ̃(aj) = 0, the odd (respectively,
even) Petrovskii class of singularity (fλ̃, aj) in the homology group H̃n−1(Vλ∩Bj , Vλ∩∂Bj)
of the corresponding Milnor fiber is equal to the image of the odd (respectively, even)
Petrovskii class of initial singularity (f, 0) under the obvious map
H̃n−1(Vλ, ∂Vλ)→ H̃n−1
(
Vλ, Vλ ∩ (B \Bj)
)
≡ H̃n−1(Vλ ∩Bj , Vλ ∩ ∂Bj). (2.2)
2. For any imaginary critical point aj ∈ B of fλ̃ with fλ̃(aj) = 0, the images of the odd and
even local Petrovskii classes under the map (2.2) are equal to 0.
3. If deformation Φ has the form (1.2), then the corresponding principal fundamental solu-
tion E(F ) is holomorphically sharp at point λ̃ of the wavefront in component C if and only
if the images of the local Petrovskii class of fλ under all maps (2.2) corresponding to all
such critical points aj are equal to zero in all groups H̃N−3(Vλ ∩Bj).
Statements 1 and 2 of this proposition follow immediately from the construction of Petrovskii
classes, and Statement 3 from the multisingularity version of Proposition 2.2 also proved in [18].
If f and g are two germs of functions
(
Cn,Rn, 0
)
→ (C,R, 0) with df(0) = 0 = dg(0), which
are Diff0-equivalent (i.e., can be transformed one to another by the composition with a local
diffeomorphism
(
Cn,Rn, 0
)
→
(
Cn,Rn, 0
)
), and Φ and Γ are some their real versal deforma-
tions, then there is a one-to one correspondence between the local connected components of
the complements of the discriminants of these deformations. This correspondence is defined by
maps inducing deformations equivalent to Φ and Γ one from the other in accordance with the
definition of versal deformations (see [3] and map θ in (1.4)). The Petrovskii conditions are re-
spected by this correspondence. Therefore to prove Statement 2 of Theorem 1.14 for a parabolic
singularity class it is enough to present a component of the complement of the discriminant
set of an arbitrary real versal deformation of an arbitrary singularity of this class, such that
the local Petrovskii condition will be satisfied in this component at all the discriminant points
of its boundary except for the most singular points corresponding to the parabolic singularity
class itself. For instance, we can consider only singularities f(ξ) given by the polynomials from
Table 1, and their monomial versal deformations of form
Φ(ξ, λ) ≡ f(ξ) +
∑
α∈M
λαξ
α, (2.3)
where M ⊂ Zn+ is a finite set of multiindices α = (α1, . . . , αn), ξα ≡ ξα1
1 · · · ξαn
n .
If Φ(ξ1, . . . , ξn;λ) is a versal deformation of function f(ξ1, . . . , ξn) of the form (2.3), then
Φ(ξ1, . . . , ξn;λ) +
(
ξ2n+1 + ξ2n+2
)
and Φ(ξ1, . . . , ξn;λ) −
(
ξ2n+1 + ξ2n+2
)
are versal deformations
respectively of functions f(ξ1, . . . , ξn) +
(
ξ2n+1 + ξ2n+2
)
and f(ξ1, . . . , ξn)−
(
ξ2n+1 + ξ2n+2
)
in n+ 2
variables, and have the same discriminant sets in the spaces of parameters λ = {λα}. The
homology groups (both absolute and modulo the boundary) of the middle dimensions of Milnor
fibers of these three functions corresponding to the same non-discriminant parameter values λ
are naturally isomorphic to one another. These isomorphisms preserve the Petrovskii condition
12 V.A. Vassiliev'
&
$
%−
−
− −+
'
&
$
%
'
&
$
%
��
�
��
�
��
�
��
H
HHH
HHH
HHHH
−
−
−
+
Figure 3. +X9 (left), X1
9 (right).
(see [20, Section V.1.7]). Therefore proving Theorem 1.14 (in its Case (a)) we can assume that
for singularity classes P8 we have n = 5, Q = ξ24 − ξ25 , and for singularities X9 or J10 we have
n = 3, Q = ξ23 .
Proposition 2.4. If function f :
(
Cn,Rn, 0
)
→ (C,R, 0) is given by one of normal forms from
Table 1, the dimension n is odd and the positive inertia index of quadratic form Q also is odd,
then the boundary of the odd Petrovskii class of any non-discriminant small perturbation fλ of f
is a non-trivial element of group Hn−2(Vλ ∩ ∂B).
Scheme of the proof. A basis in Hn−1(Vλ) can be composed of vanishing cycles ∆i, i =
1, . . . , µ(f), see, e.g., [4, 20]. By the Poincaré duality and exact sequence of pair (Vλ, ∂Vλ), the
desired non-triviality of the boundary for a morsification fλ is equivalent to the following con-
dition: vector Π(λ) of intersection indices of the odd Petrovskii class with these basic vanishing
cycles is not an integer linear combination of rows of intersection matrix {〈∆i,∆j〉} of these
cycles.
It is enough to prove the latter property for an arbitrary single non-discriminant real morsi-
fication fλ. Indeed, by the exact formulas for the local Petrovskii classes (see [20, Sections V.1
and V.4]) expressing them in terms of intersection indices and the Morse indices of real critical
points, vectors Π(λ) for morsifications fλ from the neighbouring components of the complement
of the discriminant differ by adding or subtracting a row of the intersection matrix.
So, it remains to calculate all these data for one arbitrary morsification of any of our singu-
larity classes. Vectors Π(λ) for these morsifications follow after that by the above-mentioned
explicit formulas, and give the promised result.
The intersection indices of vanishing cycles for corank 2 singularities X9 and J10 can be found
by the Gusein-Zade–A’Campo method, see [12] and [1], starting, e.g., from the morsifications
discussed in Proposition 3.1 below. For singularity class P 1
8 (represented by function ξ31 + ξ32 +
ξ33 +Q) these indices follow from the main theorem of [9], and for class P 2
8 they were calculated
in [21, Section 7]. �
Remark 2.5. Statement 1 of Theorem 1.14 follows immediately from this proposition and
Proposition 2.2.
3 Proof of Statement 2 of Theorem 1.14
for parabolic singularities of corank 2
3.1 Examples
Proposition 3.1. Any function ϕ(ξ1, ξ2) in two variables defined by one of normal forms in-
dicated in Table 1 for singularities of types +X9, X1
9 , X2
9 , J3
10, or J1
10 has an arbitrarily small
perturbation, whose set of zeros and domains of constant signs of values look topologically as
New Examples of Irreducible Local Diffusion of Hyperbolic PDE’s 13
�
�
�
�
�
�
�
�
�
�
@
@
@
@
@
@
@
@
@
@
−
+
− +
− −
−
Figure 4. X2
9 (left), J3
10 (right).
− − −+
−
Figure 5. J1
10.
shown in Figs. 3 (left and right), 4 (left and right), or 5 respectively. (The non-closed curves in
these pictures are assumed to be continued to the infinity in R2.)
In particular, any versal deformation of ϕ contains perturbations with this topological picture.
Proof is elementary.
For any of these five functions ϕ(ξ1, ξ2) let us add a small positive constant to its perturbation
shown in the corresponding picture in such a way that all critical values at the minima points
will remain negative. All the other critical values will become positive, therefore the obtained
function is non-discriminant. Denote this function by ϕ̃ and define function f̃ : C3 → C by
f̃(ξ1, ξ2, ξ3) ≡ ϕ̃(ξ1, ξ2) + ξ23 ; (3.1)
this is a perturbation of function
f ≡ ϕ(ξ1, ξ2) + ξ23 . (3.2)
Proposition 3.2. If f is the function (3.2), where ϕ is one of our five polynomials from Table 1,
and f̃ is the perturbation of f just defined, then any Morse perturbation fλ of f lying in the
same component of the complement of the discriminant variety of a deformation of f as f̃ can
have no more than one pair of imaginary critical points (so that the number of its real critical
points is equal to either µ(f) or µ(f)− 2).
Proof. The sum of local indices of vector field grad fλ at all real critical points of fλ with
negative (respectively, positive) critical values is an invariant of components of the complement
of the discriminant. For our five perturbations these sums are equal to: 4 and −3 for +X9, 3
and −4 for X1
9 , 2 and −5 for X2
9 , 3 and −5 for J3
10, 4 and −4 for J1
10. The index of any Morse
critical point is equal to 1 or −1, hence fλ should have at least 7 real critical points in the case
of any X9 singularity and at least 8 in the case of J10. �
14 V.A. Vassiliev
Proposition 3.3. Let f be the function (3.2) where ϕ is given by one of formulas of Table 1
for one of five singularity classes +X9, X1
9 , X2
9 , J3
10 or J1
10; let C ⊂ Rl be the component
of the complement of discriminant variety Σ in the parameter space of a deformation of f
containing the corresponding perturbation (3.1); let λ̃ ∈ Σ be an arbitrary discriminant point in
the boundary of this component C which is sufficiently close to the origin in the parameter space
of the deformation, and all critical points of fλ̃ with zero critical value are not parabolic (i.e.,
do not belong to the same singularity class as f). Then for any parameter value λ ∈ C, which is
sufficiently close to λ̃, and any critical point aj ∈ B of fλ̃ with critical value 0 the corresponding
localized odd Petrovskii class in H̃2(Vλ ∩Bj , Vλ ∩ ∂Bj) is equal to 0.
Statement 2 (Case (a)) of Theorem 1.14 for these five singularity classes follows immediately
from this proposition and Proposition 2.3.
Proof of Proposition 3.3 takes the remaining part of this section. Everywhere in it we assume
that fλ̃ satisfies the conditions of this proposition.
We can and will assume that our deformation is versal, otherwise we can expand it to a versal
one. Let l be the dimension of its parameter space.
Let µj , j = 1, . . . , be the Milnor numbers of all critical points aj of fλ̃ with critical value 0.
Suppose that for one of these points, say a1, the localized Petrovskii class inH2(Vλ∩B1, Vλ∩∂B1),
where λ ∈ C is very close to λ̃, is non-trivial. In Section 3.2 we will show that in this case
component C should contain functions fλ satisfying at least one of two conditions described
there; in Sections 3.3 and 3.4 we present a combinatorial program proving that such functions
actually do not exist.
3.2 Conditions
By Statement 2 of Proposition 2.3, critical point a1 mentioned in the previous paragraph is real.
Lemma 3.4. We can assume that fλ̃ has only one this critical point a1 with critical value 0
(i.e., if there are several such critical points of fλ̃, then there is another point λ̃′, λ̃′ ≈ λ̃, in the
boundary of C with unique such critical point and non-trivial localization of the Petrovskii class
at neighboring points of C).
Proof. Suppose first that all critical points of fλ̃ with critical value 0 are real. If λ̃ is sufficiently
close to the origin in parameter space Rl of our versal deformation of function f , then a small
open ball U in this space with the center at point λ̃ is also the parameter space of a versal
deformation of the multisingularity of function fλ̃, formed by all these function germs (fλ̃, aj).
A miniversal deformation of such a multisingularity splits into the product of independent versal
deformations of all singularities composing it, in particular its number of parameters is equal
to
∑
µj . Therefore our component C ∩ U of the complement of the discriminant set of this
versal deformation of multisingularity fλ̃ is ambient diffeomorphic to the direct product of some
components Cj of the complements of discriminant sets of all these singularities (fλ̃, aj), and
additionally of space Rl−
∑
µj . This diffeomorphism is realized by the restriction to C ∩ U of
a map inducing a deformation equivalent to our versal deformation of fλ̃ from this miniversal
one. (More precisely, this map is the projection map of a trivial fiber bundle, all whose fibers are
diffeomorphic to Rl−
∑
µj and consist of functions, obtained one from another by diffeomorphisms
of the argument space.) Point λ̃ is sent by this diffeomorphism to the product of origins in the
space Rl−
∑
µj and in all parameter spaces Rµj of these deformations. Let us shift for any j 6= 1
the corresponding origin point of Rµj to a non-discriminant point inside the corresponding com-
ponent Cj . The obtained point of the product corresponds via our diffeomorphism to a point λ̃′
in the boundary of our component C ∩ U , such that fλ̃′ has a single critical point with critical
value 0, with the same simple singularity class as (fλ̃, a1), and the localized Petrovskii class for
a neighboring point λ′ ∈ C again will be not equal to zero in group H̃n−1(Vλ′ ∩B1, Vλ′ ∩ ∂B1).
New Examples of Irreducible Local Diffusion of Hyperbolic PDE’s 15
Suppose now that our function fλ̃, λ̃ ∈ ∂C, has imaginary critical points with value 0. By
Proposition 3.2 there can be only one pair of such points, and these two critical points should
be Morse. In this case we consider a very similar splitting of a neighbourhood of point λ̃, in
which one factor R2 is not the parameter space of a real singularity of fλ̃, but the real part of
the product of parameter spaces of both its complex conjugate critical points; the rest of the
consideration remains as previously. �
So, we can and will assume that fλ̃ has only one critical point a1 ∈ B with critical value 0.
By Proposition 1.13 this critical point a1 is simple. A neighbourhood of point λ̃ in Rl can
be considered as the base of a versal deformation of critical point a1 of function fλ̃. Then by
Theorem 1.8 we may assume that either critical point a1 is Morse or it is of type A2 and our
component C is the “bigger” one with respect to the corresponding cuspidal edge.
If this point is Morse, then component C contains a point λ such that function fλ is Morse
and satisfies the following
Condition 3.5. The intersection index of the Petrovskii class in H̃2(Vλ, ∂Vλ) with a vanishing
cycle in H̃2(Vλ) defined by the segment in R1 ⊂ C1 connecting the non-critical value 0 with
either the smallest positive or the largest negative critical value of fλ is not equal to 0.
Indeed, this is true for all values λ ∈ C which are sufficiently close to λ̃ and correspond to
Morse functions.
Accordingly to the explicit formulas for the Petrovskii classes (see [18, 20]) this condition is
equivalent to the following one.
Condition 3.5′. Either the positive inertia index of the critical point of fλ with the smallest
positive critical value is odd, or the positive inertia index of the critical point of fλ with the
largest negative critical value is even.
In the second case of A2 singularity (assuming that Condition 3.5 is not satisfied for all points
of C) our component C contains points λ arbitrarily close to λ̃, such that fλ is Morse and
Condition 3.6.
1) fλ has at least two real critical points with positive critical values;
2) positive inertia index of the quadratic part of the critical point of fλ with the smallest
positive critical value is even;
3) positive inertia index of the quadratic part of the critical point of fλ with the next smallest
positive critical value is exactly by 1 smaller than that for the previous one;
4) intersection index in H2(Vλ) of two vanishing cycles defined by simplest paths a
0
q q� con-
necting these critical values with 0 in C1 is equal to 1 or −1;
5) the intersection index of the odd Petrovskii class in H2(Vλ, ∂Vλ) with the vanishing cycle
corresponding to the smallest positive critical value (respectively, to the next smallest one)
is equal to 0 (respectively, to 1 or −1).
It remains to prove that our component C does not contain points λ satisfying either of these
two sets of conditions. To do it we apply the program [23] counting all topological types which
the non-discriminant morsifications of function f can have.
16 V.A. Vassiliev
3.3 Description of the program
For an extended description of this program see [20, Section V.8] (although the version of the
program quoted there is now obsolete, for the actual version see [23]). Let us remind its basic
ideas.
Any generic real morsification fλ of a function singularity f of corank ≤ 2 in n variables is
characterized by a set of its topological invariants, including
a) intersection indices of appropriately oriented and ordered vanishing cycles in Hn−1(Vλ)
corresponding to all its critical points (including the imaginary ones),
b) intersection indices of both local Petrovskii classes with all these vanishing cycles,
c) Morse indices of all real critical points (more precisely, the positive inertia indices of their
quadratic parts) ordered by increase of their critical values,
d) the number of negative critical values, and
e) the number n of variables (in fact only its residue modulo 4 is important).
Any possible set of such data is called a virtual morsification.
If we know these data for an actual morsification fλ, then we can calculate them for all
morsifications which can be obtained from fλ by the standard surgeries, such as Morse surgeries
of death/birth of real critical points of neighbouring indices, the change of the order in R1 of
critical values at distant critical points, the passage of values at two distant imaginary critical
points through the real axis, the jump of a critical value through 0 (this is the unique surgery
changing the component of the complement of the discriminant), and the change of the choice of
paths defining the cycles vanishing in imaginary critical points. Our algorithm starts from the
virtual morsification corresponding to the initial perturbation fλ ∈ C and applies to it all possi-
ble chains of transformations of virtual morsifications modelling the possible surgeries of actual
morsifications. The newborn virtual morsifications are compared with all found previously ones,
and are added to the list if they are new. A priori this algorithm can provide virtual morsifi-
cations not corresponding to any actual one; however it surely finds all virtual morsifications
corresponding to all actual ones.
3.4 Modification of program [23] for our purposes
To apply this program to any of our five singularities, we calculate (by hands) data (a)–(d)
for real morsification fλ from our component C described in Section 3.1, put these data to
our program, switch out in it the surgery of jumping of critical values through 0 (since we
are interested only in the morsifications from component C) and additionally include alarm
operators aborting the program and typing the corresponding message if a newborn virtual
morsification satisfies one of two conditions listed in Section 3.2.
More precisely, we take program [23] and do the following:
a) make in lines 1–7 changes described in lines 12–18 (the Milnor number of allX9 singularities
is equal to 9, and that of J10 is equal to 10),
b) put the command NPOZC = 4 in line 37 for singularities +X9 and J1
10, NPOZC = 3 for X1
9
and J3
10, and NPOZC = 2 for X2
9 : this is the number of critical points with negative critical
values for our starting morsification specifying component C,
c) uncomment operators L(MD + 1) = 1 and L(MDD) = 1 in lines 115, 116 (i.e., we disable
the surgeries of crossing the discriminant, which correspond to the jumps of real critical
values through 0),
New Examples of Irreducible Local Diffusion of Hyperbolic PDE’s 17
d) insert the next text immediately after operator 2318 CONTINUE:
IF(PC1(NPOZC).NE.0) GOTO 7343
NPO1=NPOZC+1
IF(PC1(NPO1).NE.0) GOTO 7343
NPO2=NPOZC+2
IF(PC1(NPO2).NE.0.AND.INDC(NPO2).LT.INDC(NPO1)) GOTO 7343
(3.3)
(the first two operators “IF” here check Condition 3.5 above, and the third one detects
items 3 and (partly) 5 of Condition 3.6; on the address 7343 the program types the
word ALARM and data of the virtual morsification for which one of these conditions was
detected, and then terminates its work);
e) calculate the intersection indices of vanishing cycles of morsification fλ described in Sec-
tion 3.1 using the Gusein-Zade–A’Campo method, and substitute all obtained non-zero
indices 〈∆i,∆j〉, i < j, into the subroutine DATA at the very end of the program: e.g., if
〈∆3,∆5〉 = 1 then we insert operator C(3, 5) = 1 there. Also, if the positive inertia index
of the quadratic part of the ith (in the order of increase of critical values) critical point of
our morsification is equal to some number q 6= 2, then we put INDC(i) = q in the last part
of this subroutine. These indices for the critical points of functions (3.1) for our corank 2
singularities shown in Figs. 3–5 are equal to 3 at minima of ϕ̃, 1 at maxima, and 2 at
saddlepoints.
All programs modified in this way and corresponding to our five corank 2 parabolic sin-
gularities can be found by the address https://drive.google.com/drive/folders/1L5p_
HOvrBbcyBv-o4Ov4S_sZBbJytdxr?usp=sharing.
The result of their work is negative: in all five cases the program detects that component C
does not contain points satisfying either of Conditions 3.5 or 3.6. This proves Proposition 3.3,
and hence also Statement 2 of Theorem 1.14 for our five singularities of corank 2.
4 Singularities P8
In the case of singularities of corank > 2 we need to apply a different program [22]. Indeed, we
generally cannot predict the Morse indices of the newborn real critical points in the Morse birth
surgery, if we know only the data of the virtual morsification before the birth: we can predict
only parities of these indices. In the case of functions of corank 2, when the choice is between the
pairs of neighboring Morse indices (0, 1) or (1, 2) of the new critical points of essential part ϕλ of
the morsification, this is enough to obtain the exact values of the indices, but for the functions
of greater coranks we need to use a program in which the Morse indices take only the values
odd/even.
Fortunately, this is enough for our problem concerning the simplest singularities of corank 3.
Other changes and features in these cases are as follows.
4.1 Singularity class P 1
8
This class is represented by function f ≡ ϕ(ξ1, ξ2, ξ3) + ξ24 − ξ25 , where ϕ = ξ31 + ξ32 + ξ33 . We
consider the standard monomial versal deformation of this function depending on parameter
λ = (λ1, . . . , λ8),
fλ(ξ) ≡ f(ξ) + λ1 + λ2ξ1 + λ3ξ2 + λ4ξ3 + λ5ξ1ξ2 + λ6ξ1ξ3 + λ7ξ2ξ3 + λ8ξ1ξ2ξ3. (4.1)
https://drive.google.com/drive/folders/1L5p_HOvrBbcyBv-o4Ov4S_sZBbJytdxr?usp=sharing
https://drive.google.com/drive/folders/1L5p_HOvrBbcyBv-o4Ov4S_sZBbJytdxr?usp=sharing
18 V.A. Vassiliev
The stratum of points of type P8 is represented in it by the {λ8} axis. The multiplica-
tive group of positive numbers acts on parameter space R8 of this deformation: any num-
ber t sends a function fλ(x) to t3fλ(x/t). In other words, t(λ1, λ2, λ3, λ4, λ5, λ6, λ7, λ8) =(
t3λ1, t
2λ2, t
2λ3, t
2λ4, tλ5, tλ6, tλ7, λ8
)
. This action preserves the discriminant and allows us
not to take care of the smallness of the perturbations except for parameter λ8.
Let us construct a perturbation of function f , contained in component C of the complement
of the discriminant of this deformation, which will satisfy Statement 2 of Theorem 1.14.
Function ϕ has a convenient morsification ϕ̃ ≡ ξ31 + ξ32 + ξ33 − 3(ξ1 + ξ2 + ξ3) with eight real
critical points: namely, it are all points with ξi = ±1. Let be f̃ ≡ ϕ̃+ ξ24 − ξ25 . The intersection
indices of vanishing cycles in H4
(
f̃−1(0)
)
related with these critical points can be calculated by
the method of [9].
Consider family ϕ(τ) of perturbations of ϕ depending on parameter τ ∈ [0, 1],
ϕ(τ) ≡ ϕ(ξ1, ξ2, ξ3) + 6τ(ξ1ξ2 + ξ1ξ3 + ξ2ξ3)− (3 + 12τ)(ξ1 + ξ2 + ξ3). (4.2)
All functions of this family are invariant under the permutations of coordinates ξ1, ξ2, ξ3, and
have the fixed critical point (1, 1, 1). This family connects the morsification ϕ̃ ≡ ϕ(0) with
a function having a singularity of class D−4 at this point. The topological type of morsifications
does not change along this path (except for the very last point), in particular the intersection
indices of vanishing cycles and the Morse indices of critical points of the corresponding functions
f(τ) ≡ ϕ(τ) + ξ24 − ξ25 for nearly final values of τ ≈ 1 will be the same as for τ = 0.
A neighbourhood of the final function ϕ(1) in the parameter space of the versal deforma-
tion (4.1) can be considered as the parameter space of a versal deformation of the corresponding
multisingularity consisting of this point of type D−4 and four distant Morse points, so we can
deform these critical points independently.
Four Morse critical points of ϕ(τ), τ = 1−ε, tending to the collision in this point of type D−4 ,
consist of three points with signature (2, 1) and equal critical values, and one minimum point
with a slightly smaller critical value tending to −24 when τ tends to 1. We can perform two
standard Morse surgeries over them, first colliding this minimum point with one point of sig-
nature (2, 1), and then returning these critical points to the real domain as two points with
signatures (2, 1) and (1, 2) and critical values slightly greater than these at two points of sig-
nature (2, 1) not participating in these surgeries. Composition of these two surgeries realizes
the passage
A
A
AA
�
�
��
− ↔ A
A
AA
�
�
��
+
between the perturbations of a D−4 -singularity. Other four
critical points of ϕ(1−ε) can be perturbed very little during this surgery.
The resulting function has exactly three critical points of signature (2, 1), which have the
smallest critical values among all critical points of this function. Let us add a constant to this
function in such a way that these three critical values of the obtained function ϕ will remain
negative, and all the other ones become or remain positive. We claim that the component of
the complement of the discriminant of the deformation (4.2) containing function f ≡ ϕ+ ξ24− ξ25
satisfies an analog of Proposition 3.3: the local Petrovskii condition holds in this component at
all discriminant points of its boundary except for the most singular points of type P 1
8 .
The proof of this claim almost repeats that of Proposition 3.3.
In particular, the exact analogs of Proposition 3.2 and Lemma 3.4 hold in this case together
with their proofs: sums of indices of the vector field gradϕ over the critical points with negative
(respectively, positive) critical values are equal to −3 and 3.
The topological data of the obtained morsification f can be easily derived from these for
morsification ϕ(1−ε) (e.g., with the help of our program [22]). Then we substitute these data
to a modification of our program, see the concluding subroutine DATA in program mP81 from
the folder quoted in Section 3.4. This program is obtained from program [22] by almost the
New Examples of Irreducible Local Diffusion of Hyperbolic PDE’s 19
same changes by which we obtained in Section 3.4 similar programs for corank 2 singularities
from program [23], with only the following exceptions. The operator detecting cuspidal edge
(see the fifth line in (3.3)) should be slightly changed, because we generally do not know the
integer values of Morse indices of critical points. Instead we write the command
IF(PC1(NPO2).NE.0.AND.C(NPO1,NPO2).EQ.1) GOTO 7343
checking items 4 and 5 of Condition 3.6 (we use a definite choice of orientations of vanishing
cycles corresponding to real critical points, which allows us to fix the sign of their intersection
index). Also, we write INDC(q) = −1 in the concluding subprogram DATA for all numbers q of
critical values of the function ϕ corresponding to critical points with odd positive inertia indices
of quadratic parts; we write NPOZC = 3 in line 37 and activate operators L(MD + 1) = 1 and
L(MDD) = 1 in lines 115 and 116. At the beginning of this subprogram we preserve operator
N = −1 (which means that number n = 5 of variables is as odd as the number 3 considered
previously) but then write N2 = 1 (which means that number n(n − 1)/2 for n = 5 is even,
unlike the case of n = 3).
The obtained program proves that our component C of the complement of the discriminant
does not contain any morsifications satisfying Conditions 3.5 or 3.6 from Section 3.2, thus proving
Statement 2 of Theorem 1.14 for singularity class P 1
8 .
4.2 Singularity P 2
8
In this case we take initial morsification ϕλ̃(ξ1, ξ2, ξ2) + ξ24 − ξ25 , where function ϕλ̃ in three
variables is described in Section 7 of [21] (and is called f2 there). This function ϕλ̃ has four
critical points with signature of the quadratic part equal to (2, 1) and critical values equal
to 0, 0, 0, and 1. In addition it has four critical points with signature (1, 2); the values at three
of them are slightly greater than 0, and the value at the fourth one is also equal to 1. Let
us subtract a very small constant from this function, making critical values at the first three
critical points negative, and leaving the remaining five values positive. According to [21], the
intersection matrix of corresponding vanishing cycles is then expressed by formula (5) from [21]
with X = 0, Y = 1, Z = 0, W = −2. Let us substitute these topological data into the same
version of program [22] which was used for P 1
8 , again write NPOZC = 3 in line 37, and run the
obtained program mP82.
It gives us a disappointing answer: one can see the word ALARM in the print-out of this
program, followed by the data of a virtual morsification, for which Condition 3.5′ is satisfied:
namely, the critical point with the smallest positive critical value has an odd positive inertia
index of the quadratic part.
We will show now that this is not dangerous for us. Indeed, there are the following two
possible interpretations of this calculation.
1. All chains of admissible virtual surgeries leading to the virtual morsifications satisfying
Condition 3.5′, are fake (i.e., cannot be realized by chains of real surgeries arising along any
generic paths in the parameter space of a versal deformation). In this case the component of the
complement of the discriminant, which contains our morsifications, is the desired one. Indeed,
let us modify our program mP82 in such a way that it checks only Condition 3.6 but forgives
Condition 3.5′ (i.e., we disable the first two operators IF in (3.3)). The obtained program assures
us that the morsifications satisfying this Condition 3.6 indeed do not occur in this component
(while the morsifications satisfying Condition 3.5′ do not occur by our conjecture).
2. If the conjecture of the previous paragraph is wrong, and our component contains real
morsifications satisfying Condition 3.5′, then there is a neighbouring component satisfying the
assertion analogous to Proposition 3.3 (i.e., such that the localized Petrovskii condition is sat-
isfied in this component at all discriminant points of its boundary except for the most singular
20 V.A. Vassiliev
points of class P 2
8 ). Indeed, any morsification of f from our component has at least three critical
points with negative critical values and odd positive inertia indices, and also at least three points
with positive critical values and even negative inertia indices (see the proof of Proposition 3.2).
Moreover, quantities of all critical points with negative (or positive) critical values are odd (and
hence equal to 3 or 5). Therefore if we have a real morsification in this component, satisfying
Condition 3.5′, then we can add to it a constant function in such a way that the resulting mor-
sification has exactly four critical points with negative values and odd positive inertia indices,
and four critical points with positive critical values and even positive inertia indices.
Any component of the complement of the discriminant, which contains such a morsification, is
a desired one. Indeed, no Morse surgery can be performed over functions inside this component,
hence all morsifications from it have the same signatures, and Conditions 3.5′ or 3.6 cannot be
satisfied for them.
Acknowledgements
This work was supported by Russian Science Foundation grant 16-11-10316.
References
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https://doi.org/10.1007/BF01883883
https://doi.org/10.1007/BF01883883
https://doi.org/10.1007/978-3-662-06798-7
https://doi.org/10.1007/978-0-8176-8340-5
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https://doi.org/10.1007/BF02394570
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https://doi.org/10.1007/BF02028301
https://doi.org/10.1007/BF02028301
https://doi.org/10.24033/bsmf.1576
https://doi.org/10.1007/BF01405164
New Examples of Irreducible Local Diffusion of Hyperbolic PDE’s 21
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sharing.
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https://drive.google.com/file/d/1RtiqiuBBNYtf218zNRK8SIoFhA6GRb38/view?usp=sharing
1 Introduction
1.1 Hyperbolic PDE's of higher orders (after ABG70, ABG73, see also GP,APLT)
1.2 Davydova condition at non-singular points of a wavefront
1.3 Cuspidal edges and generating functions
1.4 Other simple singularities
1.5 Generating families of wavefronts and versality
1.6 Parabolic singularities and main result
2 Topological reformulations
2.1 Sharpness and the local Petrovskii condition
2.2 Functoriality of local Petrovskii classes under adjacencies of singularities
3 Proof of Statement 2 of Theorem 1.14 for parabolic singularities of corank 2
3.1 Examples
3.2 Conditions
3.3 Description of the program
3.4 Modification of program pro2 for our purposes
4 Singularities P8
4.1 Singularity class P81
4.2 Singularity P82
References
|
| id | nasplib_isofts_kiev_ua-123456789-210601 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:03:35Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Vassiliev, Victor A. 2025-12-12T10:37:40Z 2020 New Examples of Irreducible Local Diffusion of Hyperbolic PDE's. Victor A. Vassiliev. SIGMA 16 (2020), 009, 21 pages 1815-0659 2020 Mathematics Subject Classification: 35L67; 58K05; 32-04 arXiv:1909.13211 https://nasplib.isofts.kiev.ua/handle/123456789/210601 https://doi.org/10.3842/SIGMA.2020.009 Local diffusion of strictly hyperbolic higher-order PDE's with constant coefficients at all simple singularities of corresponding wavefronts can be explained and recognized by only two local geometrical features of these wavefronts. We radically disprove the obvious conjecture extending this fact to arbitrary singularities: namely, we present examples of diffusion at all non-simple singularity classes of generic wavefronts in odd-dimensional spaces, which are not reducible to diffusion at simple singular points. This work was supported by Russian Science Foundation grant 16-11-10316. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications New Examples of Irreducible Local Diffusion of Hyperbolic PDE's Article published earlier |
| spellingShingle | New Examples of Irreducible Local Diffusion of Hyperbolic PDE's Vassiliev, Victor A. |
| title | New Examples of Irreducible Local Diffusion of Hyperbolic PDE's |
| title_full | New Examples of Irreducible Local Diffusion of Hyperbolic PDE's |
| title_fullStr | New Examples of Irreducible Local Diffusion of Hyperbolic PDE's |
| title_full_unstemmed | New Examples of Irreducible Local Diffusion of Hyperbolic PDE's |
| title_short | New Examples of Irreducible Local Diffusion of Hyperbolic PDE's |
| title_sort | new examples of irreducible local diffusion of hyperbolic pde's |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210601 |
| work_keys_str_mv | AT vassilievvictora newexamplesofirreduciblelocaldiffusionofhyperbolicpdes |