On geometry of fronts in wave propagations

Susumu Tanabé

Abstract

We give a geometric descriptions of (wave) fronts in wave propagation processes. Concrete form of defining function of wave front issued from initial algebraic variety is obtained by the aid of Gauss-Manin systems associated with certain complete intersection singularities. In the case of propagations on the plane, we get restrictions on types of possible cusps that can appear on the wave front.

0. Introduction.

In this note, we intend to develop a preparatory theory toward asymptotic analysis around singular loci that appear in wave propagation process with initial data whose C^{∞} singularities are located on certain smooth algebraic variety. For this purpose, we study the geometry of the (wave) front i.e. the singular support S.S. of the solutions to the Cauchy problem where it loses C^{∞} smoothness. We give a concrete expression of defining function of the wave front in terms of the initial data (Theorem 10). We impose conditions on the initial data so that it is defined by a quasihomogeneous polynomial with isolated singularities. The key trick is to understand the (wave) front as a discriminantal set for a deformation of complete intersection singularity. This point of view has been exposed in [21].

As it is well known, the fundamental solutions of hyperbolic operators are expressed by means of certain kinds of Gel’fand-Leray integrals. Thus the geometry of singular support (i.e. front) of the solutions is reduced to the study of singular support of these types of integrals. Homological and cohomological approach to the analysis of sharp-diffuse type of these integrals has been initiated by [1]. V.A.Vasiliev [20] uses F.Pham’s approach [15] analyzing homology classes to describe the ramification character of certain kinds of Gel’fand-Leray integrals, consequently to clarify the sharp-diffuse type (see §4 for the definition) of the solutions to hyperbolic operators in the neighborhood of singular loci of the wave front, under the condition that one knows the type of singularities of the singular loci in question. Our method furnishes such existing results with a supplementary tool while we derive informations, global in space and time, uniquely from the initial data. In the sense that global behaviour of solutions to Cauchy problem is investigated, this paper is a continuation of [16], in which a similar setting of question has been studied by means of analysis of lagrangian variety.

AMS Subject Classification: Primary 35L25, 58G17, 33C75; Secondary 32S40, 78A05, 33C20.

Partially supported by Max Planck Institut für Mathematik,Bonn.

To pursue further this study, we propose to make use of the Gauss-Manin system associated with fibre bundle structure that naturally arises in integration. Our main tool is concrete expressions of the overdetermined differential systems obtained from non trivial relations between base elements of a Brieskorn lattice for complete intersection singularities (Proposition 5, Theorem 7).

Thorem 11 and the illustration given in Section 4 serve as a motivation why we propose to carry out asymptotic analysis by means of the Gauss-Manin systems. They justify our proposition in the sense that they demonstrate the possibility to interprete solutions to the Gauss-Manin systems as generalized hypergeometric functions. This procedure is supported by the general theory of the Gauss-Manin systems for isolated complete intersection singularities [6]. More systematic explanation of this situation from singularity theoretical point of view is given in [2] and [19].

Section 3 is consacrated to illustrate how one can apply the theory developed in Section 2 to situations on the plane. Our theorem 12 says that cusps that can appear on wave fronts issued from an initial front defined by an polynomial of degree k are at worst (2k − 1, 8k − 2). We do not come into a detailed study of all possible cusps in terms of coefficients (g_{1}, ⋯, g_{k−1}) of the defining polynomial. After the establishment of an algorithm, such kind of detailed study can be achieved by anybody who tries to write down conditions like that in Remark 1.

1. Preliminaries on Cauchy problem.

In this section we prepare fundamental notations and lemmata to develop our studies in further sections. First let us regard the following Cauchy problem:

(C.P.) \{ \begin{matrix}[D_{t}^{2} − Δ_{x}]u(t, x) & = & 0 \\ D_{t}u(0, x) & = & v(x) \\ u(0, x) & = & 0 \end{matrix}

where t ∈ R, x ∈ R^{n}, D_{t} = \frac{∂}{i∂t}, Δ_{x} = ∑_{j=1}^{n}(\frac{∂}{i∂x_{j}})^{2}, i = √ − 1. We will study the Cauchy problem (C.P.) under the following conditions (C.1), (C.2), (C.3) imposed on the initial data.

In order to describe these conditions, we use notation χ_{q}^{ϵ}(z) (ϵ = ±1) which stands for the following distributions:

χ_{q}^{ϵ}(z) = χ_{q}(z + i0) + ϵχ_{q}(z − i0),

where

χ_{q}(z) = z^{q}/Γ(q) q ∉ Z or q negative integer
χ_{q}(z) = \frac{z^{q}}{q} (− log z + C_{q}) q positive integer

where C_{0} = 0, C_{q} = C_{q−1} + 1/q. Remark that

\frac{d}{dz} χ_{q}(z) = χ_{q−1}(z).

(C.1) The initial data are given by a distribution of finite order with singular support (see Definition 2.2.3 of [10]) located on cotangent bundle of a smooth algebraic surface S := {x ∈ R^{n} : F (x) − s = 0} defined by a real polynomial F (x),

v(x) = g(x)χ_{q}^{ϵ}(F (x) − s)

with a smooth function g(x).

Further we shall denote the singular support of a distribution v(x) by S.S.v(x). We impose several technical conditions also. These conditions will be used so

that the reasoning on the isolated complete intersection singularities can be applied to our (C.P.)

(C.2)(Quasihomogeneity)There exists a set of integers (w_{1}, ⋯, w_{n}), that satisfies 1) w_{i} ≠ w_{j} for certain pair 1 ≤ i ≠ j ≤ n, 2) G.C.D.(w_{1}, ⋯, w_{n}) = 1 and 3) for a positive integer w(F),

( ∑_{1≤j≤n} w_{j}x_{j} \frac{∂}{∂x_{j}})F (x) = w(F)F (x),

holds.

(C.3)The following R[x]− module is of finite dimension:

R[x]/I

where I = ⟨F (x), \frac{∂F}{∂x_{1}} (x), …, \frac{∂F}{∂x_{n}} (x)⟩ ( ideal generated by the entries). Let us introduce the following notations. a)The phase function ψ(x, t, z) is defined as follows:

ψ(x, t, z) = ⟨x − z, grad_{z}F (z)⟩^{2} − t^{2}|grad_{z}F (z)|^{2}.

b) The paired oscillatory integrals introduced by [5] with phase function ψ(x, t, z) of a):

I_{p}^{ϵ}(x, t, s) = ∫_{\{F (z)=s\}} H_{p}(z)χ_{p}^{ϵ}(ψ(x, t, z))\frac{dz}{dF} ,

with regular amplitude functions of pseudo-differential operator

H_{p}(z) ∼ ∑_{r=p−2}^{−∞} h_{p,r}(z) ∈ S^{p−2}(R^{n}),

in which h_{p,r}(z) is homogeneous of order r for large values of z. One shall understand I_{p}^{ϵ}(x, t, s) as the Gel’fand-Leray integral ( see 1.5 [4]) defined on the real algebraic set S = {z ∈ R^{n}; F (z) = s}.

c) The function ϕ(x, t, s) denotes the defining function of the (wave) front Σ issued from S determined by (C.P.).

Proposition 1 The following assertions hold for solutions to the Cauchy problem (C.P.) with the notations introduced in a),b) and c) as above.

d) The solution to the Cauchy problem (C.P.) u(x, t) admits an asymptotic expansion

(1.1)  u(x, t) ∼ ∑_{j=0}^{∞} I_{−n/2+q+j}^{ϵ}(x, t, s).

That is to say, for every N >> 0 there exists such C_{N} > 0 that

(1.2)  | u(x, t) − ∑_{j=0}^{N} I_{−n/2+q+j}^{ϵ}(x, t, s) |≤ C_{N} | ϕ(x, t, s) |^{q+N+1},

in the neighbourhood of S.S.u(x, t).

Proof We give only a sketch of proof while a detailed one shall appear in [19]. First of all we show that the phase function of the integrals I_{p}^{ϵ}(x, t, s) coming into a) is given by c).

In solving the Hamilton-Jacobi equation associated with the Hamiltonian τ ±|ξ| (in a symplectic coordinate with canonical symplectic form dt∧dτ +∑_{j=1}^{n} dx_{j} ∧dξ_{j}. For the symplectic geometry see Chapter XXI [10]),

\{ \begin{matrix}ṫ & = & 1 \\ τ̇ & = & 0 \\ ẋ_{j} & = & ±\frac{∂|ξ|}{∂ξ_{j}} \\ ξ̇_{j} & = & 0 \\ x_{j}(0) & = & z_{j}, \{z ∈ R^{n}; F (z) = s\}, 1 ≤ j ≤ n, \end{matrix}

we get

(1.3) x_{j} = ±t\frac{∂|ξ|}{∂ξ_{j}} + z_{j} = ±t\frac{ξ_{j}}{|ξ|} + z_{j} with z ∈ S = \{z ∈ R^{n}; F (z) = s\}.

This means that the singularities of the solutions to (C.P.) lies on the rays (1.3). These lines are interpreted as rays issued from the initial front S = {z ∈ R^{n} : F (z) = s} in the direction determined by the Hamiltonians τ^{2} − |ξ|^{2}. Consequently they are expressed by integrals with phase

(1.4) ψ(x, t, z) = ⟨x − z, grad_{z}F (z)⟩^{2} − t^{2}|grad_{z}F (z)|^{2}.

Here we remark that for every p ∈ Q and H(z) ∈ D′(R_{z}^{n}), we have

[D_{t}^{2} − Δ_{x}] ∫_{S} H(z)(ψ(x, t, z))^{p} \frac{dz}{dF} = 0.

One can prove this equality with the aid of Gauss-Ostrogradskii-Stokes’ theorem. Thus the question is how to find a series of integrals

I_{p}^{ϵ}(x, t, s) = ∫_{S} H_{p}(z)χ_{p}^{ϵ}(ψ(x, t, z))\frac{dz}{dF} , p ∈ Q

whose suitably converging sum produces a distribution u(x, t) satisfying (C.1). The possibility of an asymptotic expansion (1.2) consisting of terms like b) can be proven by well known estimates on the stationary phase ([10], Theorem 7.7.12). More precisely, let us remind the following lemma.

Lemma 2 Let (G)^{0}(x, t, s) be a residue of a smooth function G(x, t, z) after division by an Jacobi ideal generated by \frac{∂ψ(x,t,z)}{∂z_{j}}, \frac{∂F (z)}{∂z_{j}}, 1 ≤ j ≤ n, and F (z) − s, i.e.

(1.6)G(x, t, z) = G^{0}(x, t, s) + ∑_{j=1}^{n} f_{j}(x, t, s, z)\frac{∂ψ(x,t,z)}{∂z_{j}}+

+ ∑_{j=1}^{n} g_{j}(x, t, s, z)\frac{∂F (z)}{∂z_{j}} + h(x, t, s, z)(F (z) − s),

with some smooth functions h(x, t, s, z), f_{j}(x, t, s, z), g_{j}(x, t, s, z), 1 ≤ j ≤ n.

Then for every smooth function a(z) the following asymptotic estimate with some C_{N} > 0 holds in the neighbourhood of the wave front Σ = {(x, t) ∈ R^{n+1}; (ψ)^{0}(x, t, s) = 0}:

(1.5) | ∫_{S} a(z)χ_{q}^{ϵ}(ψ(x, t, z))dz − ∑_{j=0}^{N}(L_{ψ,j}a)^{0}(x, t, s)χ_{q+n/2+j}^{ϵ}((ψ)^{0}(x, t, s)) |

< C_{N} | (ψ)^{0}(x, t, s) |^{n/2+N+1+q}

with differential operators L_{ψ,j} of degree 2j. Furthermore we have:

(L_{ψ,0}a)^{0}(x, t, s) = i^{\frac{n}{2}} (2π)^{\frac{n−1}{2}} (a)^{o}(x, t, s) | det(\frac{ψ_{zz}}{2πi})^{0}(x, t, s) |^{−1/2} .

Let us briefly sketch proof of the lemma. Malgrange’s division theorem yields the decomposition (1.6) in connexion with the fact that the follwing O_{C^{n}}− module is a finite dimensional vector space under the assumption (C.3):

\frac{Ω_{C^{n}}^{n}}{dF (z) ∧ Ω_{C^{n}}^{n−1} + dψ(0, 0, z) ∧ Ω_{C^{n}}^{n−1} + d_{z}Ω_{C^{n}}^{n−1} + F (z) ∧ Ω_{C^{n}}^{n} + ψ(0, 0, z) ∧ Ω_{C^{n}}^{n}} .

Further it suffices to apply above mentioned stationary phase method.

After Lemma 2 the function (ψ)^{0}(x, t, s) given by (1.6) for G(x, t, z) = ψ(x, t, z) in (1.4) can be considered as a defining function ϕ(x, t, s) of the wave front issued from S determined by the Hamiltonian flow (1.3). The initial condition means S = {x : ϕ(x, 0, s) = 0} which in turn entails c).

It remains to justify asymptotic estimates in b) and d). This can be achieved in view of (1.5) and well known construction of an elementary solution to strictly hyperbolic Cauchy problem (see for example [8], [9]). Hence the assertion follows. Q.E.D.

We formulate a simple lemma before introducing necessary notations.

Lemma 3 Under the assumptions (C.2), (C.3) imposed on F (x) there exists a collection of at most quadratic polynomials W_{1}(x, t), ⋯ W_{m}(x, t), with m an integer smaller than 2^{n} ∏_{i=1}^{n}(\frac{1}{w_{i}}), satisfying

(1.7)  ψ(x, t, z) = ⟨z, grad_{z}F (z))⟩^{2} + ∑_{i=1}^{m} W_{i}(x, t)z^{α^{(i)}}

for ψ(x, t, z) of (1.4). Here α^{(i)} = (α_{1}^{(i)}, ⋯, α_{n}^{(i)}) ∈ (Z_{≥0})^{n} stands for multi-indiex under restriction ∑_{j=1}^{n} w_{j}α_{j}^{(i)} < 2.

The proof follows direct calculation of (1.4). Quasihomogeneous type of F (z) yields the estimate on term number ′′m′′. Let us denote by

w(z^{α^{(i)}}) = ∑_{j=1}^{n} w_{j}α_{j}^{(i)},

quasihomogeneous weight of monomial z^{α^{(i)}} for α^{(i)} ∈ N^{n}. In terms of the quasihomogeneous weight we distinguish two cases.

Case 1 If there is a term with w(z^{α^{(i)}}) = 0, let us mark it as α^{(1)} and define a polynomial

f_{1}(y(x, t), z) = ⟨z, grad_{z}F (z))⟩^{2} + ∑_{i=1}^{m} y_{i}(x, t)z^{α^{(i)}}.

Here y_{i}(x, t) = W_{i}(x, t), 1 ≤ i ≤ m, for polynomials introduced in Lemma 3, (1.7). Case 2 If all terms of (1.7) has positive weight, we shall define

f_{1}(y(x, t), z) = ⟨z, grad_{z}F (z))⟩^{2} + ∑_{i=2}^{m+1} y_{i}(x, t)z^{α^{(i−1)}} + y_{1}.

with y_{i+1}(x, t) = W_{i}(x, t), 1 ≤ i ≤ m.

For the sake of simplicity we adopt notation μ = m for Case 1 and μ = m + 1 for Case 2.

Further we define integrals

(1.8)  I_{p}(y(x, t), s) = ∫_{S} H_{p}(z)χ_{p}(f_{1}(y(x, t), z))\frac{dz}{dF}

Hence if one denotes y′ = (y_{2}(x, t), ⋯, y_{μ}(x, t)),

I_{p}^{ϵ}(x, t, s) = I_{p}(y_{1} + i0, y′(x, t), s) + ϵI_{p}(y_{1} − i0, y′(x, t), s)

on understanding that the boundary value shall be taken at y_{1} = 0 in the above mentioned Case 2. Thus it is essential to study I_{p}(y(x, t), s) of (1.8) to estimate asymptotic behaviour of I_{p}^{ϵ}(x, t, s). From now on we shall regard the integral (1.8) as a function in variables y(x, t) = (y_{1}, y_{2}(x, t), ⋯, y_{μ}(x, t)). Therefore our main concern will be to investigate the differential equations that satisfy I_{p}(y, s) corresponding to various amplitudes H_{p}(z) with the aid of Gauss-Manin connexions associated to complete intersection singularities.

2. Gauss-Manin connexions for quasihomogeneous complete intersections.

Here we propose to study the integrals I_{p}(y, s) defined in (1.8) by means of the Gauss-Manin system associated with complete intersection singularities. In effect, it is well known that the Gauss-Manin connexion can be defined on the relative de Rham cohomology groups. Instead of that here we propose to calculate it on spaces so called Brieskorn lattices (see [4], [6]).

The formulation of this section is a modification of [17], §1 adapted to our situation. Let us observe a mapping between complex manifolds X = (C^{N+K}, 0), Y = (C^{K}, 0),

f : X → Y

that defines an isolated quasihomogeneous complete intersection singularity at the origin. That is to say, if we denote

(2.1)  X_{y} := {u ∈ X; f_{0}(u) = y_{0}, …, f_{K−1}(u) = y_{K−1}},

then dimX_{y} = N ≥ 0 and the critical set of mapping f : X_{0} → Y is isolated in X_{0}. Further we assume that polynomials f_{0}(u), ⋯, f_{K−1}(u) are quasihomogeneous i.e. there exists a collection of integers v_{1}, ⋯, v_{N+K} whose greatest common divisor equals 1 and

(v_{1}u_{1} \frac{∂}{∂u_{1}} + ⋯ + v_{N+K}u_{N+K} \frac{∂}{∂u_{N+K}})f_{ℓ}(u) = p_{ℓ}f_{ℓ}(u), ℓ = 0, 1, ⋯, K − 1,

for certain integers p_{0}, ⋯, p_{K−1}. We shall call the vector field

(2.2)  E = ∑_{i=1}^{N+K} v_{i}u_{i} \frac{∂}{∂u_{i}} ,

Euler vector field and v_{1}, ⋯, v_{N+K} (resp. p_{0}, ⋯, p_{K−1}) weights of variables u_{1}, ⋯, u_{N+K} (resp. polynomials f_{0}, ⋯, f_{K−1}) i.e. v_{1} = w(u_{1}), p_{0} = w(f_{0}) etc.

In order to calculate the Gauss-Manin connexion for isolated complete intersection singularity X_{0}, we introduce two vector spaces V and F. After Greuel-Hamm [7], we look at a space whose dimension as a vector space over C is known to be the Minor number μ(X_{0}) of singularity X_{0},

(2.3) V := \frac{Ω_{X}^{N}}{df_{0} ∧ Ω_{X}^{N−1} + ⋯ + df_{K−1} ∧ Ω_{X}^{N−1} + dΩ_{X}^{N−1} + f_{0}Ω_{X}^{N} + ⋯ + f_{K−1}Ω_{X}^{N}} .

The second one will later turn out to be isomorphic to V (see Proposition 6),

(2.4)  F := \frac{Ω_{X}^{N+1}}{df_{0} ∧ Ω_{X}^{N} + ⋯ + df_{K−1} ∧ Ω_{X}^{N} + i_{E}(Ω_{X}^{N+2})}.

Here i_{E} means the inner contraction with Euler field E defined by (2.2). The third vector space associated with the singularity X_{0} is defined as follows

(2.5)  Φ := \frac{Ω_{X}^{N+K}}{df_{0} ∧ ⋯ ∧ df_{K−1} ∧ Ω_{X}^{N} + f_{0}Ω_{X}^{N+K} + ⋯ + f_{K−1}Ω_{X}^{N+K}} .

Later we define period integrals as coupling of forms of V or of Φ with base element of homology groups H_{N}(X_{y}). We remember also the definition of the Brieskorn lattice ”H from [6],

”H = \frac{Ω_{X}^{N+K}}{df_{0} ∧ ⋯ ∧ df_{K−1} ∧ dΩ_{X}^{N−1}} ,

whose rank as O_{S}− module equals the Minor number μ(X_{0}) of the singularity X_{0}. It is easy to show

Lemma 4 For f_{0}, ⋯, f_{K−1} quasihomogeneous polynomials defining an isolated complete intersection singularity,

Φ ≅ ”H/(f_{0}, ⋯, f_{K−1}).

Thus dim_{C}Φ = μ(X_{0}).

From definitions (2.4) and (2.5) we easily deduce the following.

Proposition 5 For every form ω̃_{i} ∈ F, one has the following decomposition,

(2.6)  ω̃_{i} ∧ df_{0} ∧ ⋯^{∨ℓ} ∧ df_{K−1} = ∑_{j=1}^{μ(X_{0})} P_{ij}^{(ℓ)}ϕ_{j}(u)du

with P_{ij}^{(ℓ)} ∈ C[f_{0}, ⋯, f_{K−1}] and ϕ_{j}(u)du ∈ Φ, for 1 ≤ i, j ≤ μ(X_{0}), 0 ≤ ℓ ≤ K − 1 and df_{1} ∧ ⋯^{∨ℓ} ∧ df_{K−1} = ⋀_{i≠ℓ} df_{i}.

From [17] we remember the following

Proposition 6 Under the situation and definitions as above, the mapping,

i_{E} : F → V

induces an isomorphism. Consequently dim_{C}F = dim_{C}V = μ(X_{0}).

In view of Proposition 6, let us denote the base of V by ω_{i} such that ω_{i} = i_{E}(ω̃_{i}), ω̃_{i} ∈ F, 1 ≤ i ≤ μ(X_{0}).

To make a transition from (N + K)− forms to period integrals, we introduce meromorphic N−forms ψ_{i} satisfying

df_{0} ∧ ⋯ ∧ df_{K−1} ∧ ψ_{i} = ϕ_{i}(u)du, 1 ≤ i ≤ μ(X_{0}).

Then we derive the following relation from Proposition 6,

(2.7) dω_{j} = (di_{E} + i_{E}d)(ω̃_{j}) ≡ ℓ_{j}ω̃_{j} ≡ ℓ_{j}(∑_{q=1}^{μ(X_{0})} P_{jq}^{(0)}df_{0} ∧ ψ_{q} + ⋯ +(−1)^{K−1} ∑_{q=1}^{μ(X_{0})} P_{jq}^{(K−1)}df_{K−1} ∧ ψ_{q}) mod((df_{0}, ⋯, df_{K−1})Ω_{X}^{N}, dΩ_{X}^{N}),

where ℓ_{j} denotes weight of the form ω_{j}. Hence,

(2.8)  ω_{j} = i_{E}(ω̃_{j}) ≡

∑_{i=0}^{K−1}(−1)^{i}[ ∑_{q=1}^{μ(X_{0})} P_{jq}^{(i)}p_{i}f_{i}ψ_{q} − ∑_{q=1}^{μ(X_{0})} P_{jq}^{(i)}df_{i} ∧i_{E}(ψ_{q})]mod((df_{0}, ⋯, df_{K−1})Ω_{X}^{N−1}, dΩ_{X}^{N−1}).

As a consequence

(2.9) dω_{j} = ∑_{q=1}^{μ(X_{0})}[∑_{i=0}^{K−1}(−1)^{i}(d(p_{i}P_{jq}^{(i)}f_{i}) − w(ψ_{q})P_{jq}^{(i)}df_{i})] ∧ ψ_{q} + ∑_{q=1}^{μ(X_{0})}[∑_{i=0}^{K−1}(−1)^{i}p_{i}P_{jq}^{(i)}f_{i}] ∧ dψ_{q}, mod((df_{0}, ⋯, df_{K−1})Ω_{X}^{N}, dΩ_{X}^{N}),

where w(ψ_{q}), quasihomogeneous weight of form ψ_{q}. The expression (2.9) can be simplified if one lets them couple with a vanishing N−cycle, say γ(y) and attains non trivail relations between integrals ∫_{γ(y)} ψ_{q}, instead of those between forms. One defines so called period integral I_{ϕ_{q},γ(y)}(y) taken along a vanishing cycle γ(y) whose ambiguity in homology class H_{N}(X_{y}, Z) we do not care for the moment,

(2.10) I_{ϕ_{q},γ(y)}(y) := ∫_{γ(y)} ψ_{q} = (\frac{1}{2πi})^{K} ∫_{∂γ(y)} \frac{df_{0}∧···∧df_{K−1}∧ψ_{q}}{(f_{0}−y_{0})···(f_{K−1}−y_{K−1})}

= (\frac{1}{2πi})^{K} ∫_{∂γ(y)} \frac{ϕ_{q}(u)du}{(f_{0} − y_{0}) ⋯ (f_{K−1} − y_{K−1})},

where ∂γ(y) is a cycle obtained by the aid of Leray’s coboundary operator ∂. One may consult a booklet by F.Pham [15], or a book by V.A.Vasiliev [20] on the coboundary operator.

From (2.8) we can deduce

(2.11)  ∫_{γ(y)} ω_{j} = ∑_{q=1}^{μ(X_{0})}[ ∑_{i=0}^{K−1}(−1)^{i}p_{i}y_{i}P_{jq}^{(i)}(y)]I_{ϕ_{q},γ(y)}(y).

It is easily seen from the following evident equality in view of definition (2.10),

∫_{∂γ(y)} \frac{df_{0} ∧ ⋯ ∧ df_{K−1}}{(f_{0} − y_{0}) ⋯ (f_{K−1} − y_{K−1})} ∧ df_{i} ∧ i_{E}(ψ_{q}) = 0, 0 ≤ i ≤ K − 1.

Let us compare the relation

(2.12) d ∫_{γ(y)} ω_{j} = ℓ_{j} ∑_{q=1}^{μ(X_{0})}[∑_{i=0}^{K−1}(−1)^{i}P_{jq}^{(i)}(y)dy_{i}]I_{ϕ_{q},γ(y)}(y),

obtained from (2.7) and (2.11). As a result we get equations between I_{ϕ_{q}}(y) and \frac{∂}{∂y_{ℓ}} I_{ϕ_{q}}, 0 ≤ ℓ ≤ K − 1, (we omit to specify γ(y) except necessary cases),

\frac{∂}{∂y_{ℓ}} [∑_{q=1}^{μ(X_{0})} ∑_{i=0}^{K−1}(−1)^{i}p_{i}y_{i}P_{jq}^{(i)}I_{ϕ_{q}}] = ℓ_{j} ∑_{q=1}^{μ(X_{0})}[∑_{i=0}^{K−1}(−1)^{i}P_{jq}^{(i)}(y)dy_{i}]I_{ϕ_{q}}(y), 1 ≤ j ≤ μ(X_{0}).

Thus we have obtained a system of differential equations to be understood as the Gauss-Manin connexion of the singularity X_{0}. To state the theorem in a simple form, we introduce the following notations: I_{V} = (∫_{γ(y)} ω_{1}, ⋯, ∫_{γ(y)} ω_{μ(X_{0})}), I_{Φ} = (I_{ϕ_{1},γ(y)}(y), ⋯, I_{ϕ_{μ(X_{0})},γ(y)}(y)), i.e. vectors of integrals taken along a certain vanishing cycle γ(y). We define several other μ(X_{0}) × μ(X_{0}) matrices as follows,

L_{V} = diag(ℓ_{1}, ⋯, ℓ_{μ(X_{0})})

with ℓ_{i} = w(ω_{i}), P^{(0)}(y) = (P_{jq}^{(0)}(y)), ⋯, P^{(K−1)}(y) = (P_{jq}^{(K−1)}(y)), 1 ≤ j, q ≤ μ(X_{0}), matrices consisting of elements defined in (2.6).

In summing up the above arguments and the theory due to Greuel [6], we obtain the following.

Theorem 7 1). For a quasihomogeneous mapping

f : X → Y

with isolated complete intersection singularities of dimension N like (2.1), the Gauss-Manin system satisfied by I_{Φ} is described as follows:

(2.13) d[∑_{i=0}^{K−1}(−1)^{i}p_{i}y_{i}P^{(i)}(y)I_{Φ}] = L_{V} [∑_{i=0}^{K−1}(−1)^{i}P^{(i)}(y)dy_{i}]I_{Φ}.

2). The critical value D (singular locus of the system (2.13)) of deformation X_{y} is given by D = {y ∈ Y : Δ(y) = 0} where

(2.14)  Δ(y) = det( ∑_{i=0}^{K−1}(−1)^{i}p_{i}y_{i}P^{(i)}(y)).

Let us return to the problem (C.P.) of §1. Our main concern is to understand the integral (1.8) as a sum of integrals like (2.10) for certain mapping f. To adapt our (C.P.) to the scheme explained before Theorem 7, we treat the following mapping f : X → Y for X = (C_{u}^{n+μ}, 0), Y = (C_{y}^{μ+1}, 0). Concretely, it is defined as follows,

(2.15) \{ \begin{matrix}f_{0}(u) & = & F (z) & = & y_{0} \\ f_{1}(u) & = & z_{n+μ}^{P} + ⟨z, grad_{z}F (z))⟩^{2} + ∑_{i=1}^{μ−1} z_{n+i}z^{α^{(i)}} & = & y_{1} \\ f_{2}(u) & = & z_{n+1} & = & y_{2} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ f_{i+1}(u) & = & z_{n+i} & = & y_{i+1} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ f_{μ}(u) & = & z_{n+μ−1} & = & y_{μ} \end{matrix}

with notation z = (z_{1}, ⋯, z_{n}), z′ = (z_{n+1}, ⋯, z_{n+μ−1}), u = (z, z′, z_{n+μ}). Here the power P is an integer that corresponds to the denominator of q ∈ Q.

Lemma 8 For F (z) under conditions (C.2), (C.3), the mapping (2.15) (f_{0}, ⋯, f_{μ}) defines an isolated quasihomogeneous complete intersection singularity,

X_{0} = {u ∈ X : f_{0}(u) = … = f_{μ}(u) = 0}.

Namely O_{X}− module

V = \frac{Ω_{X}^{n+μ}}{f_{0}Ω_{X}^{n+μ}+f_{1}Ω_{X}^{n+μ}+∑_{i=1}^{μ} z_{i+n}Ω_{X}^{n+μ}+dF ∧Ω_{X}^{n+μ−1}+df_{1}∧Ω_{X}^{n+μ−1}+∑_{i=1}^{μ} dz_{i+n}∧Ω_{X}^{n+μ−1}} ,

is a finite dimensional vector space.

Proof The complete intersection property follows from the fact that two polynomials F (z) and ⟨z, grad_{z}F (z)⟩ are of the same quasihomogeneous weight but with different coefficients. This is a consequence of (C.2), 1) which supposes that F (z) is not a homogeneous polynomial. The condition (C.3) entails immediately the finite dimensionality of V. Q.E.D.

To see that the components of I_{Φ} defined for the mapping (2.15) give rise to integrals of type (1.8), we prepare the following.

Lemma 9 1). Let us denote

Φ(z, z′) = \frac{Ω_{X}^{n+μ+1}}{df_{0} ∧ … df_{μ} ∧ Ω_{X}^{n} + ∑_{i=0}^{μ} f_{i}Ω_{X}^{n+μ+1}} |_{z_{n+μ}=0}.

Then the following natural isomorphism holds

Φ ≅ Φ(z, z′) ⊗ (C[z_{n+μ}]/ < z_{n+μ}^{P} >).

2). For ∂γ_{n−2} a Leray coboundary of a vanishing cycle γ_{n−2} ∈ H_{n−2}(X_{y} |_{z_{n+μ}=0} , Z) one can choose a corresponding vanishing cycle γ̃_{n−1} ∈ H_{n−1}(X_{y}, Z) such that an equality

(2.16) ∫_{∂γ_{n−2}} ϕ(z)(f_{1}(z, y_{2}, ⋯, y_{μ}, 0) − y_{1})^{\frac{r+1}{P} −1} \frac{dz}{dF} =

= ϵ(\frac{1}{2πi})^{μ} ∫_{∂γ̃_{n−1}} ϕ(z)z_{n+μ}^{r} \frac{du}{df_{0} ∧ ⋯ ∧ df_{μ}}

holds, where ϵ ∈ C^{×} such that ϵ^{P} = 1. Furthermore, the cycle ∂γ̃_{n−1} is topologically equivalent to a product of a small circle on complex z_{n+μ}− plane and ∂γ_{n−2}.

Proof The statement 1) is evident. The statement 2) is an integral version of statement 1), which can be shown by means of equality (2.10). Q.E.D.

Thus the singular locus of the integral (1.8) can be given by that of

∫_{∂γ̃_{n−1}} ϕ(z)z_{n+μ}^{r} \frac{du}{df_{0} ∧ ⋯ ∧ df_{μ}}

with γ̃_{n−1} ∈ H_{n−1}(X_{y}, Z) after substitution y_{1} = −W_{1}(x, t) (Case 1 after Lemma 3), or y_{1} = 0 (Case 2 after Lemma 3), y_{i} = W_{i}(x, t), 2 ≤ i ≤ μ. Let us remind that we denoted the quasihomogeneous weight of function f_{i} by p_{i}, 0 ≤ i ≤ μ. We define matrices P^{(i)}(y), 2 ≤ i ≤ μ for the mapping (2.15) after the master (2.6) and Theorem 7. Combining Theorem 7 with Lemma 9, we obtain the following.

Theorem 10 The defining equation of wave front (Proposition 1, c)) is given by the following polynomial:

(2.17)  ϕ(x, t, s) = det(∑_{i=0}^{μ}(−1)^{i}p_{i}y_{i}P^{(i)}(y)) |_{y_{0}=s,y_{i}=W_{i}(x,t),2≤i≤μ} .

Here the restriction shall be imposed in accordance with two cases treated just after Lemma 3 i.e. y_{1} = −W_{1}(x, t) in Case 1 and y_{1} = 0 in Case 2.

3.Fronts in the wave propagation on the plane.

In this section we illustrate how effectively our machinery explained in §2 works in studying wave propagation process on the plane (i.e. n = 2). Our main argument consists in the fact that the integrals (1.8) can be expressed as a sum of solutions to A_{μ} type the Gauss-Manin system. We shall use further notations from the former sections if they can be understood without special modifications.

Let us suppose

(3.1)  F (z) = z_{1} − G(z_{2}),

for a polynomial of degree k with real coefficients,

(3.2)  G(z_{2}) = z_{2}^{k} + g_{1}z_{2}^{k−1} + g_{2}z_{2}^{k−2} + … + g_{k−1}z_{2}.

Without loss of generality, we can choose as the initial wave front S = {x ∈ R^{2} : F (x) = 0}, i.e. s = 0. As the phase function described in Proposition 1, c) restrictricted on S we get,

(3.3)ψ(x, t, z)|_{S} = ⟨(x_{1} − G(z_{2}), x_{2} − z_{2}), (−1, G′(z_{2}))⟩^{2} − t^{2}(1 + G′(z_{2})^{2})

= (k − 1)^{2}[z_{2}^{2k} + y_{1}(x, t)z_{2}^{2k−1} + y_{2}(x, t)z_{2}^{2k−2} + … + y_{2k−1}(x, t)z_{2} + y_{2k}(x, t)],

where y_{i}(x, t), 1 ≤ i ≤ 2k are polynomials at most quadratic in (x, t) variables defined as those which make (3.3) valid. More precisely, y_{1}(x, t), …, y_{k}(x, t) are linear in x_{1}, x_{2}, t^{2}. The remaining coefficients y_{k+1}(x, t), …, y_{2k}(x, t) are linear in x_{1}, x_{2}, t^{2}, x_{1}x_{2}, x_{1}^{2}, x_{2}^{2}. Especially,

y_{1}(x, t) = (k − 1)^{−2}(2x_{2} + k − (2k − 1)g_{1}),
y_{2k}(x, t) = (k − 1)^{−2}((x_{1} + g_{k−1}x_{2})^{2} − t^{2}(1 + g_{k−1}^{2})).

Thus, in our situation the integral (1.8) is reduced to

(3.4) I_{λ}(y(x, t)) = ∫_{γ} H(G(z_{2}), z_{2})[z_{2}^{2k} + y_{1}(x, t)z_{2}^{2k−1} + … + y_{2k}(x, t)]^{λ}dz_{2},

for certain vanishing cycle γ of ψ(x, t, z)|_{S} = 0. Here we took into account the equivalence relation,

\frac{dz_{1} ∧ dz_{2}}{dF} ≡ dz_{2}.

Let us consider a series of integrals

(3.5)  K_{i}^{λ}(y) = ∫_{γ} z^{i}f_{1}(y, z)^{λ}dz,  i = 0, …, 2k + 1,

where we used notation

f_{1}(y, z) := (k − 1)^{−2}ψ(x, t, z)|_{S} = z_{2}^{2k} + y_{1}z_{2}^{2k−1} + y_{2}z_{2}^{2k−2} + … + y_{2k}.

It is evident that our object of investigation (3.4) can be interpreted as a sum of these integrals (3.5) with C[y] coefficients. That is to say, singular locus D = {y ∈ Y : Δ(y) = 0} with Δ(y) the discriminant of f_{1}(y, z) coincides with singular support of (3.4) if we consider the latter a function in y variables. Thus the analysis of integrals (3.5) entails not only geometric but also analytic properties of C^{∞} singularities of (3.4).

We remind the following statement that gives complete description of the overdetermined system (the Gauss-Manin connexion with A_{2k−1} singularities) satisfied by the integrals K_{i}^{λ}(y).

Theorem 11 ([12], [13]) 1)The integrals K_{0}^{λ}(y), …, K_{4k−2}^{λ}(y) satisfy the following over-determined system of differential equations:

(3.6)  ∑_{ℓ=0}^{2k−1} y_{2k−ℓ} \frac{∂}{∂y_{2k−ℓ}} K_{i}^{λ} + \frac{∂}{∂y_{2k}} K_{2k+i}^{λ} = λK_{i}^{λ}, 0 ≤ i ≤ 2k − 2,

(3.7) ∑_{ℓ=1}^{2k−1} ℓy_{2k−ℓ} \frac{∂}{∂y_{2k−ℓ}} K_{j}^{λ} + 2k \frac{∂}{∂y_{2k}} K_{2k+j}^{λ} = −(j + 1)K_{j}^{λ}, −1 ≤ j ≤ 2k − 2.

2) In the form of matrix, the above systems (3.6), (3.7) admit an expression,

(3.8)  Π(y) · b⃗ = a⃗,

where

Π(y) = [ \begin{matrix}y_{2k} & y_{2k−1} & ⋯ & y_{2} & y_{1} & 1 & ⋯ & 0 & 0 & 0 \\ 0 & y_{2k} & ⋯ & y_{3} & y_{2} & y_{1} & ⋯ & 0 & 0 & 0 \\ 0 & 0 & ⋯ & y_{4} & y_{3} & y_{2} & ⋯ & 0 & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ & ⋯ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & ⋯ & y_{2k−1} & y_{2k−2} & y_{2k−3} & ⋯ & y_{1} & 1 & 0 \\ 0 & 0 & ⋯ & y_{2k} & y_{2k−1} & y_{2k−2} & ⋯ & y_{2} & y_{1} & 1 \\ y_{2k−1} & 2y_{2k−2} & ⋯ & (2k − 1)y_{1} & 2k & 0 & ⋯ & 0 & 0 & 0 \\ 0 & y_{2k−1} & ⋯ & (2k − 2)y_{2} & (2k − 1)y_{1} & 2k & ⋯ & 0 & 0 & 0 \\ ⋮ & ⋮ & ⋯ & ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 2y_{2k−2} & 3y_{2k−3} & 4y_{2k−4} & ⋯ & 2k & 0 & 0 \\ 0 & 0 & ⋯ & y_{2k−1} & 2y_{2k−2} & 3y_{2k−3} & ⋯ & (2k − 1)y_{1} & 2k & 0 \\ 0 & 0 & ⋯ & 0 & y_{2k−1} & 2y_{2k−2} & ⋯ & (2k − 2)y_{2} & (2k − 1)y_{1} & 2k \end{matrix} ]

a⃗ =^{t} (λK_{0}^{λ}, λK_{1}^{λ}, ⋯, λK_{2k−2}^{λ}, 0, −K_{0}^{λ}, −2K_{1}^{λ}, ⋯, −(2k − 1)K_{2k−2}^{λ}),

b⃗ =^{t} (\frac{∂}{∂y_{2k}} K_{0}^{λ}, \frac{∂}{∂y_{2k}} K_{1}^{λ}, ⋯, \frac{∂}{∂y_{2k}} K_{4k−2}^{λ}).

To be remarked, \frac{∂}{∂y_{2k−i}} K_{j}^{λ}(y) = \frac{∂}{∂y_{2k−j}} K_{i}^{λ}(y).

In [18] we derived a partial differential operator of Fuchsian type annihilating K_{j}^{λ}(y).

By the aid of the above theorem, we get our main result on 2 dimensional wave propagation.

Let us denote by Δ(y) the discriminant of f_{1}(z, y) concretely calculated by

(3.9)  Δ(y) = detΠ(y),

for the matrix Π(y) of (3.8). In literature one often calls Π(y) the resultant of Sylvester.

Theorem 12 1). The wave front Σ of (C.P.) with initial front S = {x ∈ R^{2} : F (x) = 0} given by (3.1), (3.2), admits the following expression.

Σ = {(x, t) ∈ R^{3} : Δ(y(x, t)) = 0},

where y(x, t) = (y_{1}(x, t), ⋯, y_{2k}(x, t)) defined in (3.3).

2). Except ordinary double points, Σ has cusps of type at most (2k − 1, 8k − 2). In other words if under a suitable local diffeomorphism (x_{1}, x_{2}) → (X_{1}, X_{2}) the restriction of wave front at certain moment Σ|_{t=const.} admits as its local equation

(3.10)  {X_{1}(x)^{p_{1}} − X_{2}(x)^{p_{2}} = 0}, p_{1} ≤ p_{2},

then these powers satisfy p_{1} ≤ 2k − 1, p_{2} ≤ 8k − 2.

Proof The statement 1) follows if one applies Theorem 11 to our integral (3.5) then replace variables y by y(x, t) defined in (3.3).

To see (3.10), one shall examine the matrix Π(y). After (3.8), Δ(y) = detΠ(y) contains the highest in variable y_{2k} term y_{2k}^{2k−1}, while Δ(y) is quasihomogeneous with respect to the weight w(y_{i}) = i, 1 ≤ i ≤ 2k. According to this weight, every term of Δ(y) must be of weight 2k(2k − 1). Remark that

y_{2k}(x, t) = L(x, t)L(x, −t)

for

L(x, t) = (k − 1)^{−1}(x_{1} + g_{k−1}x_{2} − (1 + g_{k−1}^{2})^{1/2}t).

Hence L(x, t) ≠ L(x, −t) out of t = 0 and y_{2k}(x, t) can never be a constantly zero function for any choice of (g_{1}, ⋯, g_{k−1}).

The terms that may appear in the expression Δ(y) must be of degree less than 4k − 1 after concrete form of Π(y). According to the remark after (3.3), y_{i}(x, t) are at most quadratic polynomials. Suppose that there exists a set of indices 1 ≤ α_{j} ≤ 2k − 1, {α_{1}, ⋯, α_{ℓ}} ⊂ {1, …, 2k − 1}

y_{α_{j}}(x, t) = c_{α_{j}}X^{2}(x, t),

for a linear function X(x, t). Such cases do really happen if the coefficients of G(z_{2}), (g_{1}, …, g_{k−1}) satisfy certain algebraic equations of degree 4 (see Remark 1 below). Suppose that for X(x, t) in question such a set {α_{1}, ⋯, α_{M}} is maximal i.e. for complementary indices {β_{1}, ⋯, β_{2k−M−1}} = {1, …, 2k − 1}\ {α_{1}, ⋯, α_{M}}, y_{β_{i}}(x, t), 1 ≤ i ≤ 2k − M − 1 are not divided by X^{2}(x, t). Then evidently,

Δ(y) = y_{2k}^{2k−1} − const.X^{2(4k−1)}(x, t) + R(X, y_{β_{1}}, ⋯, y_{β_{2k−M−1}}, y_{2k}),

where R(X, y_{β_{1}}, ⋯, y_{β_{2k−M−1}}, y_{2k}) is a polynomial of degree ≤ 2(4k − 2) in X and ≤ (4k − 2) in y_{β_{1}}, ⋯, y_{β_{2k−M−1}}, y_{2k}. The coefficients of the polynomial R can vary in dependence of (g_{1}, ⋯, g_{k−1}). From the above expression of Δ(y) the statement 2) follows. Q.E.D.

Remark 1 On the divisibility of certain y_{α_{j}}(x, t) by a square of another function, one can calculate, in principle, the condition to be satisfied by (g_{1}, ⋯, g_{k−1}). For example a sufficient condition so that y_{ℓ}(x, t_{0}), ℓ < k be expressed as a square of a linear function in x_{2} at t = t_{0} ≠ 0 sounds,

(∑_{i+j=ℓ−1}(2k − ℓ − 2(k − i)(k − j))g_{i}g_{j})^{2} = 4[∑_{i+j=ℓ−2}(k − i)(k − j)g_{i}g_{j}][∑_{i+j=ℓ}(2(k − i)(k − j) − 2k + ℓ + 1)g_{i}g_{j} − t_{0}^{2} ∑_{i+j=ℓ−2}(k − i)(k − j)g_{i}g_{j}].

Conditions corresponding to other cases have similar form. We omit them in view of purely technical character of such conditions.

4. An illustration by Gauss’ hypergeometric function.

In this section, we mention a little about analytic aspects of our theory. Namely we illustrate how to determine sharp-(diffuse) type at singular locus of fronts ((2,3) cusp ) in an adequate way by means of the Gauss-Manin system of A_{2} type that admits solutions expressed through Gauss hypergeometric functions.

We say that a distribution u(x) with C^{∞} singular locus on Σ = {x : ϕ(x) = 0} has sharp (resp. diffuse) front from the side Σ_{+} = {x : ϕ(x) > 0} in approaching to a point of Σ if and only if there exists (resp. does not exist) a C^{∞} extension of u(x) across Σ. The distribution u(x) has lacuna inside Σ_{+} iff u(x) ≡ 0 on Σ_{+}, see [14]. This notion has been introduced in connexion with the situation that a diffuse front can never be a front of a lacuna component. On the other hand, sharpness of a front is a necessary condition for presence of lacuna. As for details and typical examples, see [5].

Though the degree of a phase function f_{1}(y(x, t), z) defined in (3.3) cannot be odd, an observation in terms of A_{2} type singularity is useful because of the adjacency property of A_{2} singularity to almost all more complicated singularities. In other words, one can expect that the solutions to (C.P.) are expressed by means of period integrals of A_{2} type around the generic points on the singular loci of the wave front. Let us look at the integrals

K_{i}^{λ}(y) = ∫_{Γ} z^{i}(z^{3} + y_{1}z + y_{0})^{λ}dz, i = 0, 1,

where Γ a double loop of Pochhammer (cf. [3]) i.e. one dimensional regularized cycle. Then from Theorem 11 (or from [18]) one can deduce the following differential equations for K_{0}^{λ}(y), K_{1}^{λ}(y).

(4.1)_{0}  [(27y_{0}^{2} + 4y_{1}^{3})(\frac{∂}{∂y_{0}} )^{2} + 27(1 − 2λ)y_{0} \frac{∂}{∂y_{0}} + 27(λ^{2} − 1/9)]K_{0}^{λ}(y) = 0

(4.1)_{1}  [(27y_{0}^{2} + 4y_{1}^{3})(\frac{∂}{∂y_{0}} )^{2} + 27(1 − 2λ)y_{0} \frac{∂}{∂y_{0}} + 27(λ^{2} − 4/9)]K_{1}^{λ}(y) = 0

The equation (4.1)_{0} has two independent solutions U_{1,λ}(y), U_{2,λ}(y) expressed by Gauss’ hypergeometric function:

\begin{aligned}U_{1,λ}(y) & = & y_{1}y_{0}^{−λ+1/3} ·_{2} F_{1}(\frac{−λ}{2} − \frac{1}{6}, \frac{−λ}{2} + \frac{1}{3}, \frac{2}{3}, \frac{−4y_{1}^{3}}{27y_{0}^{2}} ) \\ U_{2,λ}(y) & = & y_{1}^{2}y_{0}^{−λ−1/3} ·_{2} F_{1}(\frac{−λ}{2} + \frac{1}{6}, \frac{−λ}{2} + \frac{2}{3}, \frac{4}{3}, \frac{−4y_{1}^{3}}{27y_{0}^{2}} ) \end{aligned}

The solutions to (4.1)_{1} are written down in a similar way. As these two solutions U_{1,λ}(y), U_{2,λ}(y) correspond to two independent basis γ_{1}, γ_{2} of the twisted homology group, we can write

U_{j,λ}^{ϵ}(y) = ∫_{γ_{j}} χ_{λ}^{ϵ}(z^{3} + y_{1}z + y_{0})dz. (j = 1, 2)

These solutions ramify around the singular locus Σ = {y ∈ R^{2} : Δ_{2}(y) = 0}. Here Δ_{2}(y) = 27y_{0}^{2} +4y_{1}^{3}. We shall remark that ∫_{γ_{i}} z^{j}χ_{λ}^{ϵ}(z^{3} +y_{1}z +y_{0})dz and U_{i,λ}^{ϵ}(y) have same sharp diffuse-type because of a relation

(\frac{∂}{∂y_{1}} )^{j}U_{i,λ}^{ϵ}(y) = (\frac{∂}{∂y_{0}} )^{j} ∫_{γ_{i}} z^{j}χ_{λ}^{ϵ}(z^{3} + y_{1}z + y_{0})dz.

It shall be noted that sharp-diffuse type does not change by differentiation or integration of a distribution. To study the asymptotic behavior of U_{i,λ}^{ϵ} around Σ, we remind here the connexion formula for Gauss’ hypergeometric function:

2F_{1}(α, β, γ, x) = \frac{Γ(γ)Γ(γ−α−β)}{Γ(γ−α)Γ(γ−β)} ·_{2} F_{1}(α, β, α + β − γ + 1, 1 − x) +\frac{Γ(γ)Γ(α+β−γ)}{Γ(α)Γ(β)}(1 − x)^{γ−α−β} ·_{2} F_{1}(γ − α, γ − β, γ − α − β + 1, 1 − x).

These are based on a relation between 4 expressions from Kummer’s 24 integrals, that goes back to Gauß(cf. [3],Chapitre I (10)). In applying this connexion formula to our U_{1,λ}(y) and U_{2,λ}(y), we get

2F_{1}(−\frac{λ}{2} − \frac{1}{6}, −\frac{λ}{2} + \frac{1}{3}, \frac{2}{3}, \frac{−4y_{1}^{3}}{27y_{0}^{2}} ) =

= \frac{Γ( \frac{2}{3} )Γ(λ+ \frac{1}{2} )}{Γ( \frac{λ}{2} + \frac{5}{6} )Γ( \frac{λ}{2} + \frac{1}{3} )} ·_{2} F_{1}(−\frac{λ}{2} − \frac{1}{6}, −\frac{λ}{2} + \frac{1}{3}, −λ + \frac{1}{2}, 1 + \frac{4y_{1}^{3}}{27y_{0}^{2}} )

+\frac{Γ( \frac{2}{3} )Γ(−λ− \frac{1}{2} )}{Γ(− \frac{λ}{2} − \frac{1}{6} )Γ(− \frac{λ}{2} + \frac{1}{3} )}(1 + \frac{4y_{1}^{3}}{27y_{0}^{2}} )^{λ+ \frac{1}{2}} ·_{2} F_{1}(\frac{λ}{2} + \frac{5}{6}, \frac{λ}{2} + \frac{1}{3}, λ + \frac{3}{2}, 1 + \frac{4y_{1}^{3}}{27y_{0}^{2}} ),

2F_{1}(−\frac{λ}{2} + \frac{1}{6}, −\frac{λ}{2} + \frac{2}{3}, \frac{4}{3}, \frac{−4y_{1}^{3}}{27y_{0}^{2}} ) =

= \frac{Γ( \frac{4}{3} )Γ(λ+ \frac{1}{2} )}{Γ( \frac{λ}{2} + \frac{3}{2} )Γ( \frac{λ}{2} + \frac{2}{3} )} ·_{2} F_{1}(−\frac{λ}{2} + \frac{1}{6}, −\frac{λ}{2} + \frac{2}{3}, −λ + \frac{1}{2}, 1 + \frac{4y_{1}^{3}}{27y_{0}^{2}} )

+\frac{Γ( \frac{4}{3} )Γ(−λ−12)}{Γ(− \frac{λ}{2} + \frac{1}{6} )Γ(− \frac{λ}{2} + \frac{2}{3} )}(1 + \frac{4y_{1}^{3}}{27y_{0}^{2}} )^{λ+ \frac{1}{2}} ·_{2} F_{1}(\frac{λ}{2} + \frac{3}{2}, \frac{λ}{2} + \frac{2}{3}, λ + \frac{3}{2}, 1 + \frac{4y_{1}^{3}}{27y_{0}^{2}} ).

It is evident that the second term in the above expressions ramifies around the singular locus Σ, if λ is not a half integer. Hence we obtain the following four classifications of sharp-diffuse type. We write down here the classification only for U_{1,λ}^{ϵ} while U_{2,λ}^{ϵ} behaves in a parallel manner.

Case 1. λ = −1, ϵ = −1.

U_{1,−1}^{−}(y) = 0 in y_{0} > 0, Δ_{2}(y) > 0

U_{1,−1}^{−}(y) = const.y_{1}|y_{0}|^{4/3} ·_{2} F_{1}(\frac{1}{3}, \frac{5}{6}, \frac{2}{3}; −\frac{4y_{1}^{3}}{27y_{0}^{2}} ) in y_{0} < 0, Δ_{2}(y) > 0

U_{1,−1}^{−}(y) = const.y_{1}|y_{0}|^{7/3}Δ_{2}(y)^{−1/2} ·_{2} F_{1}(1, \frac{1}{6}, \frac{1}{2}; 1 + \frac{4y_{1}^{3}}{27y_{0}^{2}} ) in y_{0} ≠, Δ_{2}(y) < 0 Hence U_{1,−1}^{−}(y) has sharp front in approaching to a generic point on Σ from the

side Δ_{2}(y) > 0, while it has diffuse front in approaching to the singular point of Σ (i.e. the origin). It has diffuse front from the side Δ_{2}(y) < 0.

Case 2. λ = −1, ϵ = +1.

U_{1,+1}^{−}(y) =

= const.y_{1}|y_{0}|^{7/3}Δ_{2}(y)^{−1/2} ·_{2} F_{1}(1, \frac{1}{6}, 1; 1 + \frac{4y_{1}^{3}}{27y_{0}^{2}} ) + hol(1 + \frac{4y_{1}^{3}}{27y_{0}^{2}} )

in y_{0} ≠ 0, Δ_{2}(y) > 0

U_{1,+1}^{−}(y) = 0 in Δ_{2}(y) < 0

Hence U_{1,+1}^{−}(y) has diffuse front in approaching to Σ from the side Δ_{2}(y) > 0. It has always lacuna ( a fortiori sharp front) from the side Δ_{2}(y) < 0.

Case 3. λ = −\frac{3}{2}, ϵ = −1.

U_{1,−3/2}^{−}(y) = 0 in y_{0} > 0

U_{1,−3/2}^{−}(y) = const.y_{1}|y_{0}|^{11/6}Δ_{2}(y)^{−1} ·_{2} F_{1}(\frac{3}{4}, \frac{−1}{12}, 0; 1 + \frac{4y_{1}^{3}}{27y_{0}^{2}} ) in y_{0} < 0 Hence U_{1,−3/2}^{−}(y) has sharp front in approaching to a generic point of Σ from

any side if y_{0} > 0. It has diffuse front from any side of Σ if y_{0} < 0. In approaching to a singular point of Σ, U_{1,−3/2}^{−}(y) has always diffuse front.

Case 4. λ = −\frac{3}{2}, ϵ = +1.

U_{1,−3/2}^{+}(y) =

= c_{1}y_{1}y_{0}^{11/6}p.v.(1 + \frac{4y_{1}^{3}}{27y_{0}^{2}} )^{−1} ·_{2} F_{1}(\frac{3}{4}, \frac{−1}{12}, 0; 1 + \frac{4y_{1}^{3}}{27y_{0}^{2}} )+

+c_{2}y_{1}y_{0}^{11/6} ·_{2} F_{1}(\frac{7}{12}, \frac{13}{12}, 2; 1 + \frac{4y_{1}^{3}}{27y_{0}^{2}} ) in y_{0} > 0. U_{1,−3/2}^{+}(y) =

= c′_{1}y_{1}y_{0}^{11/6}δ(1 + \frac{4y_{1}^{3}}{27y_{0}^{2}} ) ·_{2} F_{1}(\frac{3}{4}, \frac{−1}{12}, 0; 1 + \frac{4y_{1}^{3}}{27y_{0}^{2}} )

+c′_{2}y_{1}y_{0}^{11/6} ·_{2} F_{1}(\frac{7}{12}, \frac{13}{12}, 2; 1 + \frac{4y_{1}^{3}}{27y_{0}^{2}} ) in y_{0} < 0.

Hence we conclude that U_{1,−3/2}^{+}(y) has diffuse front in approaching to a generic point of Σ from any side if y_{0} > 0. It has sharp front from any side of Σ if y_{0} < 0. In approaching to a singular point of Σ, U_{1,−3/2}^{−}(y) has always diffuse front.

Remark 2 The above results yield not only known pictures but also add supplementary details to the figures drawn in [5]. For example we can add an arrow ”d” (indicating diffuse front) directed to the singular point of cusp from the right side to figure 3. λ = +1, ϵ = −1. Furthermore, we can add two arrows ”s” directed to the singular points appearing in swallow’s tail (”back bone”) from the inside to figure 5. integer,+.

References

[1] M.F.Atiyah,R.Bott,L.Gårding, Lacunas for hyperbolic differential operators with constant coefficients, II, Acta.Math., 131 (1973), pp. 145-206.

[2] A.G.Aleksandrov and S.Tanabé Gauss-Manin connexions, logarithmic forms and hypergeometric functions, Geometry from the Pacific Rim: Proceedings of the Pacific Rim Geometry Conference held in Singapore Dec. 1994, Walter de Gruyter, (1997), pp.1-21.

[3] P.Appel and J.Kampé de Feriet, Fonctions hypergeometriques et hypersphériques, Paris, Gauthier-Villars (1926).

[4] E.Brieskorn, Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscripta Math. 2 (1970), pp. 103-161.

[5] L.Gårding, Sharp fronts of paired oscillatory integrals, Publ. RIMS, Kyoto Univ. 12 suppl. (1977), pp. 53-68.

[6] G.-M. Greuel, Der Gauß-Manin Zusammenhang isolierter Singularitäten von vollständigen Durchschnitte , Math. Ann. 214 (1975), no.3 , pp.235-266.

[7] G.-M.Greuel, H.A.Hamm, Invarianten quasihomogener vollständiger Durchschnitten, Invent.Math. 49 (1978), pp. 67- 86.

[8] Y.Hamada, The singularities of the solutions of the Cauchy problem , Publ. RIMS, Kyoto Univ. 5 (1969), pp. 21-40.

[9] Y.Hamada, J.Leray, C.Waschal, Systèmes d’équations aux dérivées partielles à caractéristiques multiples:problème de Cauchy ramifié, hyperbolicité partielle, , J. Math. Pures et Appl., 55 (1976), pp. 297 - 352.

[10] L.Hörmander , The Analysis of Linear Partial Differential Operators, vol I (1984), Springer Verlag.

[11] E.Leichtnam, Problème de Cauchy ramifié linéaire pour des données à singularités algébriques , Supplément au Bull.Soc.math.France, 121 (1993),fascicule 2.

[12] Kh.M.Malikov, Overdeterminacy of the differential systems for versal integrals of type A,D,E, (Russian) Differentsial’nye Uravneniya 18 (1982), no.8, pp.1394-1400.

[13] V.P.Palamodov, Deformations of complex spaces, Encyclopedia of Mathematical Sciences 10, Several Complex Variables IV, pp. 105-194. Springer, 1990.

[14] I.G.Petrovski, On the diffusion of waves and the lacunas for hyperbolic equations , Mat.Sb.,17(59), 1945, pp. 289-370.

[15] F.Pham, Introduction à l’étude topologique des singularités de Landau, Gauthier-Villars, 1967.

[16] S.Tanabé, Lagrangian variety and the condition for the presence of sharp front of the fundamental solution to Cauchy problem, Scientific Papers of the College of Arts and Sci.Univ.Tokyo, 42,(1992),No.2, pp. 149-159.

[17] S.Tanabé, Transformée de Mellin des intégrales fibres de courbe espace associées aux singularités isolées d’intersection complète quasihomogènes, preprint MPIM Bonn, (1998), 28 pages.

[18] S.Tanabé, Connexion de Gauss-Manin associée à la déformation universelle des singularités A_{μ} et son application au XVIe problème de Hilbert. (submitted to Annales de l’Institut Fourier,Grenoble, 32 pages).

[19] S.Tanabé, An application of Gauss-Manin systems to the asymptotic analysis around singular locus in wave propagations. (submitted to Annales de l’Institut Fourier,Grenoble, 32 pages).

[20] V.A.Vasiliev, Ramified integrals, singularities and Lacunas,Kluwer Academic Publishers, Dordrecht, 1995.

[21] A.N.Varchenko, On normal forms of nonsmoothness of solutions of hyperbolic equations Izvestia USSR Acad.Ser.Math.,51, 1987, pp. 652-665.

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E-mail: tanabe@mpim-bonn.mpg.de