Characteristics, Bicharacteristics, andGeometric Singularities of Solutions of PDEs

—Lecture II: Singularities of Solutions of PDEs

Luca Vitagliano

University of Salerno, Italy

XXII International Fall Workshop on Geometry and PhysicsEvora, September 02–05, 2013

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Introduction

In the first lecture, I reviewed the classical theory of characteristicCauchy data for PDEs. The concept of a PDE itself has a nice intrinsic,geometric formulation which is manifestly coordinate independent.The key concept in such geometric formulation is that of jet spaces.Basically, a jet space is a manifold whose points are Taylor polynomi-als of functions. Characteristics can be defined intrinsically within jetspaces. As usual, also in this context, the geometric approach clarifies,unifies, and allows for interesting generalizations.

Plan of the Second LectureI will review the fundamentals of the jet space approach to PDEs. Iwill also give geometric definitions of

singularities of solutions,characteristics of PDEs,

and discuss their relationship, already sketched in the first lecture.

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Jet Spaces

Let π : E −→ M be a fiber bundle, and (x,u) a bundle chart on it. A(local) section σ of π is locally given by

σ : u = f (x).

Jet spaces formalize geometrically the concept of Taylor polynomial of asection of π.

∂|I|f

∂xI (x0) =∂|I|f ′

∂xI (x0), |I| ≤ k.

The k-th jet space is the manifold Jk of classes of tangency up to theorder k. It is coordinatized by (x, . . . ,uI , . . .), |I| ≤ k. A section σ of πcan be prolonged to a section jkσ of Jk −→ M. jkσ is locally given by

jkσ : uI =∂|I|f

∂xI (x), |I| ≤ k,

and encodes “derivatives of σ up to the order k”.

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The Cartan Distribution

There is a tower of fiber bundles

M Eπoo J1π1oo · · ·oo Jkπkoo Jk+1πk+1oo · · ·oo

Any tangent space to the image of a holonomic section jkσ is an R-plane.

Definition: Cartan Distribution

C : Jk 3 θ 7−→ Cθ := 〈R-planes through θ〉 ⊂ Tθ Jk.

In local coordinates

C = 〈. . . , Di, . . . , ∂/∂uI , . . .〉|I|=k ,

where Di := ∂∂xi + ∑|I|<k uIi

∂∂uI

are total derivatives. Dually

Ann C = 〈. . . , duI − uIidxi, . . .〉|I|<k.

Holonomic sections are integral sections of the Cartan distribution.

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Partial Differential Equations

Jet spaces allow a coordinate-free definition of PDEs.

Definition: System of PDEs

A system of r (nonlinear) PDEs of order k imposed on sections of the bundleπ is a codimension r submanifold E ⊂ Jk. A solution of E is a sectionσ of π such that im jkσ ⊂ E .

In local coordinates

E : F (x, . . . ,uI , . . .) = 0, |I| ≤ k,

and a section σ : u = f (x) is a solution iffF (x, . . . , ∂|I|f/∂xI , . . .) = 0.

Definition: Cartan Distribution of a PDE

C(E) := C ∩ TE .

Solutions of E identify (via prolongations) with integral sections of C(E).

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Jets of Submanifolds

There are many cases when a PDE is imposed on a submanifold. Jetsof submanifolds can be defined similarly to jets of sections of bundles.

Let E be an (n + m)-manifold. A submanifold L ⊂ E is locally givenby

L : u = f (x), (x,u) a divided chart.

Two n-submanifolds L, L′ are tangent at e0 ∈ L ∩ L′ up to the order k if

∂|I|f

∂xI (e0) =∂|I|f ′

∂xI (e0), |I| ≤ k.

The k-th jet space (of n-submanifolds of E) is the manifold Jk(E, n) ofclasses of tangency up to the order k.

Remark: First Order PDEs Imposed on Submanifolds

J1(E, n) = Gr(TE, n). Accordingly, a first order PDE imposed on n-submanifolds of E is a submanifold E ⊂ Gr(TE, n), and a solution isan n-submanifold L ⊂ E whose tangent spaces belong to E .

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Singular Solutions of PDEs

Some kind of singularities of solutions can be treated using jets.

Remark: Horizontal Integral Manifolds of C

Sections of π (locally) identify with (locally) maximal integral n-manifolds of C that are horizontal with respect to πk : Jk −→ Jk−1,but generic maximal integral n-manifolds need not to be horizontal.

Definition: Singular Sections

A singular section of π is a maximal integral n-manifold N of C.

Singular solutions of a PDE E are singular sections contained into E .

Singular points are singularities of the map πk|N : N −→ Jk−1:

sing N := θ ∈ N : rank dθπk|N < n

and can be classified along the Thom-Boardmann theory.

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Examples of Singular Solutions

u

ux

x

E

Example

Let E : u2x − 9

4 x = 0.The curve

u2 − x3 = 0u2

x − 94 x = 0

is a singular solution.

Example

Let E : u2x + x2 − 1 = 0.

The curvex = sin 2tu = 1

2 t + 18 sin 4t

ux = cos 2t

is a singular solution.

u

ux

E

x

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Singular R-planes

Singular R-planes are the infinitesimal version of singular sections.

Definition: Singular R-planes

A singular R-plane is the tangent space to a singular section at a singu-lar point. The type of a singular R-plane K at θ (also called, the type of thesingular point θ) is type K := dim ker dπk|K.

Remark: Metaplectic Structure of C

(Singular) R-planes have a useful characterization in terms of the meta-plectic structure of C:

Ω : C× C −→ TJk/C, (X, Y) 7−→ [X, Y] + C.

In local coordinates

Ω = ∑|I|<k

duIi ∧ dxi ⊗ ∂

∂uI.

(Singular) R-planes are n-dimensional, isotropic subspaces of Ω, i.e., sub-spaces K ⊂ C such that Ω(ξ, η) = 0 for all ξ, η ∈ K.

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Singularity Loci of Singular Sections

Let N be a singular section. The singular locus sing N is stratified bysingular points of different type. For simplicity, assume

sing N is a smooth submanifold,type θ is constant along sing N,ker dπk|Tθ N is transversal to Tθ sing N.

It follows that the type of singular points is codim sing N.

Definition: Fold Type Singularities

A singular section N has a fold-type singularity if codim sing N = 1.

N

sing N

T N

k

ker d k T N

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Fold-Type Singularity Equation

The shape of (fold-type) singularities is governed by a suitable PDE.

Let E ⊂ Jk be a PDE.

Definition: Fold-Type Singularity Equation

The fold-type singularity equation associated to E is

Σ1E := (dπk)(K) : K is a type 1 singular R-plane tangent to E.

N

sing N

K

k

Jk-1

K d k(K)

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Shape of Fold-Type Singularities

Since dim K = n− 1 for all K ∈ Σ1E , then

Σ1E ⊂ Gr(TJk−1, n− 1) = J1(Jk−1, n− 1)

is a first order PDE for (n− 1)-submanifolds of Jk−1.

RemarkIf N is a singular solution of E with a fold-type singularity, thenπk(sing L) is a solution of Σ1E . Thus, Σ1E describes the shape of fold-type singularity loci of solutions of E .

RemarkSolutions of Σ1E are automatically integral submanifolds of the Cartandistribution on Jk−1. In their turn, integral (n − 1)-submanifolds ofJk−1 are naturally interpreted as Cauchy data for E . Thus, Σ1E is a PDEimposed on Cauchy data for E .

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Symbol of a Non-Linear PDE

Characteristic covectors of a PDE can be defined in terms of jets.

Proposition

The bundle πk : Jk −→ Jk−1 is an affine bundle with model vector bundle

SkT∗M⊗Jk−1 VE −→ Jk−1.

Quasi-linear k-th order PDEs are affine subbundles of πk : Jk −→ Jk−1.

Definition: Symbol of a Non-Linear PDE

The symbol of a PDE E ⊂ Jk at a point θ ∈ E is

gθ := TθE ∩ ker dθπk ⊂ SkT∗x M⊗VeE.

If, locally, E : F (x, . . . ,uI , . . .) = 0, and v = vi1···ik dxi1 · · · dxik ⊗ ∂∂u ∈

SkT∗x M⊗VeE, then v ∈ gθ iff

1I!

∂F

∂ui1···ik

∣∣∣∣θ

· vi1···ik = 0.

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Characteristic Covectors of a Non-Linear PDE

Definition: Characteristic Covectors

A non-zero covector p ∈ T∗x M is characteristic for E at θ ∈ E if thereexists ξ ∈ VeE such that

pk ⊗ ξ ∈ gθ .

In local coordinates, p = pidxi is characteristic iff

rank

(1I!

pi1 · · · pik∂F

∂ui1···ik

∣∣∣∣θ

)< m. (CC)

For a determined (r = m), quasi-linear system E locally given by

E : Ai1···ikui1···ik = g,(CC) reduces to

detA(p) = 0, A(p) := pi1 · · · pikAi1···ik .

RemarkThe classical definition is recovered.

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Characteristics and Fold-Type Singularities

Recall that singularities of solutions occur along characteristic surfaces. Thisinformal statement can be made rigourous and coordinate-free withinthe jet space approach to PDEs.

E ⊂ Jk a PDE, and θ = (jkσ)(x) ∈ E ,p ∈ T∗x M, p 6= 0,kerθ p := (dx jk−1σ)(kerp) ∈ Gr(TJk−1, n− 1).

Theorem

If E is formally integrable, then the equation of fold-type singularities is “dual” to the manifold of char-acteristic covectors, i.e.,

Σ1E = kerθ p : p is characteristic for E at θ.

Moreover, when interpreted as an equation forCauchy data, Σ1E is the equation for compatible,characteristic Cauchy data.

ker p

ker p

Jk-1

(dxjk-1 )(TxM)

TxMx

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An Example: The 2D Klein-Gordon Equation

The equation for characteristic surfaces does not contain a full infor-mation on the original equation: On the other hand, the equation forsingularities may contain a full information on the original equation.

The 2D flat Klein-Gordon equation is

EKG : utt − uxx + m2u = 0.

Coordinatize J1(J1, 1) by x, t, u, ux, ut, t′, u′, u′x, u′t, with f ′ := d f /dx.Then

Σ1EKG :

(t′)2 = 1 characteristic surfaceu′ = ux + t′utu′x = m2u + t′u′t

.

Eliminating ux one gets a system for the Cauchy data t, u, ut:(t′)2 = 1 characteristic surfaceu′′ − t′′ut − 2t′u′t = m2u

.

Σ1EKG contains the mass parameter m.Luca Vitagliano Singularities of Solutions of PDEs 16 / 21

Bicharacteristics

Let E ⊂ Jk be a determined system of PDEs, and N a singular solutionwith a fold-type singularity along sing N. Interpret N as a wave prop-agating in the space-time M. Then the projection sing N of sing N onM is the wave-front, and it is a characteristic surface.

Theorem

sing N is equipped with a canonical field of directions.

Corollary

The wave-front sing N is foliated by 1-dimensional manifolds.

Definition: Bicharacteristic Lines on Wave Fronts

The 1-dimensional leaves of sing N are bicharacteristics.

Wave-fronts propagate along bicharacteristics!

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Conclusions

The theory of PDEs can be formulated in a manifestly coordinate-freeway on jet spaces. One can give very general, geometric definitions ofsingularities of solutions, and characteristics. Fold-type singularities aredual to characteristics. Their shape is governed by a fold-type singular-ity equation which describes compatible, characteristic Cauchy data.

The wave-front of a singular solution is a characteristic surface foliatedby bicharacteristic lines, i.e., wave-fronts propagate along bicharacteristics.I will not discuss bicharacteristics in full generality. I will only con-sider the special case when the symbol does only depend on points ofthe base manifold M (e.g., linear equations). In this case, characteristicsurfaces are solutions of a first order, scalar PDE, which can be treatedwith the method of characteristics. Bicharacteristics are then character-istics of the characteristic equation.

From a physical point of view, the passage from the wave-front to itsbicharacteristics can be interpreted as the passage from wave optics (thedynamics of wave-fronts) to geometric optics (the dynamics of rays), orfrom wave (quantum) mechanics to classical mechanics.

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References

Generalities on Jet Spaces

A. V. Bocharov et al., Symmetries and conservation laws for differentialequations of mathematical physics, (I. S. Krasil’shchik, and A. M. Vino-gradov eds.) Transl. Math. Mon. 182, AMS, Providence - RI, 1999.

I. S. Krasil’shchik, V. V. Lychagin, and A. M. Vinogradov, Geometry ofjet spaces and nonlinear partial differential equations, Adv. Stud. Cont.Math. 1, Gordon and Breach Science Publishing, New York, 1986.

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References

Geometric Singularities of Solutions of PDEs

M. Bachtold, Fold-type solution singularities and characteristic varietiesof nonlinear PDEs, Zurich, 2009, PhD Thesis.

A. P. Krishchenko, The structure of singularities of the solutions of quasi-linear equations, Uspekhi Mat. Nauk. 31 (1976) 219.

A. P. Krishchenko, Folding of R-manifolds, Vestnik Moskov. Univ. Ser. IMath. Mech. 1 (1977) 17.

F. Lizzi et al., Eikonal type equations for geometrical singularities of so-lutions in field theory, J. Geom. Phys. 14 (1994) 211.

V. V. Lychagin, On singularities of the solutions of differential equations,Dokl. Akad. Nauk. SSSR 251 (1980) 794.

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Reference

Geometric Singularities of Solutions of PDEs

V. V. Lychagin, Geometric singularities of the solutions of nonlinear dif-ferential equations, Dokl. Akad. Nauk. SSSR 261 (1980) 1299.

V. V. Lychagin, Singularities of multivalued solutions on nonlinear dif-ferential equations, and nonlinear phenomena, Acta Appl. Math. 3 (1985)135.

V. V. Lychagin, Geometric theory of singularities of solutions of nonlineardifferential equations, J. Soviet Math. 51 (1990) 2735.

A. M. Vinogradov, Many-valued solutions, and a principle for the classi-fications of nonlinear differential equations, Dokl. Akad. Nauk. SSSR 210(1973) 11.

A. M. Vinogradov, Geometric singularities of solutions of nonlinear par-tial differential equations, in: Differential Geometry and its Applications(Brno, 1986), Math. Appl. (East Europeand Ser.) 27, Reidel, Dordrecht,1987, p. 359.

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