Characteristics, Bicharacteristics, and Geometric Singularities of Solutions of PDEs
aa r X i v : . [ m a t h . DG ] N ov CHARACTERISTICS, BICHARACTERISTICS, AND GEOMETRICSINGULARITIES OF SOLUTIONS OF PDES
LUCA VITAGLIANO
Abstract.
Many physical systems are described by partial differential equations (PDEs). Determin-ism then requires the Cauchy problem to be well-posed. Even when the Cauchy problem is well-posedfor generic Cauchy data, there may exist characteristic Cauchy data. Characteristics of PDEs playan important role both in Mathematics and in Physics. I will review the theory of characteristics andbicharacteristics of PDEs, with a special emphasis on intrinsic aspects, i.e., those aspects which areinvariant under general changes of coordinates. After a basically analytic introduction, I will pass toa modern, geometric point of view, presenting characteristics within the jet space approach to PDEs.In particular, I will discuss the relationship between characteristics and singularities of solutions andobserve that: “wave-fronts are characteristic surfaces and propagate along bicharacteristics”. This re-mark may be understood as a mathematical formulation of the wave/particle duality in optics and/orquantum mechanics. The content of the paper reflects the three hour minicourse that I gave at theXXII International Fall Workshop on Geometry and Physics, September 2–5, 2013, Évora, Portugal.
Contents
Introduction 11. Characteristic Cauchy Data for PDEs 31.1. Cauchy Problems 31.2. Characteristic Cauchy Data 51.3. Examples 82. Singularities of Solutions of PDEs 112.1. PDEs and Jet Spaces 112.2. Singular Solutions 142.3. Fold-type Singularities 153. Bicharacteristics and the Hamilton-Jacobi Theory 193.1. Contact Geometry of First Jets of Functions 193.2. First Order Scalar PDEs 203.3. Hamilton-Jacobi Theory 21Conclusions 24Acknowledgments 24References 24
Introduction
Many physical systems are described by partial differential equations (PDEs). Determinism thenrequires the Cauchy problem to be well-posed. Even when the Cauchy problem is well-posed for genericCauchy surfaces, there may exist characteristic Cauchy data . Roughly speaking, characteristic Cauchydata are those for which the Cauchy problem is ill-posed, in the sense of non-existence or non-uniquenessof corresponding solutions. Surprisingly enough, characteristic Cauchy data play an important role both in the (mathematical) theory of PDEs and in Theoretical Physics. From a mathematical point ofview, characteristics of PDEs are related to intermediate integrals, classifications of PDEs, singularitiesof solutions (besides Cauchy problems). From a physical point of view, if one interprets independentvariables as space-time coordinates and dependent variables as fields, then a characteristic Cauchysurface may be understood as the wave-front of a “bounded” disturbance in the fields, propagating in thespace-time. Characteristic Cauchy surfaces are often themselves described by a first order, scalar PDE.In their turn first order, scalar PDEs can be integrated with the method of characteristics. Namely,the integration problem can be reduced to the integration problem for a system of ordinary differentialequations (ODEs) whose solutions “foliate” solutions of the original PDE. Accordingly, characteristicsurfaces are foliated by lines: characteristic lines in Cauchy terminology, bicharacteristic lines inHadamard terminology. From a physical point of view, one concludes that a wave-front propagatesalong bicharacteristics . Notice that the transition between the three different “mathematical regimes”PDEs ⇓ st order scalar PDEs and characteristics ⇓ ODEs and bicharacteristics (1)formalizes in rigorous terms the transition between three different “physical regimes”:Fields and field equations ⇓ wave fronts and wave optics ⇓ light rays and geometric opticsEven more, the equation for characteristic surfaces is often an Hamilton-Jacobi equation. It is wellknown that the Hamilton-Jacobi equation is the short wave-length limit of the Schrödinger equation.Actually, interpreting the Hamilton-Jacobi equation as an equation for the wave-front of a wave-function propagating in the space-time (see, e.g., [36]), one can infer the Schrödinger equation accordingto the analogy:wave optics/geometric optics = wave mechanics/classical mechanicsIn this sense transition (1) is also analogous to the transition from quantum mechanics to classicalmechanics, summarized in the scheme: Schrödinger equation ⇓ Hamilton-Jacobi equation ⇓ Hamilton equation (2)Accordingly, the quantizing , i.e., reversing the arrows in (2), is analogous to “ reconstructing a PDEfrom its (bi)characteristics ”. For certain specific classes of PDEs the reconstruction can be actuallyaccomplished, and, in a sense, quantization is not ambiguous . The aim of this paper is reviewing thetheory of (bi)characteristics of PDEs and its physical interpretation. In particular, I will describe insome details the transition (1) focusing on intrinsic aspects, i.e., those aspects which are independentof the choice of coordinates. Differential geometry will be then the natural language.The paper is divided into three sections. In the first section, I discuss Cauchy problems and char-acteristic Cauchy data. I conclude with some examples from Mathematical Physics. This section is
HARACTERISTICS AND SINGULARITIES OF SOLUTIONS OF PDES 3 basically analytic and makes use of local coordinates. However, most of the results therein are actuallyindependent of the choice of coordinates. In the second section, I present the geometric setting forPDEs and their characteristics, specifically, jet spaces. Characteristics of PDEs has a nice, intrinsicdefinition in terms of jets. The geometric setting clarifies the relationship between characteristics andsingularities of solutions. In the last section, I focus on bicharacteristics. Often characteristic surfacesare governed by a first order scalar PDE E . The geometry underlying such PDEs is contact geometrywhich is at the basis of the method of characteristics. It may happen that E is an Hamilton-Jacobiequation. There is a symplectic version of the method of characteristics for Hamilton-Jacobi equationsbased on the Hamilton-Jacobi theorem. This motives me to review the Hamilton-Jacobi theory. Iconclude speculating about the possibility of extending the Hamilton-Jacobi theory to field theory in acovariant way, thus opening the road through a rigorous, covariant, Schrödinger quantization of gaugetheories . 1. Characteristic Cauchy Data for PDEs
Cauchy Problems.
The evolution of many physical systems, especially (but not only) in classi-cal physics, is described by a system of (sometimes non-linear) partial differential equations (PDEs).Determinism requires that the full evolution of the system is anambiguosly determined by the initialconfiguration. From a mathematical point of view this means that the Cauchy problem for the corre-sponding PDE should be well-posed, i.e., there should be ( existence ) a unique ( uniqueness ) solution forany set of (physically admissible) Cauchy data. The most general way to understand a set of Cauchydata is “a general hypersurface Σ in the space of independent variables + derivatives of the dependentvariables normal to Σ along Σ itself”. Even if the Cauchy problem is well-posed for generic Cauchydata, existence or uniqueness may fail for special Cauchy data usually referred as characteristic Cauchydata . Nonetheless, characteristic Cauchy data have a nice physical interpretation. In this section I willrecall some basic facts about the Cauchy problem, characteristics of PDEs and their physical inter-pretation. For simplicity, I will mainly focus on the case of determined systems of quasi-linear partialdifferential equations. I will use local coordinates everywhere, and I will conclude with few examples,mainly from Mathematical Physics.1.1.1. Cauchy problems in normal form.
Let u = ( u , . . . , u m ) be a vector valued function of the n real variables x = ( x , . . . , x n ) . I will often ineterpret the x ’s as space-time coordinates, and the u ’sas components of a field propagating on the space-time. Put u I := ∂ | I | ∂x I u = ∂ ℓ ∂x i · · · ∂x i ℓ u , where I = i · · · i ℓ is a multi-index (denoting multiple partial derivatives) and | I | := ℓ (the numberof derivatives). Sometimes, it is convenient to split space-time coordinates into “space coordinates” x = ( x , . . . , x n − ) + a time coordinate t = x n . In this case I use the following notation for multiplespace-time derivatives: u ℓ,J := ∂ | J | + ℓ ∂ x J ∂t ℓ u . Consider the system of m PDEs in m unknown functions ∂ k u ∂t k = f ( t, x , . . . , u ℓ,J , . . . ) (3) LUCA VITAGLIANO and the initial data problem ∂ k u ∂t k = f ( t, x , . . . , u ℓ,J , . . . ) ∂ ℓ u ∂t ℓ (cid:12)(cid:12)(cid:12)(cid:12) t = t = h ℓ ( x ) ℓ < k, | J | + ℓ ≤ k (4)where f and the h ℓ ’s are analytic vector valued functions of their arguments. A system of m PDEs in m unknowns functions is in normal form if it is of the kind (3) Cauchy-Kowalewski Theorem (see, for instance, [9]) asserts that, locally, there exists a unique solutionof the Cauchy problem (3) + (4). Since the initial data in (4) completely determine the Taylor seriesof u at points of the initial surface t = t , the proof basically consists in checking convergence of theseries.1.1.2. General Cauchy problems.
Often, e.g. in relativistic theories, there is no preferred “space +time splitting” of the space-time. In this case, it is generically advisable not to break the covarianceby an arbitrary choice of a time coordinate. Thus, a Cauchy problem is better posed on a generichypersurface in the space-time. Namely, consider a system of PDEs in m unknown functions u in thegeneral form F ( x, . . . , u I , . . . ) = 0 , | I | ≤ k (5)and a generic (Cauchy, i.e., initial) hypersurface Σ : z ( x ) = 0 , where F are smooth functions with independent differentials and z is a smooth function with non-vanishing gradient. The first normal derivative of u at a point ( x , . . . , x n ) of Σ is ∂ u ∂z := ∂z∂x u + · · · + ∂z∂x n u n . Put ∂ ℓ u ∂z ℓ := ∂∂z · · · ∂∂z u . An initial data problem (Cauchy problem) on Σ can be posed as follows: F ( x, . . . , u I , . . . ) = 0 ∂ ℓ u ∂z ℓ (cid:12)(cid:12)(cid:12)(cid:12) z =0 = h ℓ ( x ) | I | ≤ k, ℓ < k. (6)If Problem (6) could be recast in the normal form (4), then, under additional analiticity condition,I could apply the Cauchy-Kowalewski Theorem and get existsence and uniqueness of solutions. Forsimplicity, I assume, from now on, that system (5) is(1) weakly determined , in the sense that it consists of precisely m equations.(2) quasi-linear , i.e., F = A j ··· j k · u j ··· j k + g , (7)where, for all multi-indexes j · · · j k , A j ··· j k = A j ··· j k ( x, . . . , u J , . . . ) , | J | < k is an m × m matrix valued function, and g = g ( x, . . . , u J , . . . ) , | J | < k is a vector valued function. HARACTERISTICS AND SINGULARITIES OF SOLUTIONS OF PDES 5
Notice that quasi-linearity is a condition invariant under a change of (both independent and depen-dent) coordinates. In particular, it is easy to see (by induction on k ) that if x = ( x , . . . , x n ) ¯ x =(¯ x , . . . , ¯ x n ) is a diffeomorphism, then F = ¯ A j ··· j k · ¯ u j ··· j k + ¯ g , where the ¯ u I ’s are derivatives of u with respect to the ¯ x ’s, ¯ A i ··· i k = ∂ ¯ x i ∂x j · · · ∂ ¯ x i k ∂x j k A j ··· j k and ¯ g = ¯ g (¯ x, . . . , ¯ u J , . . . ) , | J | < k. In particular, the coefficients A i ··· i k of the highest order (linear) term transform as a contravariantsymmetric tensor under a change of (independent) coordinates. The contravariant tensor A = ( A i ··· i k ) is called the (principal) symbol of the quasi-linear operator F (see Subsection 2.3.2 for an intrisicdefinition of the symbol). Remark 1.
Limiting the discussion to determined, quasi-linear systems of PDEs is not really re-strictive for physical applications. Indeed, such systems are particularly relevant in Physics, sinceEuler-Lagrange PDEs are precisely of this form.
Now, choose independent coordinates adapted to Σ , i.e., complete z to a system of coordinates ( z, y , . . . , y n − ) . Then y = ( y , . . . , y n − ) can be understood as internal coordinates on Σ . In the newcoordinates, Eq. (5) becomes ∂z∂x j · · · ∂z∂x j k A j ··· j k · ∂ k u ∂z k = f ( z, y , . . . , ¯ u ℓ,J , . . . ) , ℓ < k, | J | + ℓ ≤ k. (8)for a suitable f , where ¯ u ℓ,J = ∂ | J | + ℓ u ∂ y J ∂z ℓ . If det (cid:18) ∂z∂x j · · · ∂z∂x j k A j ··· j k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) z =0 = 0 (9)then (8) can be clearly recast in the normal form (4), around Σ . Notice, however, that the A j ··· j k willgenerically depend on the u J , | J | < k . Accordingly, unequality (9) is actually a condition on Σ andinitial data on it, rather then a condition on the sole Σ .1.2. Characteristic Cauchy Data.
Characteristic covectors and characteristic Cauchy data.
One is thus led to consider the m × m matrix A ( p ) = p j · · · p j k A j ··· j k for an arbitary co-vector p = p i dx i . More precisely, A ( p ) is a matrix-valued, homogeneous polynomialfunction on cotangent spaces to the space-time (see [1] for a nice example). Notice that, in general,it does also depend on the space-time point x and on derivatives of the field at the point x up to theorder k − . For simplicity, I assume, temporarily, that A ( p ) is generically invertible, i.e., rank A ( p ) = m somewhere, and therefore, almost everywhere, in the space of the p ’s. Notice that det A ( p ) is ahomogenous polynomial in the p i ’s. Therefore, if the equation det A ( p ) = 0 LUCA VITAGLIANO is compatible, then it determines a closed, nowhere dense, conic subset of the space of the p ’s, called(up to projectivization) the characteristic variety of the equation (5) (see, for instance, [6]). Points ofthe characteristic variety are called characteristic covectors and play an important role for differentaspects of the theory of PDEs, namely: the Cauchy problem and singularities of solutions (as discussedbelow), the classification of PDEs [39], the method of intermediate integrals [19] (for finding solutionsof a PDE by integrating lower order PDEs).An hypersurface Σ : z = 0 such that det A ( dz ) | z =0 = det (cid:18) ∂z∂x j · · · ∂z∂x j k A j ··· j k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) z =0 = 0 (10)is called a characteristic (Cauchy) surface (see, for instance, [9]). Beware, however, that this is an abuseof terminology (mutuated by the theory of linear PDEs ). Indeed, as already remarked, (10) is actuallya condition on Σ and initial data on it. Accordingly, we should rather speak about characteristic(Cauchy) data . The initial value problem may not be well-posed (i.e., there may be no existence anduniqueness, even for analytic data), in general, on characteristic surfaces. In particular, initial dataare “ constrained ” on a characteristic surface, in the sense that not all initial data on a characteristicsurface are admissible , i.e., are compatible with the PDE. To see this, let Σ be characteristic, and q = m − rank A ( dz ) | z =0 > . As a minimal regularity condition, I assume q to be constant on Σ . Thenthere is a non zero [ q, m ] matrix M = M ( z, y , . . . , ¯ u ℓ,J , . . . ) such that ( M · A ( dz )) | z =0 = 0 . It follows that Eq. (8) may only possess solution if ( M · f )( z, y , . . . , ¯ u ℓ,J , . . . ) | z =0 = 0 . (11)This last equation may be interpreted as a system of (generically non-linear) PDEs constraining theinitial data ∂ ℓ u ∂z ℓ (cid:12)(cid:12)(cid:12)(cid:12) z =0 on the characteristic surface Σ .Finally, notice that (10) may be interpreted as a first order, polynomial, PDE whose unknown is ahypersurface in the space-time. To see this, assume z to be in the form z = t − τ ( x ) , and y = x . Then(10) becomes det (cid:18)P kℓ =0 ∂τ∂ y a · · · ∂τ∂ y a ℓ B a ··· a ℓ (cid:19) = 0 (12)where B a ··· a ℓ = k ! ℓ ! A a ··· a ℓ n ··· n , a , . . . , a ℓ = 1 , . . . , n − . It should be stressed, however, that the “coefficients” B a ··· a ℓ depend in general on u and its derivativesup to the order k − . When the B a ··· a ℓ ’s do only depend on independent variables y , (e.g., whenEq. (5) is linear) Eq. (12) is a first order, scalar (inhomogeneous polynomial) PDE in the unknown τ that can be treated, for instance, with the method of characteristics. In this case, one usually refersto characteristic lines of (12) as bicharacteristics of (5). We will come back to bicharacteristics (andthe method of characteristics for scalar PDEs) in Section 3.1.2.2. Physical interpretation of characteristic surfaces.
On another hand, singularities of solutions ofa system of PDEs occur along characteristic surfaces . To clarify this sentence, let u be a fiducial,background solution of (5), and Σ be an hypersurface bounding a region Ω of the space-time. Wesearch for a (possibly singular along Σ ) solution u of (5) which agrees with u in Ω but is everywheredifferent from u outside Σ . On physical ground , I assume that all derivatives of u up to the order k − are continuous along Σ . In particular, u and u are both solutions of a Cauchy problem of HARACTERISTICS AND SINGULARITIES OF SOLUTIONS OF PDES 7 the form (6). It follows that Σ must be a characteristic surface. In other words, as already stated,singularities (e.g., wave fronts) occur along characteristic surfaces, i.e., the boundary of a disturbancein the space-time is a characteristic surface [25]. Thus, to understand how disturbaces of a specificfield (with specific field equations) propagate in the space time, one has to solve the characteristicequation (10).Under suitable conditions, a characteristic surface is actually equipped with a field of directionsthat integrates to a 1 dimensional foliation, whose leaves are traditionally referred to as bicharacteris-tics (Hadamard terminology). Accordingly, singularities of solutions propagate along bicharacteristics .From a physical point of view, one may interpret characteristic surfaces as wave-fronts and bicharac-teristics as rays . Under this interpretations the passage from characteristics to bicharacteristics is the passage from wave-optics to geometric optics (see [27] for more details, see also [17]). Alternatively,bicharacteristics can be interpreted as trajectories of particles. If one adopts this interpretation, theydescribe the motion of a particle-like counterpart of the field under consideration. This relates theprinciple of wave-particle duality to the geometric theory of PDEs .1.2.3. Characteristics of Euler-Lagrange equations.
In general, A ( p ) may be non-invertible everywhereon the space of p ’s. In this case, initial data are constrained on every Cauchy hypersurface and theCauchy-Kowalewski theorem fails. Let r be the maximum rank of A ( p ) on the space of p ’s. Then rank A ( p ) = r almost everywhere in the space of p ’s. In this case, one define a characteristic surfaceto be an hypersurface Σ : z = 0 such that rank A ( dz ) | z =0 = rank (cid:18) ∂z∂x j · · · ∂z∂x j k A j ··· j k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) z =0 < r. In this general case, characteristic surfaces still play a role in Cauchy problems and the theory ofsigularity propagation, but I will not enter this here. However, notice that the general situation doesoccur for the field equation of a gauge theory as I briefly discuss now. Indeed, let (5) be the Euler-Lagrange (EL) equations determined by a variational principle Z L ( x , . . . , x n , . . . , u J , . . . ) d n x, | J | ≤ ℓ where L ( x , . . . , x n , . . . , u J , . . . ) d n x is a Lagrangian density depending on derivatives of the fields upto the order ℓ . Then F = X | J |≤ ℓ ( − ) | J | D J ∂L∂ u J , where, for J = j · · · j r , D J := D j ◦ · · · ◦ D j r , and D i := ∂∂x i + X I u Ii · ∂∂ u I is the i -th total derivative. Then F = X j ≤···≤ j ℓ X k ≤···≤ k ℓ ∂ L∂ u j ··· j ℓ ∂ u k ··· k ℓ · u j ··· j ℓ k ··· k ℓ + g where g = g ( x , . . . , x n , . . . , u J , . . . ) , | J | < ℓ. For a gauge invariant Lagrangian det A ( p ) = det X j ≤···≤ j ℓ X k ≤···≤ k ℓ ∂ L∂ u j ··· j ℓ ∂ u k ··· k ℓ p j · · · p j ℓ p k · · · p k ℓ = 0 for all p ’s (see examples below). LUCA VITAGLIANO
Characteristics of fully non-linear equations.
Finally, I briefly discuss the case when Eq. (5) isnot quasi-linear. In this case, a careful use of the inverse function theorem shows that the Cauchyproblem is well posed on any hypersurface
Σ : z = 0 such that det (cid:18)X j ≤···≤ j k ∂z∂x j · · · ∂z∂x j k ∂ F ∂ u j ··· j k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) z =0 = 0 , Accordingly, all the above considerations remain valid up to a substitution A j ··· j k −→ ( j · · · j k )! k ! ∂ F ∂ u j ··· j k , where ( i · · · i k )! /k ! is a suitable combinatorial coefficients that accounts for the fact that u j ··· j ℓ = u j σ (1) ··· j σ ( ℓ ) for every permutation σ of { , . . . , ℓ } . Specifically, let j = 1 , . . . , n appear N j times in the multi-index j · · · j k . Then ( j · · · j k )! := N ! · · · N n ! . If the matrix X j ≤···≤ j k p j · · · p j k ∂ F ∂u j ··· j k is generically invertible on the space of p ’s, then Σ : z = 0 is a characteristic surface if det (cid:18)X j ≤···≤ j k ∂z∂x j · · · ∂z∂x j k ∂ F ∂u j ··· j k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) z =0 = 0 . Examples.
Klein-Gordon and wave equations on a curved space-time.
Let g = g ij dx i dx j be a Riemannian,or pseudo-Riemannian metric on an open subset U of R n . Consider the following linear equation g ij ∇ i ∇ j u = 0 , (13)where u is an unknown function on U , and ∇ is the Levi-Civita connection of g . Eq. (13) is the ELequation coming from the action functional − Z ( g ij ∇ i u ∇ j u ) p | det g | d n x. The symbol of the operator F = g ij ∇ i ∇ j is A = g − =: ( g ij ) . Accordingly, A ( p ) = g ij p i p j = g − ( p , p ) , which is generically non-zero. The characteristic variety is the quadric A ( p ) = g ij p i p j = g − ( p , p ) = 0 , and characterictic surfaces Σ : z = 0 are defined by g ij ∂z∂x i ∂z∂x j (cid:12)(cid:12)(cid:12)(cid:12) z =0 = g − ( d z, d z ) | z =0 = 0 . In particular, if g is Riemannian then F = ∆ is the (curved) Laplacian, which is an elliptic operator, F = 0 is the (curved) Laplace equation, and there are no characteristic surfaces. On another hand,if g is Lorentzian, F = (cid:3) is the (curved) d’Alambertian, which is a hyperbolic operator, F = 0 isthe (curved) wave equation, and characteristic surfaces are precisely the null hypersurfaces. In thiscase, one concludes that wave fronts are light-like hypersurfaces. Notice that the (curved) Klein-Gordon operator (cid:3) + m has the same symbol as the d’Alambertian, and, therefore, the Klein-Gordonequation has the same characteristic surfaces as the wave equation. HARACTERISTICS AND SINGULARITIES OF SOLUTIONS OF PDES 9
Dirac equation on Minkowski space-time.
Let η = η µν dx µ dx ν be the Minkowski metric on R .The Dirac equation is the linear, first order (system of) PDE(s) given by ( i γ µ ∂ µ − m ) u = 0 , where u = ( u , u , u , u ) is a 4-component (complex) spinor and γ , γ , γ , γ are the × Diracmatrices. The symbol of the operator F = i γ µ ∂ µ − m is A = ( i γ µ ) . Accordingly, A ( p ) = i γ µ p µ which is generically invertible. The characteristic variety is defined by the 4-th order algebraic equation det A ( p ) = det( i γ µ p µ ) = 0 . An easy computation (first performed by G. Racah in the 30th’s [37]) shows that det A ( p ) = ( η µν p µ p ν ) = ( η − ( p , p )) = 0 , and characteristic surfaces Σ : z = 0 are defined by (cid:18) η µν ∂z∂x µ ∂z∂x ν (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z =0 = ( η − ( d z, d z )) | z =0 = 0 . Therefore, characteristic surfaces of the Dirac equations are precisely null surfaces in the Minkowskispace-time (and Racah himself interpreted this result in terms of the Heisenberg principle).1.3.3.
Maxwell equations on a curved space-time.
Let g = g ij dx i dx j be a Lorentzian metric on an opensubset U of R . The (vacuum) Maxwell equations in U read F j = g ik ∇ k ( ∇ i u j − ∇ j u i ) = 0 , where u = ( u , u , u , u ) are the components of a differential -form (the electromagnetic potential)on U . Maxwell equations are the EL equations coming from the action functional − Z g ik g jℓ ∇ [ i u j ] ∇ [ k u ℓ ] p | det g | d x. Now F j = (cid:0) g ik δ ℓj − g ℓk δ ij (cid:1) ∂ u ℓ ∂x k ∂x i + · · · , where the dots · · · denote lower order terms. Accordingly, A ( p ) ℓj = (cid:0) g ik δ ℓj − g ℓk δ ij (cid:1) p k p i , i.e., A ( p ) = g − ( p , p ) I − p ♯ ⊗ p , where p ♯ := g − ( p , − ) . Notice that A ( p ) is never invertible. Indeed, rank A ( p ) is generically ratherthen . This corresponds to the fact that gauge freedom is parametrized by arbitrary function onthe space-time. In this (degenerate) case, characteristic surfaces Σ : z = 0 are defined by rank A ( dz ) | z =0 < . But rank A ( p ) < iff p is a null covector, and, in this case, rank A ( p ) = 1 whenever p = 0 . Oneconcludes that the characteristic surfaces of Maxwell equations in curved space-time are again nullhypersurfaces [45].Notice that the degeneracy of the matrix A ( p ) can be cured by gauge fixing. For instance, forMaxwell equations in the Lorentz gauge ( ∇ i u i = 0 ) A ( p ) = g − ( p , p ) I , which is generically invertible, and degenerates iff g − ( p , p ) = 0 again [45].1.3.4. Einstein equations.
Let U be an open subset of R . The (vacuum) Einstein equations in U read Ric [ u ] = 0 , where u = u ij dx i dx j is an unknown Lorentzian metric on U and Ric [ u ] = R ij [ u ] dx i dx j is its Riccitensor. Einstein equations are the EL equations coming from the action functional Z u ij R ij [ u ] p | det u | d x. The symbol of the Ricci operator
Ric has been first computed by Levi-Civita in the 30th’s. One has R ij [ u ] = (cid:16) δ [ mi u k ][ ℓ δ n ] j − u n [ m u k ] ℓ u ij (cid:17) ∂ u kℓ ∂x m ∂x n + · · · where the dots · · · denote lower order terms. Accordingly, A ( p ) kℓij = (cid:16) δ [ mi u k ][ ℓ δ n ] j − u n [ m u k ] ℓ g ij (cid:17) p k p ℓ , which should be understood as entries of a × matrix (the pairs ij and kℓ are to be ordered,for instance, lexicographically). Levi-Civita proved that A ( p ) is never invertible, and rank A ( p ) isgenerically rather then . This corresponds to the fact that the gauge freedom is parametrized by arbitrary functions on the space-time. Finally, rank A ( p ) < iff u − ( p , p ) = u ij p i p j = 0 , and, in this case, rank A ( p ) = 4 whenever p = 0 . One concludes that the characteristic surfaces ofEinstein equations are null hypersurfaces with respect to the unknown metric u . This is a typicalexample when (10) is a condition on Σ and initial data on it (in this case, the metric on it) and notonly on Σ itself. Notice that, from a physical point of view, the outcome of this and the previous threesubsections is that the phase velocity of gravitational, electromagnetic, Dirac, and Klein-Gordon fieldis the speed of light. An unphysical, fully non-linear example.
Consider the scalar PDE in two independent variables x, y : u yyy − ( u xxy ) + u xxx u xyy = 0 . (14)Eq. (14) is a third order Monge-Ampère equation [5]. For p = pdx + qdy , one has A ( p ) = u xyy p − u xxy p q + u xxx pq + q , which is generically non-zero. Accordingly, a hypersurface Σ : z = z ( x, y ) is characteristic iff u xyy z x − u xxy z x z y + u xxx z x z y + z y = 0 . Notice that z x = 0 , otherwise z x = z y = 0 . Therefore, one can search for Σ in the form Σ : x = τ ( y ) ,which gives u xyy + 2 u xxy τ y + u xxx τ y − τ y = 0 depending on (constrained) initial data on Σ . HARACTERISTICS AND SINGULARITIES OF SOLUTIONS OF PDES 11 Singularities of Solutions of PDEs
PDEs and Jet Spaces.
Most of the considerations done in the previous lecture are indepen-dent of the choice of coordinates. This suggests that there is an intrinsic, geometric theory capturingthe concept of characteristics of a system of PDEs. This is actually the case. The aim of this sec-tion is to provide a gentle introduction to basics of the geometric theory of (nonlinear) PDEs, theircharacteristics, and (fold-type) singularities of their solutions. In particular, I will present a rigorous,mathematical version of the physical considerations in Subsection 1.2.2. The main results will be pre-sented without a proof and the interested reader should refer to the bibliography for details. Indeed,a deeper analysis would show that many branches of Mathematics enter the intrinsic theory of PDEs,namely: differential geometry and differential topology, commutative algebra and algebraic geometry,homological algebra and algebraic topology.I begin with a geometric framework for PDEs, namely, jet spaces (for more details about jet spaces,see [4, 19]).2.1.1.
Jets of sections.
Let π : E −→ M be a fiber bundle, and let ( x , . . . , x n , u ) be a bundle charton E , i.e., ( x , . . . , x n ) are coordinates on M , and u = ( u , . . . , u m ) are fiber coordinates on E . the x ’swill be interpreted as independent variables, and u as a set of dependent variables. From a physicalpoint of view, M will be often interpreted as the space-time and sections of π as configurations of afield on it. I want to discuss PDEs imposed on sections of π . To do this in a way which is manifestlyindependent of coordinates (and any other auxiliary structure on π , e.g., a connection) it is necessaryto introduce jet spaces.Two local sections σ and σ of π , locally given by σ , : u = f , ( x , . . . , x n ) , are tangent up to the order k at a point x ∈ M if the k -th order Taylor polynomials of f and f coincide at x ≡ ( x , . . . , x n ) : ∂ | I | f ∂x I ( x , . . . , x n ) = ∂ | I | f ∂x I ( x , . . . , x n ) , | I | ≤ k. Tangency up to the order k at x is a well defined equivalence relation. In particular, it is independentof coordinates. Denote by J kx π the set of equivalence classes. Finally put J k π := a x ∈ M J kx π. It is called the k -th jet space of the bundle π and can be given a canonical structure of smooth manifoldas follows. First of all, for a local section σ of π , denote by [ σ ] kx its class of tangency up to the order k at the point x ∈ M . It is a point of J k π , which is called the k -th jet of σ at x , and can be intepretedas (an intrinsic version of) the k -th order Taylor polynomial of σ at x . Notice that J π identifiescanonically with E . Moreover, there are canonical surjections π k,ℓ : J k π −→ J ℓ π, k ≥ ℓ which consist in forgetting derivatives of order higher than ℓ . There are also surjections π k : J k π −→ M, [ σ ] kx x. Let U be a bundle coordinate domain in E = J π . On π − k, ( U ) there are coordinates ( x , . . . , x n , . . . , u I , . . . ) given by u I ([ σ ] kx ) := ∂ | I | f ∂x I ( x , . . . , x n ) , | I | ≤ k, where σ is a local section of π which in coordinates look as σ : u = f ( x , . . . , x n ) . (15)It is easy to see that the J k π , with these coordinates, are smooth manifolds, and the π k,ℓ ’s (and,consequently, the π k ’s) are fiber bundles.A section σ of π can be “prolonged” to a section j k σ of π k , its k -th jet prolongation , by putting ( j k σ )( x ) := [ σ ] kx . If σ looks locally as (15), then j k σ looks locally as j k σ : u I := ∂ | I | f ∂x I ( x , . . . , x n ) , | I | ≤ k. Thus j k σ is a coordinate free version of “partial derivative functions of σ up to the order k ”. Noticethat not all sections of π k are of the form j k σ . The latter are sometimes called holonomic sections .2.1.2. The Cartan Distribution.
In the following, I will denote simply by J k the space of k -th jets ofsections of π , if there is no risk of confusion. There is a canonical structure on J k , namely a distribution,which, in a sense, encodes the “differential relations among the jet coordinates u I ”. Let us fix a point θ ∈ J k . If θ is the k -th jet at a point x ∈ M of a section σ of π , then, clearly, im j k σ ∋ θ . Consider thetangent space R [ σ ] ⊂ T θ J k to im j k σ at θ . Any subspace of T θ J k of the form R [ σ ] is called an R -plane at θ . Notice that R [ σ ] does only depend on the ( k + 1) -th jet θ ′ = [ σ ] k +1 x of σ . Accordingly, it will bedenoted by R θ ′ . The correspondence π − k +1 ,k ( θ ) −→ { R -planes at θ } , θ ′ R θ ′ , is a bijection that allows to construct jet spaces inductively from lower order ones. R -planes at θ span a distinguished subspace C θ in T θ J k and the correspondence C : θ θ isa smooth distribution on J k often called the Cartan distribution . The Cartan distribution is locallyspanned by vector fields D i := ∂∂x i + X | I | This shows that C is not involutive (and, therefore, not integrable ). Actually,in general, the Cartan distribution possesses many different (locally) maximal integral submanifolds(even of different dimensions) through any point. For instance, fibers of π k,k − and images of holonomicsections are both maximal integral submanifolds and there are more maximal integral submanifolds ofdifferent kinds. However, if a maximal integral submanifold is horizontal with respect to the projection π k,k − , then it is the image of a holonomic section. In this sense the Cartan distribution “detects”holonomic sections .2.1.3. Differential Equations. Jet spaces formalize in a coordinate free way the concept of partialderivatives. Accordingly, they allow a coordinate free definition of system of PDEs. Specifically, a system of (non-linear) PDEs of the order k imposed on sections of the bundle π (in the following,simply a PDE) is a submanifold E of J k . Indeed E looks locally as E : F ( x , . . . , x n , . . . , u I , . . . ) = 0 , | I | ≤ k, (16)which is a system of PDEs in the analytic sense up to the interpretation of the u I ’s as partial derivativesof the u ’s. In view of (16), it is meaningful to say that a a system of PDEs E ⊂ J k is (weakly) determined (i.e., the number of equations coincides with the number of dependent variables) if codim E = m . The HARACTERISTICS AND SINGULARITIES OF SOLUTIONS OF PDES 13 coordinate free definition of solutions of E should be now clear: a solution of E is a section σ of π such that j k σ takes values in E . Indeed, if σ is locally given by (15), then the condition im j k σ ⊂ E reads locally F (cid:16) x , . . . , x n , . . . , ∂ | I | f / ∂x I , . . . (cid:17) = 0 , | I | ≤ k which is a system of PDEs imposed on the f ’s. On a PDE E ⊂ J k one can consider the distribution C ( E ) : θ θ ( E ) := C θ ∩ T θ E . Under suitable regularity conditions on E , C ( E ) is a smooth distribution.It is then clear that if a maximal integral submanifold of C ( E ) is horizontal with respect to π k,k − ,then it is the image of j k σ for some solution σ of E . In other words, smooth solutions of E are in one-to-one correspondence with maximal integral submanifolds of C ( E ) satisfying a suitable horizontalityconditions . The main point here is that, relaxing this horizontality condition, one can describe, inpurely geometric terms, solutions with (specific type of ) singularities .2.1.4. Jets of Submanifolds. Notice that, in differential geometry, one often wishes to impose condi-tions on submanifolds of a given manifold and those conditions locally look like differential equations.Typical examples are: Lagrangian submanifolds in a symplectic manifold, Legendrian submanifoldsin a contact manifold, totally geodesic submanifolds in a Riemannian manifold, etc. As I have al-ready discussed, characteristic surfaces themselves are submanifolds satisfying suitable “differentialconditions”. Accordingly, one speaks about PDEs imposed on submanifolds . Jets of sections can begeneralized to jets of submanifolds . The latter provide a coordinate free formalism for PDEs imposedon submanifolds. In the following, I will only need first jets of submanifolds, which can be defined asfollows.Let E be a smooth manifold. Fix a positive integer n and let dim E = n + m . Consider n -dimensionalsubmanifolds of E . Tangency at a fixed point e ∈ E is an equivalence relation on the set of submanifolds(through e ). Denote by J e ( E, n ) the set of equivalence classes. Notice that points in J e ( E, n ) canbe naturally identified with n -dimensional subspaces of T e E . Accordingly, J e ( E, n ) identifies with theGrassmannian Gr( T e E, n ) . Put J ( E, n ) := a e ∈ E J e ( E, n ) . It identifies with the Grassmanian bundle Gr( T E, n ) . For an n -dimensional submanifold L ⊂ E , denoteby [ L ] e its tangency class at e ∈ L . It is a point of J ( E, n ) which is called the first jet of L at e .Notice that if L and L are n -dimensional submanifolds of E through the same point e , then there isa (divided) chart ( x , . . . , x n , u ) on E which is adapted to both, i.e., such that, in local coordinates, L , : u = f , ( x , . . . , x n ) , for some functions f , = f , ( x , . . . , x n ) of the x ’s. Moreover, L and L have the same jet at e , i.e.,are tangent at e ≡ ( x , . . . , x n , u ) , iff: f ( x , . . . , x n ) = f ( x , . . . , x n ) = u ∂ f ∂x i ( x , . . . , x n ) = ∂ f ∂x i ( x , . . . , x n ) , i = 1 , . . . , n. In this sense first jets of submanifolds are a coordinate free version of first order Taylor polynomialsof submanifolds. Using charts adapted to submanifolds one can coordinatize J ( E, n ) in an obviousway. I leave the details to the reader. An n -dimensional submanifold L of E can be prolonged to an n -dimensional submanifold L (1) of J ( E, n ) by putting L (1) = { [ L ] e : e ∈ L } . I leave to the reader to check that L (1) is a coordinate free version of “partial derivatives of L up tothe order ”. First jets of submanifolds are equipped with a Cartan distribution playing the same role as in the previous subsection. A system of first order PDEs imposed on n -dimensional submanifoldsof E is a submanifold E ⊂ J ( E, n ) . A solutions of E is an n -dimensional submanifolds L of E suchthat L (1) ⊂ E .Finally, notice that if E has the structure of a bundle π : E −→ M over an n -dimensional manifold M , then J π is an open and dense submanifold in J ( E, n ) .2.2. Singular Solutions. Multi-valued sections. Solutions with singularities (e.g., shock waves) may have physical mean-ing. For instance, in field theory, charges are often interpreted as singularities of the fields. Therefore,it is interesting from both a mathematical and physical point of view, to study how do singularitiesof solutions propagate. We already mentioned some facts about the propagation of singularities ofsolutions in the first section. Here I show that certain kinds of singularities can be effectively treatedin geometric terms within the jet space approach to PDEs ([39, 20, 21, 28, 29, 30, 40, 31]).Let π : E −→ M be a fiber bundle as above, and L an n -dimensional, locally maximal integralsubmanifold of the Cartan distribution on J k . It is easy to see that L is almost everywhere horizontalwith respect to π k,k − [4]. Consequently, L is almost everywhere , and locally, the image of a holonomicsection of π k . However, L doesn’t need to be the image of a holonomic section everywhere . In particular, L may project under π k,k − , and, therefore, under π k, , to a submanifold with singularities. Denoteby sing L the (nowhere dense) subset of L where the singularity occur, i.e., sing L := { θ ∈ L : d θ ( π k,k − | L ) is not injective } . The subset sing L ⊂ L will be referred to as singularity locus of L . A tangent space T θ L to L at apoint θ ∈ sing L is called a singular R -plane . Singular R -planes may be characterized in terms of the metaplectic structure on C . Namely, the correspondence Ω : C × C ∋ ( X, Y ) Ω( X, Y ) := [ X, Y ] + C ∈ T J k / C is a well-defined bilinear map (the metaplectic structure). A subspace V of C θ , θ ∈ J k , is isotropic iff Ω( ξ, η ) = 0 for all ξ, η ∈ V . (Singular) R -planes are n -dimensional isotropic subspaces V . If ( d θ π k,k − ) | V is not injective, then V is singular. A typical example of singular section is the following:let n = m = k = 1 . In J consider the smooth submanifold L : ( u − x = 0 u x − x = 0 . It is easy to see that L is a locally maximal integral submanifold of the Cartan distribution. However,the singularity locus of L is sing L = ( x = 0 , u = 0 , u x = 0) = ∅ . The projection of L to J is the subset L : u − x = 0 which has a singularity in the origin and may be interpreted as the image of the multi-valued section. σ : u = ± x / The other way round, the multivalued section σ possesses a singularity in the origin, but the singularityis resolved after the first jet prolongation . More generally, the multi-valued section σ : u = ± x k +1 / possesses a singularity in the origin which is resolved after the k -th jet prolongation.Now, let n, m, k be arbitary. The above considerations suggest the following definition: a multivalued(or singular) section of π is an n -dimensional, locally maximal integral submanifold of C . HARACTERISTICS AND SINGULARITIES OF SOLUTIONS OF PDES 15 Multi-valued solutions of PDEs. Similarly, let E ⊂ J k be a PDE. Then a multivalued solutionof E is an n -dimensional, locally maximal integral submanifold of C ( E ) . Notice that singularitiesof a multivalued section L are not singularities of the submanifold L (which is always assumed tobe smooth). Rather they are singularities of the smooth map of manifolds π k,k − : L −→ J k − .Singularities of smooth maps are usually classified along the Thom-Boardman theory [16]. Here, Iwill only consider the simplest one among Thom-Boardman singularities. Namely, I assume that dπ k,k − | L has constant rank r along sing L , and that sing L ⊂ L is a smooth submanifold transversalto ker( dπ k,k − | L ) . In particular, dim sing L = r. It follows that the projections of sing L on lower order jets are also smooth submanifolds. Let type sing L = n − r. For type sing L = 1 , dim sing L = n − and one speaks about fold-type singularities . Tangent spacesto multivaled sections with fold-type singularities at points of their singular locus are called type singular R -planes . Fold-type singularities of solutions are intimately related with characteristics. Inthe following, I will only consider fold-type singularities (see, for instance, [2] and references therein).2.3. Fold-type Singularities. Shapes of fold-type singularities. Let E ⊂ J k be a PDE, and L ⊂ J k a multivalued section witha fold-type singularity along the singular locus sing L . Moreover, let θ ∈ sing L , and let S := T θ L be the type singular R -plane tangent to L at θ . If S is tangent to E then, in a sense, L is amultivalued solution of E up to the order . Notice that S , being tangent to E , cannot be arbitrary.Put θ := π k,k − ( θ ) . Clearly, S := ( d θ π k,k − )( S ) = ( d θ π k,k − )( T θ sing L ) is an n − dimensional subspace of T θ J k − . As such it can be understood as a point in J ( J k − , n − .Define Σ E := { S : S is a type singular R -plane tangent to E at θ ∈ E}⊂ J ( J k − , n − . It can be interpreted as a first order PDE for n − dimensional submanifolds of J k − . Notice that if L is a multivalued solution of E with fold-type singularity, then π k,k − (sing L ) is a solution of Σ E . Inthis sense, Σ E describes the “shape” of fold-type singularities of solutions of E .More precisely, let ( x , t ) be (divided) coordinates on M , and, as in Section 1, denote by u ℓ,J , ℓ + | J | < k , coordinates on J k − corresponding to partial derivatives ∂ | J | + ℓ ∂ x J ∂t ℓ . Thus, the u ℓ,J may beinterpreted as derivatives along the initial surface t = 0 of the initial data ∂ ℓ u /∂t ℓ , ℓ < k . Search forsolutions N of Σ E in the form N : (cid:26) t = τ ( x ) u ℓ,J = τ ℓ,J ( x ) , | I | < k . Then, in general, Σ E constraints both τ and τ ℓ,J , looks locally like Σ E : f (cid:18) x , . . . , ∂τ∂ x i , . . . , ∂τ ℓ,J ∂ x i , . . . (cid:19) = 0 , and can therefore be interpreted as a PDE for the Cauchy data, i.e., the datum of 1) a Cauchy surface Σ : t = τ ( x ) together with 2) initial data ∂ ℓ u /∂t ℓ = τ ℓ, ∅ ( x ) , ℓ < k , on it. When E is a determinedsystem of quasi-linear equations, then a Cauchy surface can only be part of a solution of Σ E if it is a characteristic surface of E (see below). This result relates the theory of multivalued solutions and thetheory of characteristic surfaces.2.3.2. The Symbol of a Differential Equations. Now, I want to relate fold-type singularities of solutionswith characteristics of a PDE. It will be useful to have an intrinsic definition of characteristic covectorsfor a generic system of (generically fully nonlinear) PDEs. I will present the new definition in the nextsection. It will generalize (and, to some extent, clarify) the analytic definition given for determined,quasi-linear systems. Here I provide some geometric preliminaries.The bundle π k,k − : J k −→ J k − , k > , is actually an affine bundle modelled over the vector bundle S k T ∗ M ⊗ J k − V E −→ J k − whose fiber at θ ∈ J k − is the vector space S k T ∗ x M ⊗ V e E , e = π k − , ( θ ) , x = π k − ( θ ) = π ( e ) (here V e E is the π -vertical tangent bundle to E ). In local coordinates, the affinestructure in π − k,k − ( θ ) looks as follows. Let θ ∈ J k and π k,k − ( θ ) = θ , and let θ have jet coordinates ( x , . . . , x n , . . . , u I , . . . ) ≡ θ, | I | ≤ k. Take v ∈ S k T ∗ x M ⊗ V e E , and let v = v i ··· i k dx i · · · dx i k ⊗ ∂∂ u . One can use the v i ··· i k ’s as coordinates in S k T ∗ x M ⊗ V e E . Then θ + v ∈ π − k,k − ( θ ) have jet coordinates ( x , . . . , x n , . . . , u J , . . . , u i ··· i k + k !( i ··· i k )! v i ··· i k , . . . ) ≡ θ + v, | J | < k, (17)(see Subsection 1.2.4 for the meaning of the combinatorial coefficient ( i · · · i k )! /k ! ). As a consequence,the vertical bundle V J k −→ J k of π k,k − is isomorphic to the vector bundle J k × J k − Σ k − −→ J k .In local coordinates, the isomorphism looks as ∂∂ u i ··· i k ( i · · · i k )! k ! dx i · · · dx i k ⊗ ∂∂ u . Let E ⊂ J k be a PDE locally given by (16). According to (17), E has a quasi-linear local descriptioniff it is an affine subbundle of J k −→ J k − . This remark provides an intrinsic definition of quasi-linear equations. Now, let E be generic, θ ∈ E , θ := π k,k − ( θ ) , e = π k, ( θ ) , and x = π k ( θ ) . Put g θ := V θ J k ∩ T θ E . In view of the affine structure in the fibers of π k,k − , g θ can be understood as asubspace of S k T ∗ x M ⊗ V e E . It is easy to see that g θ consists of v ∈ S k T ∗ x M ⊗ V e E , such that g θ : X i ≤···≤ i k ∂ F ∂ u i ··· i k · v i ··· i k = 0 In particular, if E is determined, and quasi-linear then g θ : A i ··· i k · v i ··· i k = 0 , where A = ( A i ··· i k ) is the symbol of F . For this reason g θ is called the symbol of E at θ . In thecase when E is a linear equation Du = 0 , D being a linear differential operator, the symbol g θ doesonly depend on x = π k ( θ ) . If, moreover, M is an Euclidean space, the symbol can be understood as ahomogeneous fiber-wise polynomial function σ ( D ) on T ∗ M . In this case, it can be defined analyticallyas σ ( D )( p ) = 0 , σ ( D )( p ) := e − p · x D ( e p · x ) , x ∈ M, p ∈ T x M, and plays an important role in quantization (see, e.g., [15]). HARACTERISTICS AND SINGULARITIES OF SOLUTIONS OF PDES 17 Characteristic Covectors of a PDE. Let E , θ , e , x be as in the above subsection. A non zerocovector p ∈ T ∗ x M is a characteristic covector for E at θ if there exists a non zero ξ ∈ V e E such that p · · · · · p ⊗ ξ ∈ g θ . If p = p i dx i and ξ = ξ ∂/∂ u , this means that the system of linear equations X i ≤···≤ i k p i · · · p i k ∂ F ∂ u i ··· i k (cid:12)(cid:12)(cid:12)(cid:12) θ · ξ in the m unknowns ξ has non-trivial solutions. In other words rank X i ≤···≤ i k p i · · · p i k ∂ F ∂ u i ··· i k (cid:12)(cid:12)(cid:12)(cid:12) θ < m. (18)Notice that for underdetermined systems, i.e., codim E ≤ m , every covector is characteristic, and,therefore, only the determined and overdetermined cases ( codim E ≥ m ) need to be considered. For adetermined system, condition (18) coincides with condition det X i ≤···≤ i k p i · · · p i k ∂ F ∂ u i ··· i k (cid:12)(cid:12)(cid:12)(cid:12) θ = 0 which I already considered in Section 1. For quasi-linear, determined systems, p = p i dx i is character-istic iff det A ( p ) = 0 . One concludes that the notion introduced here is a coordinate free version of the one introduced inSection 1. Notice that the characteristic condition (18) depends on the point θ in E and, in general,changes from point to point.2.3.4. Characteristic Covectors and Fold-type Singularities. In Subsection 1.2.2, I presented an (infor-mal) argument showing that characteristics are actually related to singularities of solutions: namely,singularities of solutions of quasi-linear, determined systems of PDEs occur along characteristic sur-faces. Here I present a rigorous, intrinsic argument which applies to generic nonlinear PDEs. First Ineed few remarks. Let θ = [ σ ] kx ∈ E . The set of characteristic covectors at θ is the characteristic variety at θ . In this way, one gets a (possibly singular) fiber bundle over E whose fibers are, by definition,characteristic varieties. Call it the characteristic bundle of E . Now, consider the R -plane R θ ⊂ T θ J k − corresponding to θ . The projection d θ π k − identifies R θ with T x M . Thus, if p is a non-zero covectorin T ∗ x M , one can understand its kernel as an ( n − -dimensional subspace in R θ . I denote it by ker θ p .Specifically, ker θ p := ( d x j k − σ )(ker p ) . It holds the following proposition: if E is a formally integrablePDE then the equation of fold-type singularities is “dual” to the characteristic bundle in the followingsense: Σ E = { ker θ p : p is a characteristic covector of E at θ ∈ E} . (19)(for this part of the statement see, for instance, [2, Theorem 5.2], Theorem 5.2. Formal integrabilityroughly means that: if a Taylor polynomial of the order k + ℓ is a solution of E up to the order ℓ , thenit can be “completed” to a formal solution , i.e., a solution in the form of a (possibly non-converging)Taylor series. If, moreover, E is determined, quasi-linear and A ( p ) is generically invertible (i.e., E can be generically recast in normal form), then Σ E is locally equivalent to (12) (which constraints theshape of a Cauchy surface Σ ) + (11) (which constraints the initial data on Σ ) (see [40]). Concluding,the equation for fold-type singularities is an equation for Cauchy data (including both the Cauchysurface and the initial data on it), telling us that 1) fold-type singularities may only occur along acharacteristic surface and that 2) initial data on a characteristic surface may not be assigned arbitrarily. An Example: the 2D Klein-Gordon Equation. The equation for characteristic surfaces of a quasi-linear system does not contain a full information on the original equation. In fact, it does only dependon its symbol. However, the equation for singularities may contain a full information on the originalequation. I briefly illustrate this phenomenon with a simple example. For details, see [39, 40] (see also[26], where more examples from Mathematical Physics can be found).Consider the Klein-Gordon equation on the -dimensional Minkowski space-time: E KG : u tt − u xx + m u = 0 . I already showed that characteristic surfaces of the Klein-Gordon equation are null hypersurfaces. Anhypersurface Σ : z ( t, x ) = 0 is a characteristic surface iff ( z t − z x ) | z =0 = 0 This shows that z t = 0 so that Σ is actually of the form Σ : t = τ ( x ) , with τ x = 1 which is the -dimensional eikonal equation. The fold-type singularity equation Σ E KG ⊂ J ( J , can be easily computed, using, for instance, (19). Coordinatize J by x, t, u, u x , u t . Since characteristicsurfaces are of the form Σ : t = τ ( x ) we can interpret x as independent variable, and coordinatize J ( J , by x, t, u, u x , u t , t ′ , u ′ , u ′ x , u ′ t , where f ′ means df /dx . Then Σ E KG : ( t ′ ) = 1 u ′ = u x + t ′ u t u ′ x = m u + t ′ u ′ t . Eliminating u x , one gets the following second order system for the Cauchy data t, u, u t : (cid:26) ( t ′ ) = 1 u ′′ − t ′′ u t − t ′ u ′ t = m u . Notice that Σ E KG contains the mass parameter m . Actually, it can be proved, by purely geometricmethods, that Σ E KG contains a full information about E KG . More generally, understanding when aPDE can be reconstructed from its singularity equation is an interesting open problem (in some sense,as I already outlined in the introduction, analogous to quantization) that has been first addressed byVinogradov in simple situations [39, 40].2.3.6. Bicharacteristics of Determined Systems of PDEs. Let E ⊂ J k be a determined system of PDEs,and L a multi-valued solution with a fold-type singularity along sing L . If one interprets L as a wavepropagating in the space-time, then it is natural to intepret π k (sing L ) ⊂ M as its wave-front . Recallthat the wave-front is a characteristic surface. It can be shown that sing L is equipped with a canonicalfield of directions, i.e., a -dimensional distribution. Accordingly, the wave-front of L is foliated by -dimensional submanifolds [21, 19]. In the case when E is a linear system, this is a classical result, the -dimensional leaves of L are called bicharacteristics , and one usually says that wave-fronts propagatealong bicharacteristics . I will not discuss this result in full generality, which would require too muchspace. Instead, I will consider, in the next section, the case when the symbol g θ at θ ∈ E does onlydepend on x = π k ( θ ) . In this case the wave-front is a solution of a genuine first order PDE in onedependent variable, which locally looks like (12), where the B ’s does only depend on the x ’s and τ .We are thus led to consider the class of first order PDEs in one dependent variable. This will be themain topic of the last section. Notice once again that, from a physical point of view, the passage from π k (sing L ) to its bicharacteristics can be interpreted as the passage from a wave optics (the dynamicsof wave-fronts) to a geometric optics (the dynamics of rays). HARACTERISTICS AND SINGULARITIES OF SOLUTIONS OF PDES 19 Bicharacteristics and the Hamilton-Jacobi Theory Contact Geometry of First Jets of Functions. Jets of functions. In this section, I focus on first order PDEs in one dependent variable. Theequations for characteristic surfaces of determined, quasi-linear systems whose symbol does only dependon independent variables (for instance, linear systems) are precisely of this kind. For the sake ofsimplicity, I will suppose that the equation under consideration is imposed on a real function f ona manifold M . This is always true locally. More generally, one could consider equations imposedon sections of a bundle with one dimensional fibers or on -codimensional submanifolds of a givenmanifold. Similar results as the one presented in this lecture hold for these (unparameterized) cases.Notice that a real function f on a manifold M can be understood as a section of the trivial bundle π M : M × R −→ M . In the following, the first jet space of π M will be denoted by J ( M ) . A first orderPDE in one dependent variable is then a hypersurface E in J ( M ) . The most remarkable property of J ( M ) is that it is equipped with a natural contact structure . Recall that a contact structure on a n +1 dimensional manifold N is an hyperplane distribution with non-degenerate, associated metaplecticstructure. A contact structure can be presented as the kernel distribution ker α of a contact form -form α such that dα is non degenerate on ker α . The contact structure in J ( M ) is given by theCartan distribution . A contact -form on J ( M ) can be defined as follows. First of all, notice thatthere is a canonical isomorphism (of bundles over M ) J ( M ) ≃ T ∗ M × R given by [ f ] x ( d x f, f ( x )) . Denote by u : M × R −→ R the canonical function on M × R , i.e., theprojection onto the second factor, and by θ the tautological -form on T ∗ M . Abusing the notation, Idenote by the same symbols u and θ , the pull-backs on J ( M ) . The -form α := du − θ ∈ Λ ( J ( M )) is a contact form. Indeed, it is easy to see that its coordinate description in jet coordinates is α = du − u i dx i . Moreover, ker α is precisely the n -dimensional Cartan distribution, and dα is non-degenerate over it.The contact geometry of J ( M ) is intimately related to the symplectic geometry of T ∗ M .3.1.2. Jacobi Algebra of a Contact Manifold. Recall that functions on a symplectic manifold forma Poisson algebra equipped with a morphism of Lie algebras into infinitesimal syplectomorphisms.Similarly, functions on a contact manifold form a Jacobi algebra equipped with a morphism of Liealgebras into infinitesimal contactomorphisms . Let us illustrate this in the simple case of the contactmanifold J ( M ) . In this case, a contactomorphism is nothing but a diffeomeorphism J ( M ) −→ J ( M ) preserving the Cartan distribution. Similarly, an infinitesimal contactomorphism is a vector field X over J ( M ) whose flow preserves the Cartan distribution. In other words, L X α = λα for some function λ ∈ C ∞ ( J ( M )) . A smooth function f on J ( M ) determines an infinitesimal contactomorphism and vice-versa as follows.Let ∂/∂u be the vector field on J ( M ) determined by the canonical coordinate vector field on R andthe identification J ( M ) ≃ T ∗ M × R . The vector field ∂/∂u is transversal to the Cartan distribution,so that T J ( M ) = h ∂/∂u i ⊗ C . Consider the -form δf := df − ∂f∂u α. It is easy to see that δf ∈ Ann( ∂/∂u ) so that there exists a unique vector field Y f in the Cartandistribution such that i Y f dα = δf. The vector field X f := Y f − f ∂∂u is an infinitesimal contactomorphism and every infinitesimal contactomorphism X is of the form X = X f , with f = − α ( X ) . Notice that ∂/∂u = − X . Finally, for any two smooth functions f, g on J ( M ) one has X fg = f X g + gX g − f gX [ X f , X g ] = X { f,g } with { f, g } := X f ( g ) − X ( f ) g This shows that smooth functions on J ( M ) equipped with the bracket {− , −} form a Jacobi algebraisomorphic (as a Lie algebra) to the Lie agebra of infinitesimal contactomorphisms. In local coordinates, X f = ∂f∂u i ∂∂x i − (cid:18) ∂f∂x i + u i ∂f∂u (cid:19) ∂∂u i + (cid:18) u i ∂f∂u i − f (cid:19) ∂∂u and { f, g } = ∂f∂u i ∂g∂x i − ∂g∂u i ∂f∂x i + u i (cid:18) ∂f∂u i ∂g∂u − ∂g∂u i ∂f∂u (cid:19) − f ∂g∂u + g ∂f∂u . First Order Scalar PDEs. (Bi)characteristic Foliation and the Method of Characteristics. Now, let E ⊂ J ( M ) be a (codi-mension ) PDE. As a minimal regularity condition, I assume that the Cartan distribution C ( E ) on E is regular (i.e., constant dimension). It then follows that dim C ( E ) = 2 n − . Since dα is a symplectic form on C , it must degenerate on C ( E ) along a field of directions ℓ ( E ) ⊂ C ( E ) on E . Integral manifolds of ℓ ( E ) foliate E and are called characteristic lines of E , or, bicharacteristics if E is the equation for characteristic surfaces of a determined, quasi-linear system (whose symbol doesonly depend on independent variables). If E is assigned as the zero locus of a function F on J ( M ) ,i.e., E : F = 0 , then ℓ ( E ) is spanned by the vector field Y F .The key remark here is that ℓ ( E ) is tangent to every (multivalued) solution of E . Therefore, solu-tions themselves are foliated by -dimensional leaves. More generally, it can be proved that solutionsof the fold-type singularity equation of a determined system of PDEs are foliated by -dimensionalleaves . One concludes that singularities of solutions of determined systems of PDEs propagate alongbicharacteristics .The existence of characteristic lines suggests a way to solve the Cauchy problem for E . Namely,let Σ ⊂ M be an hypersurface and µ a smooth function on it. Search for a solution f of E suchthat f | Σ = µ , i.e., understand (Σ , µ ) as Cauchy data for E . If Σ is not a characteristic surface for E ,then the goal can be achieved as follows. First, notice that there exists a unique ( n − -dimensionalsubmanifold N in E such that 1) N projects to the graph of µ under π , , 2) N is integral for C ( E ) , 3) N is transversal to ℓ ( E ) [4] (see also [41] for the case of an Hamilton-Jacobi equation). The submanifold N encodes the information about Σ , µ , and derivatives of µ along Σ . The union of characteristic linespassing through N is, by construction, an n -dimensional integral manifold of C ( E ) horizontal withrespect to fibers of π , . As such, it is the image of the first jet prolongation of a solution of E agreeingwith the Cauchy data (Σ , µ ) . Notice that, if E : F = 0 , and Σ is in the form Σ : z = 0 , then thecondition of not-being characteristic is ∂F∂u i ∂z∂x i (cid:12)(cid:12)(cid:12)(cid:12) z =0 = 0 . HARACTERISTICS AND SINGULARITIES OF SOLUTIONS OF PDES 21 In this case, solving the assigned Cauchy problem amounts to solve the following system of ODEs: ˙ x i = ∂F∂u i ˙ u i = − ∂F∂x i − u i ∂F∂u ˙ u = u i ∂F∂u i , with initial data on N . This is nothing but the classical method of characteristics to solve st orderscalar PDEs.3.2.2. An example. Consider the following Cauchy problem in two independent variables x , x : u − ∂u∂x ∂u∂x = 0 ,u | x =0 = ( x ) . Then F = u − u u and Σ : z ( x , x ) = x = 0 is (almost everywhere) non-characteristic since ∂F∂u i ∂z∂x i (cid:12)(cid:12)(cid:12)(cid:12) z =0 = − u | x =0 = − x = 0 almost everywhere . The Cauchy data determine a -dimensional integral manifold N for C ( E ) which is parametrically givenby N : x = sx = 0 u = s u = 2 su = s/ . (20)Indeed, one may check that this is the unique choice of N satisfying all the required properties. Onealso has Y F = − u ∂∂x − u ∂∂x − u u ∂∂u − u ∂∂u − u ∂∂u . So characteristic lines may be computed integrating equations ˙ x = − u ˙ x = − u ˙ u = − u u ˙ u = − u ˙ u = − u , with (parametric) initial conditions given by (20). Integrating and eliminating the parameters (andthe higher derivatives) one gets the solution u = (4 x + x ) . Hamilton-Jacobi Theory. Hamilton-Jacobi equations. It may happen that E is the preimage of an hypersurface H of T ∗ M under the canonical projection J ( M ) −→ T ∗ M . If E is locally given by E : { F = 0 , then F canbe chosen such that ∂F∂u = 0 , i.e., F = F ( x , . . . , x n , u , . . . , u n ) . In other words, F is the pull-backof a function H = H ( x , . . . , x n , p , . . . , p n ) on T ∗ M . In this case, E is precisely the Hamilton-Jacobi equation associated to the Hamiltonian system ( T ∗ M, H ) . Finding characteristics of E is thenthe same as finding characteristics of H , i.e., the degeneracy lines of the restriction to H of the canonical symplectic form Ω := − dθ on T ∗ M . In their turn, characteristics of H are trajectories ofthe Hamiltonian vector field X H defined by i X H Ω = dH. Thus the method of characteristics to solve a Cauchy problem for an Hamilton-Jacobi equation consistsin integrating the Hamilton equations with suitable initial data.3.3.2. Hamilton-Jacobi theorem. I conclude this section reviewing the Hamilton-Jacobi theory of aHamiltonian system ( T ∗ M, H ) . First, I specialize the geometric definition of PDE, (multivalued)solutions, and the method of (bi)characteristics to this context.As already noticed, if one is interested in first order PDEs, in one independent variable, of the form H ( x , . . . , x n , u , . . . , u n ) = E , E being a constant, then one may understand them geometrically ashypersurfaces H in T ∗ M , M being a manifold (of independent variables) coordinatized by x , . . . , x n .Any such hypersurface will be referred to as a Hamilton-Jacobi equation . The cotangent bundle T ∗ M comes equipped with its canonical symplectic structure Ω . Locally, Ω = dp i ∧ dx i , where the p i ’s arecotangent coordinates conjugate to the x i ’s. The symplectic form Ω plays here a similar role as theCartan distribution in the general theory of PDEs. The geometric definition of solutions of H is clear:a solution is a (local) function f on M such that df takes values into H . Notice that the image of df is a Lagrangian submanifold of T ∗ M horizontal with respect to the projection T ∗ M −→ M , andviceversa: every Lagrangian submanifold horizontal with respect to T ∗ M −→ M is locally the imageof df for some local function f on M . Similarly as in the general case, one could be interested in multivalued (singular) solutions . As in the general case, a geometric definition is obtained relaxing thehorizontality condition from the definition of a solution. Thus, a multivalued solution of a HamiltonJacobi equation H ⊂ T ∗ M is a Lagrangian submanifold L ⊂ T ∗ M such that L ⊂ H . Example 2. Let M = R , and H : p + q = E , E > . Then H is a multivalued solution, correspondingto the multivalued function implicitly defined by (cid:18) dfdq (cid:19) + q = E. Now notice that every Hamilton-Jacobi equation H comes equipped with a canonical field of direc-tions ℓ ( H ) : the degeneracy distribution of Ω | H . Namely, for p ∈ H ℓ ( H ) p := { ξ ∈ T p H : i ξ Ω | H = 0 } . One of the key points here is the following version of the Hamilton-Jacobi Theorem : ℓ ( H ) is tangent toevery multivalued solution of H . In particular, ℓ ( H ) restricts to a field of (bi)characteristic directionson any (multivalued) solution. When H : H = E for some function H on T ∗ M , then the Hamiltonianvector field X H of H is tangent to H and generates the field of directions ℓ ( H ) . Therefore, to find (mul-tivalued) solutions of the Hamilton-Jacobi equation H , it is enough to start from an ( n − -dimensionalsubmanifold N ⊂ H such that 1) N is isotropic, 2) N is transversal to X H , and then move N alongthe flow of X H . The n -dimensional submanifold swept in this way is, by construction, Lagrangian,and, therefore, is a multivalued solution. This procedure specializes the method of characteristics toHamilton-Jacobi equations. Thus, one can get solutions of the Hamilton-Jacobi equation H : H = E ,from solutions of the Hamilton equations (for the flow of X H ). The converse is also true to someextent: any multivalued solution L of the Hamilton-Jacobi equation is invariant under the flow of X H :restricting X H to L reduces by n the number of degrees of freedom in the Hamilton equations, andsimplifyies the integration problem.The geometric version of the Hamilton-Jacobi theorem recalled above (which is actually a simpleremark in geometric terms) has been generalized by the Geometric Mechanics community to manydifferent contexts: Lagrangian mechanics [7], non-holonomic systems [18, 12, 35, 8, 22], almost Poisson HARACTERISTICS AND SINGULARITIES OF SOLUTIONS OF PDES 23 manifolds [10], mechanics on Lie algebroids [23], field theory [11, 13, 14] and higher derivative fieldtheory [42, 43, 44]. Another key aspect of the Hamilton-Jacobi theory is the role played by completeintegrals of the Hamilton-Jacobi problem. Complete integrals of the Hamilton-Jacobi problem. Consider the family of Hamilton-Jacobiequations H = E . It is sometimes collectively referred to as the Hamilton-Jacobi problem . A (local) complete integral of the Hamilton-Jacobi problem is then a (local) Lagrangian foliation F of T ∗ M whose leaves are multivalued solutions. Notice that having a complete integral amounts to havingan n -parameter family of solutions depending on the parameters in an essential way . Now, supposethat the space of leaves of F is a smooth manifold Q and that the canonical map ϕ : T ∗ M −→ Q is a submersion. Let q , . . . , q n be coordinates in Q . Thus, they are precisely the n -parametersparameterizing athe complete integral. Interpret the q i ’s as (local) functions on T ∗ M . Since fibersof ϕ are contained into the level surfaces of H , one can always choose one of the q i ’s to be H itself.Applying the Hamilton-Jacobi theorem to the Hamilton-Jacobi equations q i = c i , with c i ’s constant,one sees that the q i ’s are actually n independent and Poisson-commuting functions on T ∗ M (see, forinstance, [41]). It follows that ( T ∗ M, Ω , H ) is a (locally) integrable Hamiltonian system! This resultclarifies the use of the Hamilton-Jacobi problem to integrate Hamilton equations.On another hand, let F be a complete integral of the Hamilton-Jacobi problem such that: 1) thespace of leaves of F is a smooth manifold Q with local coordinates q , . . . , q n and the canonical map ϕ : T ∗ M −→ Q is a submersion, 2) the leaves of F are all graphs of (closed) -forms, i.e., sections of T ∗ M −→ M . It follows thet there is a diffeomorphism Φ : M × Q ≃ T ∗ M , locally given by: Φ ∗ ( x i ) = x i , Φ ∗ ( p i ) = ∂W∂x i , for some local function W = W ( x , . . . , x n , q , . . . , q n ) . Clearly, M × Q inherits a symplectic structure Ω F := Φ ∗ (Ω) and an Hamiltonian function H F := Φ ∗ (Ω) . It is easy to see that the local functions on M × Q defined by P i := ∂W∂q i are conjugate to the q i ’s, i.e., Ω F = dP i ∧ dq i , moreover, the Hamiltonian system ( M × Q, Ω F , H F ) is canonically isomorphic to ( T ∗ M, Ω , H ) , but H F does not depend on the P i ’s. Therefore, in the coordinates . . . , q i , . . . , P i , . . . , the Hamilton equationson ( M × Q, Ω F , H F ) look particularly simple ˙ q i = 0˙ P i = − ∂H F ∂q i = const . In this sense W generated a canonical transformation that simplifies the original problem . In thisrespect, see [32], where a quantum version of the last result is also proposed. It is an interesting issuedeveloping Hamilton-Jacobi techniques for the computation of quantum propagators (see, e.g., [46]).3.3.4. Hamiltonian dynamics of boundary data. In the case when the Hamiltonian system ( T ∗ M, Ω , H ) comes from a regular Lagrangian system, with Lagrangian L ∈ C ∞ ( T M ) , then one can choose Q = M and there is a canonical choice for W , namely W ( x, q ) = Z t t L ( γ ( t ) , ˙ γ ( t )) dt where γ is the solution of the Euler-Lagrange equations such that γ ( t ) = x , and γ ( t ) = q . In thiscase, the diffeomorphism Φ − : T ∗ M ≃ M × Q “ transforms the symplectic manifold of initial data of Hamilton equations into a symplectic manifold of boundary data of the Euler-Lagrange equations ”. Inparticular, there is a Hamiltonian system on boundary data (see [32]). Rovelli [38] showed that thisconsiderations can be generalized (in a covariant way) to any classical field theory. In particular he wasable to write down a Hamilton-Jacobi equation on the space of boundary data of the field equationsand show that the action functional provide a canonical solution. He also used the Hamilton-Jacobiequation to perform a transition to the quantum regime . In the case of Einstein gravity, he obtainedthe Wheeler-De Witt equation (see also [3]). Unfortunately, Rovelli’s theory is rather far from beingfully general and mathematically rigorous. More recently, I and G. Moreno proved [34] that whateverthe Lagrangian field theory one starts from (any number of dependent and independent variables,derivatives, and gauge symmetries), there is a canonical Hamiltonian system on the space of boundarydata (see also [33]) which is, in a sense, equivalent to the Euler-Lagrange equations. We achieved thisresult in full rigour within the jet space (and, in particular, ∞ -jet space) approach to PDEs. However,it is not clear what is the precise relation to Hamilton-Jacobi theory. In particular, it is not clear inwhat sense the action provide a complete integral of the Hamilton-Jacobi problem, nor if one couldactually quantize along this lines. Notice that, in this formalism, characteristic Cauchy data, andsingularities of solutions should play a distinguished role. Clarifying these issues is, in my opinion, aninteresting open problem. Conclusions Consider a determined system of quasi-linear PDEs governing the dynamics of a field in the space-time. The boundary of a disturbance in the field, i.e., a wave-front, is a characteristic surface in thespace-time. In their turn, wave-fronts propagate along bicharacteristics and bicharacteristics are oftentrajectories of a Hamiltonian system. I just described a mathematically rather precise way to pass fromwaves to rays, or from fields to particles. The possibility of making this passage may be understoodas a manifestation of the quantum-mechanical wave-particle duality. Accordingly, the transitionfield equations = ⇒ characteristic surfaces = ⇒ bicharacteristics (21)may be understood as analogous to the transitionquantum mechanics = ⇒ short wave-lenght limit = ⇒ classical mechanics. (22)Notice that in both transitions one progressively lose information. Therefore, it should be expectedthat, performing the inverse transitions (in particular, quantizing ) requires additional information.Vinogradov conjectured (see the appedix of [4]) that part of this information is actually contained inthe singularity equations. The idea that the geometric theory of PDEs can account for quantization isintriguing and worth to be explored. Acknowledgments. I thank the scientific committee of the XXII International Fall Workshop onGeometry and Physics for the opportunity to give the mini-course on which this review is based. I alsothank Beppe Marmo for suggesting me the topic of the mini-course, for carefully reading a preliminaryversion of this paper, and for all its valuable comments. References [1] I. G. Avramidi, Matrix general relativity: a new look at old problems, Class. Quant. Grav. (2004) 103–120,e-print: arXiv:hep-th/0307140.[2] M. Bächtold, Fold-type solution singularities and characteristic varieties of nonlinear PDEs, Zurich, 2009, PhDThesis.[3] P. G. Bergmann, Hamilton-Jacobi and Schrödinger Theory in Theories with First-Class Hamiltonian Constraints, Phys. Rev. (1966) 1078.[4] A. V. Bocharov et al., Symmetries and conservation laws for differential equations of Mathematical Physics, (I. S.Krasil’shchik, and A. M. Vinogradov eds.) Transl. Math. Mon. , AMS, Providence - RI, 1999. HARACTERISTICS AND SINGULARITIES OF SOLUTIONS OF PDES 25 [5] G. Boillat, Sur l’équation générale de Monge-Ampère d’ordre supérieur, C. R. Acad. Sci. Paris I: Math. (1992)1211.[6] R. Bryant, et al., Exterior differential systems, Springer-Verlag, New York, 1991, Chapter V.[7] J. F. Cariñena, et al., Geometric Hamilton-Jacobi theory, Int. J. Geom. Meth. Mod. Phys. (2006) 1417–1458.[8] J. F. Cariñena, et al., Geometric Hamilton-Jacobi theory for nonholonomic dynamical systems, Int. J. Geom. Meth.Mod. Phys. (2010) 431.[9] R. Courant, and D. Hilbert, Methods of Mathematical Physics II, Wiley-Interscience, New York, 1962.[10] M. de Leon, D. Martín de Diego, M. Vaquero, A universal Hamilton-Jacobi theory, e-print: arXiv:1209.5351.[11] M. de Leon, J. C. Marrero, D. Martín de Diego, A Geometric Hamilton-Jacobi Theory for Classical Field Theories;e-print: arXiv:0801.1181.[12] M. de Leon, J. C. Marrero, and D. Martin de Diego, Linear almost Poisson structures and Hamilton-Jacobi equation,e-print: arXiv:0801.4358.[13] M. de Leon, et al., Hamilton-Jacobi theory in k -symplectic field theory, Int. J. Geom. Meth. Mod. Phys. (2010)1491.[14] M. de Leon, and S. Vilariño, Hamilton-Jacobi theory in k -cosymplectic field theory, e-print: arXiv:1304.3360.[15] G. Esposito, G. Marmo, and E. C. G. Sudarshan, From classical to quantum mechanics, Cambridge UniversityPress, Cambridge, 2004.[16] M. Golubitsky, and V. Guillemin, Stable mappings and their singularities, Springer-Verlag, New York, 1973.[17] V. Guillemin, and S. Sternberg, Shlomo, Geometric asymptotics, AMS, Providence, 1977.[18] D. Iglesias-Ponte, M. de Leon, and D. Martin de Diego, Towards a Hamilton-Jacobi theory for nonholonomicmechanical systems, J. Phys. A: Math. Theor. (2008) 015205.[19] I. S. Krasil’shchik, V. V. Lychagin, and A. M. Vinogradov, Geometry of jet spaces and nonlinear partial differentialequations, Adv. Stud. Cont. Math. , Gordon and Breach Science Publishing, New York, 1986.[20] A. P. Krishchenko, The structure of singularities of the solutions of quasilinear equations, Uspekhi Mat. Nauk. (1976) 219.[21] A. P. Krishchenko, Folding of R -manifolds, Vestnik Moskov. Univ. Ser. I Math. Mech. (1977) 17.[22] M. Leok, T. Ohsawa, and D. Sosa, Hamilton-Jacobi theory for degenerateLlagrangian systems with holonomic andnonholonomic constraints, J. Math. Phys. (2012) 072905.[23] M. Leok, and D. Sosa, Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids, J.Geom. Mech. (2012) 421.[24] T. Levi-Civita, Caratteristiche e bicaratteristiche delle equazioni gravitazionali di Einstein I, II, Rend. Accad Naz.Lincei (Sci. Fis. Mat. Nat.) (1930) 3, 113.[25] T. Levi-Civita, Caratteristiche dei sistemi differenziali e propagazione ondosa, Zanichelli, Bologna, 1831.[26] F. Lizzi et al., Eikonal type equations for geometrical singularities of solutions in field theory, J. Geom. Phys. (1994) 211.[27] R. K. Luneburg, Mathematical theory of optics, University of California press, Berkley, 1964.[28] V. V. Lychagin, On singularities of the solutions of differential equations, Dokl. Akad. Nauk. SSSR (1980) 794.[29] V. V. Lychagin, Geometric singularities of the solutions of nonlinear differential equations, Dokl. Akad. Nauk. SSSR (1980) 1299.[30] V. V. Lychagin, Singularities of multivalued solutions on nonlinear differential equations, and nonlinear phenomena, Acta Appl. Math. (1985) 135.[31] V. V. Lychagin, Geometric theory of singularities of solutions of nonlinear differential equations, J. Soviet Math. (1990) 2735.[32] G. Marmo, G. Morandi, and N. Mukunda, The Hamilton-Jacobi theory and the analogy between classical andquantum mechanics, J. Geom. Mech. (2009) 317.[33] G. Moreno, The geometry of the space of Cauchy data of nonlinear PDEs, CEJM , in press; e-print: arXiv:1207.6290.[34] G. Moreno, and L. Vitagliano, Covariant space + time splittings of classical field theories, in preparation. Talk deliv-ered by the second author at the “ Third Iberoamerican meeting on Geometry, Mechanics and Control ”, Salamanca,September 03–07, 2012.[35] T. Ohsawa, and A. M. Bloch, Nonholonomic Hamilton-Jacobi equation and Integrability, J. Geom. Mech. (2009)461.[36] P. Orsi, Teoria delle caratteristiche ed equazioni ondulatorie quantiche, Quaderni di Fisica Teorica , Bibliopolis,Napoli, 2001.[37] G. Racah, Caratteristiche delle equazioni di Dirac e principio di indeterminazione, il Nuovo Cimento (1932) 28.[38] C. Rovelli, Covariant Hamiltonian formalism for field theory: Hamilton-Jacobi equation on the space G , Lect. NotesPhys. (2003) 36.[39] A. M. Vinogradov, Many-valued solutions, and a principle for the classifications of nonlinear differential equations, Dokl. Akad. Nauk. SSSR (1973) 11. [40] A. M. Vinogradov, Geometric singularities of solutions of nonlinear partial differential equations, in: DifferentialGeometry and its Applications (Brno, 1986), Math. Appl. (East Europeand Ser.) , Reidel, Dordrecht, 1987, p.359.[41] A. M. Vinogradov, and B. A. Kupershmidt, The Structure of Hamiltonian Mechanics, London Math. Soc. Lect.Notes Ser. , Cambridge Univ. Press, London, 1981, p. 173.[42] L. Vitagliano, The Hamilton-Jacobi formalism for higher order field theory, Int. J. Geom. Meth. Mod. Phys. (2010) 1413.[43] L. Vitagliano, Hamilton-Jacobi diffieties, J. Geom. Phys. (2011) 1932.[44] L. Vitagliano, Geometric Hamilton-Jacobi field theory, Int. J. Geom. Meth. Mod. Phys. (2012) 1260008.[45] T. E. Whittaker, Note on the law that light rays are the null geodesics of a gravitational field, Math. Proc. CambridgePhil. Soc. (1928) 32.[46] T. E. Whittaker, On Hamilton’s principal function in quantum mechanics, Proc. Roy. Soc. Edinburgh A: Math.Phys. Sci. (1941) 1. E-mail address : [email protected]@unisa.it