Huygens Principle, Integrable PDEs, and Solitons

Transcription

1 Huygens Principle, Integrable PDEs, and Solitons Jorge P. Zubelli July 20, zubelli/ download: zubelli/ Thanks: Prof. Marco Calahorrano Joint work with Fabio Chalub - U. Lisboa

2 Overview Free Space Waves in (2+1) Dimensions (3+1) Dimensions

3 Overview Free Space Waves in (2+1) Dimensions (3+1) Dimensions Huygens principle in Hadamard s sense

4 Overview Free Space Waves in (2+1) Dimensions (3+1) Dimensions Huygens principle in Hadamard s sense The Hadamard conjecture

5 Overview Free Space Waves in (2+1) Dimensions (3+1) Dimensions Huygens principle in Hadamard s sense The Hadamard conjecture Calogero-Moser Systems

6 Overview Free Space Waves in (2+1) Dimensions (3+1) Dimensions Huygens principle in Hadamard s sense The Hadamard conjecture Calogero-Moser Systems Solitons - KdV, KP, and Friends

7 Overview Free Space Waves in (2+1) Dimensions (3+1) Dimensions Huygens principle in Hadamard s sense The Hadamard conjecture Calogero-Moser Systems Solitons - KdV, KP, and Friends Huygens for Dirac Systems

8 Overview Free Space Waves in (2+1) Dimensions (3+1) Dimensions Huygens principle in Hadamard s sense The Hadamard conjecture Calogero-Moser Systems Solitons - KdV, KP, and Friends Huygens for Dirac Systems

9 Cauchy Problem for the Wave Equation ψ def = ( t 2 )ψ = 0 (1) ψ = f (2) t=0 t ψ = g. (3) t=0

10 Domains of Dependence

11 Domains of Dependence

12 Domains of Dependence

13 Huygens Construction

14 Huygens Construction

15 Huygens Construction

16 Hadamard s Approach Huygens Property (A) Major Premise The action of phenomena produced at the instant t = 0 on the state of matter at the later time t = t 0 takes place by the mediation of every intermediary instant t = t, i.e., (assuming 0 < t < t 0 ),...

17 Hadamard s Approach Huygens Property (A) Major Premise The action of phenomena produced at the instant t = 0 on the state of matter at the later time t = t 0 takes place by the mediation of every intermediary instant t = t, i.e., (assuming 0 < t < t 0 ),... (B) Minor Premise If we produce a luminous disturbance localized in a neighborhood of 0, its effect after an elapsed time t 0 will

18 be localized in a neighborhood of the sphere centered at 0 with radius ct 0.

19 be localized in a neighborhood of the sphere centered at 0 with radius ct 0. (C) Conclusion In order to calculate the effect of our initial luminous phenomenon produced at 0 at t = 0, we may replace it by a proper system of disturbances taking place at t = t and distributed over the surface of the sphere with center 0 and radius ct.

20 be localized in a neighborhood of the sphere centered at 0 with radius ct 0. (C) Conclusion In order to calculate the effect of our initial luminous phenomenon produced at 0 at t = 0, we may replace it by a proper system of disturbances taking place at t = t and distributed over the surface of the sphere with center 0 and radius ct. (A) & (B) = (C)

21 be localized in a neighborhood of the sphere centered at 0 with radius ct 0. (C) Conclusion In order to calculate the effect of our initial luminous phenomenon produced at 0 at t = 0, we may replace it by a proper system of disturbances taking place at t = t and distributed over the surface of the sphere with center 0 and radius ct. (A) & (B) = (C)

22 Free Wave Equation Notation: (n + 1) dimensions, n = number of space dimensions and 1 indicates time dimension. space-time: (x 0, x 1,..., x n ) R n+1 where x 0 = t is the time variable. wave operator: ψ def = ( 2 t c 2 )ψ, where ψ = n x 2 i ψ. i=1

23 Free Wave Equation Notation: (n + 1) dimensions, n = number of space dimensions and 1 indicates time dimension. space-time: (x 0, x 1,..., x n ) R n+1 where x 0 = t is the time variable. wave operator: ψ def = ( 2 t c 2 )ψ, where ψ = n x 2 i ψ. i=1

24 Cauchy problem: ψ = 0 (4) ψ = f (5) t=0 t ψ = g. (6) t=0

25 focus first: n = 3. Method of Spherical Means

26 Method of Spherical Means focus first: n = 3. M r [u] def = 1 4πr 2 x x =r u(x ) ds(x ). (7) Exercise: Solution to the Cauchy problem given by ψ = d dt (tm ct [f]) + tm ct [g].

27 Consequence: At the point ( t, x) R 4 solution ψ depends only on the initial data at the points on the sphere in R 3 of radius c t centered at x. Hadamard s Method of Descent: Use the sol. of the given equation in a higher dimensional space to produce the sol. in a lower dimensional space by introducing extra dummy variables. Using solutions of the wave eq for n = 3 one obtains a solution of the wave equation for n = 2. For n = 2 the dependence is on the interior and the surface of the

28 ball. Theorem: For n > 1 odd: Solution of the wave equation depends on the initial data on the intersection of the light cone with the initial data manifold t = 0. For n even: Solution depends on the values of the data on the closure intersection of the interior part of the light cone and the initial data manifold t = 0. Idea of the proof: Use the method of spherical means indicated above for n odd, and descending to even n. Remarks on the Strict Huygens Property: Fascinating property of the solutions to the wave equation in 3 spatial dimensions

29 meaningful transmission of information instantaneous signal in 3 spatial dimensions remains instantaneous Thus our actual physical world, in which acoustic and electromagnetic signals are the basis of communication, seems to be singled out among other mathematically conceivable models by intrinsic simplicity and harmony (Courant & Hilbert, Methods of Math. Phys.)

30 meaningful transmission of information instantaneous signal in 3 spatial dimensions remains instantaneous Thus our actual physical world, in which acoustic and electromagnetic signals are the basis of communication, seems to be singled out among other mathematically conceivable models by intrinsic simplicity and harmony (Courant & Hilbert, Methods of Math. Phys.)

31 Strict Huygens Let L be a strictly hyperbolic (linear) second order operator defined on a causal domain (e.g d Alembertian + lower order terms) Let S be a space-like surface Consider the Cauchy-problem Lψ = 0 ψ S = f ν ψ S = g (8)

32 Strict Huygens Let L be a strictly hyperbolic (linear) second order operator defined on a causal domain (e.g d Alembertian + lower order terms) Let S be a space-like surface Consider the Cauchy-problem Lψ = 0 ψ S = f ν ψ S = g (8) Definition: L satisfies a strict Huygens principle if the solution to

33 every (well posed) Cauchy problem (8) depends on the initial data f, g only in the intersection of the characteristic conoid and the space-like manifold S. Remark: Equivalent definition in terms of the support of the fundamental solutions { LΨ± = δ ξ Ψ {±t>0} = 0

34 every (well posed) Cauchy problem (8) depends on the initial data f, g only in the intersection of the characteristic conoid and the space-like manifold S. Remark: Equivalent definition in terms of the support of the fundamental solutions { LΨ± = δ ξ Ψ {±t>0} = 0

35 Trivial Symmetries 1. Nonsingular coordinate transformations: x x = x(x) det(d x) 0 2. Gauge transformations: ψ ψ = λ 1 ψ L L = λ 1 Lλ, where λ = λ(x) 0.

36 3. Multiplication by a scalar function: where µ = µ(x) 0. L L = µl,

37 Hadamard s Question: Determine all the hyperbolic operators that satisfy a strict Huygens Property (HP).

38 Hadamard s Question: Determine all the hyperbolic operators that satisfy a strict Huygens Property (HP). HARD!!! Facts: The usual wave operator satisfies a strict HP iff n is odd and n > 1. If n = 3 the op + u satisfies strict HP iff u = 0.

39 Hadamard s Conjecture Conjecture: The only potentials u that can be added to so that L = + u satisfies a strict HP is (mod the trivial symmetries) u = 0.

40 Hadamard s Conjecture Conjecture: The only potentials u that can be added to so that L = + u satisfies a strict HP is (mod the trivial symmetries) u = 0. FALSE!

41 Historical Background Hadamard s conjecture Courant & Hilbert Asgeirsson Stellmacher (counter-example to Hadamard s conjec) Stellmacher and Lagnese: DARBOUX TRANSFORMATIONS RATIONAL SOLUTIONS OF THE KdV Günther - Ibragimov Berest: Iso-Huygens deformations (JMP 92) Relation to Virasoro flows and the bispectral problem

42 The symmetries that preserve HP for operators of the form + u(x 0 ) are related to the symmetries that preserve the bispectral property for u(x 0 ), i.e., the positive Virasoro flows

43 The symmetries that preserve HP for operators of the form + u(x 0 ) are related to the symmetries that preserve the bispectral property for u(x 0 ), i.e., the positive Virasoro flows related to joint work w/ F. Magri. CMP 1992

44 Lagnese & Stellmacher Notation: x 0 = t, x = (x 0, x 1, x n ) The simplest example discovered by Lagnese & Stellmacher (dimension 5 + 1): More generally L = + 2 x 2 0, (9)

45 Lagnese & Stellmacher Notation: x 0 = t, x = (x 0, x 1, x n ) The simplest example discovered by Lagnese & Stellmacher (dimension 5 + 1): L = + 2 x 2, (9) 0 More generally If L = + u(x 0 ) where u rational solution of the KdV, i.e. u is of the form u k (x 0 ) = 2 2 x 0 log ϑ k (x 0 ), (10) where ϑ k is the k-th Adler-Moser polynomial. defined by the

46 relations ϑ 0 = 1 ϑ 1 = x ϑ k+1ϑ k 1 ϑ k 1ϑ k+1 = (2k + 1)ϑ 2 k. The variable x 0 is the time variable THEOREM: (Lagnese and Stelmacher) Let ϑ k be the k-th Adler-Moser polynomial and u k as in equation (10). If the number n of spatial dimensions is odd and n 3 + 2k, then, the operator is a strict Huygens operator. L = + u k (x 0 ), (11)

47 THEOREM: (Lagnese) If L is a strict Huygens operator of the form (11), and u(x 0 ) is an analytic potential, then (mod trivial symmetries) u( ) is a rational solution of KdV.

48 KdV, KP, and Friends Korteweg-de Vries equation: ψ t + cψ x + αψ xxx + βψψ x = 0. (12) The Kadmontsev-Petviashvilli equation x (u t 6uu x u xxx ) = κu yy (13) are archetypical examples of infinite dimensional completely integrable systems.

49 They exhibit a tremendous amount of structure in terms of conservation laws and symmetries. solitary wave solutions, i.e. u t = u xxx + 6uu x. (14) u(x, t) = f(x ct). (15) exercise: Assuming that f is sufficiently smooth, and decays at together with its derivatives, it can be written as f(x) = 1 ( ) 1 2 c 2 sech2 c(x x0 ) (16)

50 The J. S. Russel Story

51

52

53 Figure 1: The term soliton was coined by M. Kruskal to denote the aspect of a solitary wave and a particle, whence the greek suffix ton

54 Solutions of the KP

55 Soliton Equations are Full of Structure

56 Soliton Equations are Full of Structure

57 Soliton Equations are Full of Structure

58 It can be shown, that the KdV admits solutions of the form u(x, t) = 2 2 x log det[a(x, t)], where A is the N N matrix whose entries are A ij = δ ij + β i κ i + κ j exp( (κ i + κ j )x + 8κ 3 i t). The asymptotic behavior along lines of slope 4κ 2 n on the (t, x)-plane displays interesting particle like interactions, between the different solitons. Furthermore, the KdV comes together with a full hierarchy of commuting Hamiltonian flows

59 It can be shown, that the KdV admits solutions of the form u(x, t) = 2 2 x log det[a(x, t)], where A is the N N matrix whose entries are A ij = δ ij + β i κ i + κ j exp( (κ i + κ j )x + 8κ 3 i t). The asymptotic behavior along lines of slope 4κ 2 n on the (t, x)-plane displays interesting particle like interactions, between the different solitons. Furthermore, the KdV comes together with a full hierarchy of commuting Hamiltonian flows

60 The KdV hierarchy {X k } k=1,3,5, X 1 (u) = u x X 3 (u) = u xxx + 6uu x.

61 Rational Solutions of KdV Airault, McKean, Moser, Adler, Ablowitz, Choodnovsky 2 The Adler-Moser Polynomials: If one iterates the Darboux process starting from u 0 = 0 one obtains from the results of the previous section that u n = 2 2 x log W n. The polynomial W n, after a suitable choice of parameters and normalization, is nothing more than the n-th Adler-Moser polynomial. After a suitable choice of parameters in the sequence of Darboux transformations can be made so that u n = 2 2 x log ϑ n (x + t 1, t 3,..., t 2n 1 ),

62 satisfies the flows of the KdV hierarchy, i.e., tk u n = X k [u n ]. Furthermore:

63 satisfies the flows of the KdV hierarchy, i.e., tk u n = X k [u n ]. Furthermore: If we look at the dynamics of the poles of the rational solutions of KdV we get Calogero-Moser systems

64 Calogero-Moser Systems ẋ = H/ y (17) ẏ = H/ x (18) H = 1 y 2 2 j + (x j x k ) 2 (19) j<k Integrability - Introduce (with Moser) A(x, y) def = diag[y 1,, y n ] + (( 1/(x j x k ) )) j k

65 Eigenvalues of A(x, y) are constants of motion & in involution F j def = Tr(A j /j) The F j Poisson commute Taking j = 2 and H = F 2 the locus defined by gradf 2 = 0 x j k (x j x k ) 3 = 0 j = 1,, n {y = 0} (20) is invariant by the flow of ẋ j = 6 j k(x j x k ) 2 j =,, n (21)

66 Miracle: If we set v def = 2 j (x x j ) 2 v = v(x, t) x j = x j (t) The requirement that v satisfies the KdV is equivalent to (20) and (21). Only the tip of the iceberg: v t = 3vv x v xxx

67 Miracle: If we set v def = 2 j (x x j ) 2 v = v(x, t) x j = x j (t) The requirement that v satisfies the KdV is equivalent to (20) and (21). v t = 3vv x v xxx Only the tip of the iceberg: Extended in many directions... Rational solutions of KP (Krichever) Rational solutions of AKNS (Sachs)

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69 AKNS: Ablowitz-Kaup-Newell-Segur-Zakharov-Shabat Obtained a way of constructing certain hierarchies of integrable evolution equations.

70 AKNS: Ablowitz-Kaup-Newell-Segur-Zakharov-Shabat Obtained a way of constructing certain hierarchies of integrable evolution equations. Integrability of the nonlinear-schröredinger equation, the mkdv, Sine-Gordon,... Set L = [ x q r x ]. (22)

71 AKNS: Ablowitz-Kaup-Newell-Segur-Zakharov-Shabat Obtained a way of constructing certain hierarchies of integrable evolution equations. Integrability of the nonlinear-schröredinger equation, the mkdv, Sine-Gordon,... Set L = [ x q r x ]. (22) The AKNS is obtained by constructing Lax pairs of the form L tk = [ ] P (k), L

72 This induces flows on the fields (q, r) { qtl = A l (q,..., xq; l r,..., xr) l, r tl = B l (q,..., xq; l r,..., xr) l. (23) flow generated by the hierarchy for l = 3 { qt3 = 1 4 (q xxx 6qrq x ), r t3 = 1 4 (r xxx 6qrr x ).

73 Rational Solutions of AKNS Set for p & q non-negative. σ = τ = q r p (x x j (t)) (24) j=1 p+q j=p+1 (x x j (t)) (25) def = (log σ) xx σ/τ (26) def = τ/σ (27)

74 Dynamics of the poles of (q, r) The dynamics of (q, r) under the first equation of the AKNS hierarchy is essentially equivalent to the dynamics of the roots x j of σ and τ in two Calogero-Moser systems as follows: Set a l = { +1 if 1 l p 1 if p + 1 l p + q and ẍ j = 8 k j a k =a j (x j x k ) 3 j = 1,, p + q

75 coupled through the constraints (a j a k )(x j x k ) 2 = 0 j = 1, 2,, p + q k j

76 Dirac operators Similar problem for Dirac operators ought to be studied. D = γ µ µ + v The Dirac matrices γ µ obey the anti-commutation rule where is the Minkowski tensor, {γ µ, γ ν } def = γ µ γ ν + γ ν γ µ = 2g µν, ((g µν )) = diag[1, 1,, 1]

77 and v is a matrix potential Notation: Hence From now on: = γ µ µ. 2 = x j denotes the j-th coordinate.

78 Huygens principle for Dirac operators Joint work F. Chalub (Lisboa) Fundamental Solutions for the free Dirac operator: ( + v)φ = δ ξ, where δ ξ denotes Dirac-delta at ξ = (t 0, y) Notation: ξ = (t 0, y): λ is the geodesic distance from (t, x) to the point λ def = (t t 0 ) 2 x y 2 = (x µ ξ µ )(x µ ξ µ ).

79 is the light cone with vertex in ξ C(ξ) def = {(t, x) λ((t, x), ξ) = 0}, Definition: Φ = Φ(x, ξ) satisfies Huygens principle iff supp Φ C(ξ), ξ Ω, (Ω a causal domain). Theorem 1. The free Dirac operator in dimension n obeys the Huygens principle if, and only if, n is odd. A few examples of Huygens ops are:

80 1. Depending only on t = x 0, Huygens in three dimensions are v = 1 t v = ( t t t t ) 2. An example of a potential depending on one space parameter. is Huygens for n = 3. v = ( 0 1 x 3 1 x 3 0 )

81 Natural Question: Can one construct a family of (strict) Huygens Dirac operators similar to the one obtained by Lagnese and Stellmacher for the wave operator?

82 Natural Question: Can one construct a family of (strict) Huygens Dirac operators similar to the one obtained by Lagnese and Stellmacher for the wave operator? Theorem 2 (Chalub & JPZ). If q(x 0 ) and r(x 0 ) are solutions of the rational solutions of AKNS constructed in (24), then q + r 2 I + q r γ (28) 2 satisfies Huygens property in dimension d + 2, if d is odd, or d + 3 if d is even.

83 Natural Question: Can one construct a family of (strict) Huygens Dirac operators similar to the one obtained by Lagnese and Stellmacher for the wave operator? Theorem 2 (Chalub & JPZ). If q(x 0 ) and r(x 0 ) are solutions of the rational solutions of AKNS constructed in (24), then q + r 2 I + q r γ (28) 2 satisfies Huygens property in dimension d + 2, if d is odd, or d + 3 if d is even. Remark: (JPZ, Jr. Diff. Eqs, 97(1):71 98, 1992) The corresponding AKNS operators are bispectral.

84 Technical Details Construction of Fundamental Solutions: LF = δ

85 Technical Details Construction of Fundamental Solutions: LF = δ consider the following problem for the wave eq E = δ ξ E = 0, t<t0 (29) where ξ = (t 0, ξ 1,..., ξ n ) where ξ = (t 0, ξ 1,..., ξ n )

86 admits the following solutions depending on the value of n. { c n δ (n 3)/2 (Υ) n odd E(x, ξ) = c n Υ (n 1)/2 + n even, (30) where Υ denotes Υ = (t t 0 ) 2 n (x i ξ i ) 2, i=1 and δ (k) denotes the k-th derivative of Dirac s delta function. The distribution that appears for n even in equation (30) S n (Υ) = Υ (n 1)/2 + Γ((n 3)/2)

87 plays an important role in what follows. It should be interpreted by evaluating at x = Υ the function S ν (x) = x ν + Γ(ν + 1) def = { x ν Γ(ν+1) x > 0 0 x < 0 (31) This function in turn has the special property of being homogeneous of degree ν. Seek solutions in the form: U ν (x, ξ)s ν p (Υ), (32) ν=0

88 where now Υ is the square of the geodesic in the metric defined by the g ij and p = def n 1. (33) 2 For odd values of n: must search for E(x, ξ) of the form E(x, ξ) = W 1 (x, ξ)υ p (34) where W 1 (x, ξ) = U j (x, ξ)υ j. j=0

89 where now Υ is the square of the geodesic in the metric defined by the g ij and p = def n 1. (33) 2 For odd values of n: must search for E(x, ξ) of the form E(x, ξ) = W 1 (x, ξ)υ p (34) where W 1 (x, ξ) = U j (x, ξ)υ j. For even n: j=0 E(x, ξ) = V (x, ξ)υ p + W 0 (x, ξ) log Υ + R, (35)

90 where W 0 (x, ξ) = U j (x, ξ)υ j p, j=p V (x, ξ) = and R is a smooth function. p 1 j=0 U j (x, ξ)υ j,

91 where W 0 (x, ξ) = U j (x, ξ)υ j p, j=p V (x, ξ) = and R is a smooth function. p 1 j=0 U j (x, ξ)υ j, Remark: The coefficients of the formal expansion (32) found by substituting this ansatz into the equation LE = g ij E xi x j + b i E xi + ue = 0 and matching the behavior of the powers of Υ.

92 This yields for r = 0, 1,... 2 g ij Υ xi xj U r + 4((r 1) p)u r + U r (LΥ uυ) = LU r 1, (36) def where we set U 1 = 0.

93 Hadamard s Criterion The question of determining which hyperbolic operators satisfy a strict Huygens principle received an indirect characterization in the work of Hadamard. Formal adjoint L to the operator L. The operator L is defined as L ψ def = xi xj (g ij ψ) xi (b i ψ) + uψ. THEOREM: [(Hadamard s Criterion)] The operator L satisfies a strict HP iff n is odd, n > 1, and the elementary solution of the

94 adjoint operator L contains no logarithmic term, i.e., W 0 (x, ξ) = 0 for all ξ and all x in the internal part of the characteristic conoid.

95 adjoint operator L contains no logarithmic term, i.e., W 0 (x, ξ) = 0 for all ξ and all x in the internal part of the characteristic conoid.

96 Riesz Kernels Marcel Riesz: unified treatment of the expression for the solution to the wave equation for different n. Basic Idea: Extending for λ C the expression Ξ(x, t, λ) def = { (x 2 0 n i x2 i )λ ( n i x2 i )1/2 < x 0 0 elsewhere. This is defined for R[λ] > 0 and the meromorphic extension takes values in an appropriate space of distributions. Normalizing Ξ by a suitable meromorphic function (of λ) one gets for non-positive integers derivatives of the δ function.

97 A Few Technicalities g a Lorentzian metric of signature +,,,...,. Square of the geodesic distance Υ(x, ξ), Characteristic conoid C(ξ), by the equation Υ(x, ξ) = 0. C(ξ) \ {ξ}: two connected components C + (ξ) and C (ξ), which are naturally associated with the forward and backward time. Open subsets of Ω as D + (ξ) and D (ξ).

98 working w/ an operator of the form L = div grad + a, + u,

99 all coefficients (real) analytic and defined in a causal domain Ω. Causal Domain: 1. Any two pts x and ξ are joined by a unique geodesic. 2. D + (x) D (ξ) is either empty or compact in Ω.

100 DEFINITION A forward Riesz kernel of L is a holomorphic mapping C λ Φ Ω λ(, ξ) D (Ω), s.t. supp[φ Ω λ(, ξ)] D + (ξ) (37) L[Φ Ω λ(, ξ)] = Φ Ω λ 1(, ξ) (38) Φ Ω 0 (, ξ) = δ ξ. (39) Remarks:

101 λ = 1 the Riesz kernel gives a distribution E + (, ξ) = Φ Ω 1 (, ξ) such that LE + (, ξ) = δ ξ Causality supp[e + (, ξ)] D + (ξ). Modern treatment of the asymptotic behavior of solutions to hyperbolic problems studies the so called asymptotics in smoothness If the operator L is analytic, this asymptotic behavior can be replaced by a convergent series. This type of argument has been used extensively in the recent work of Berest.

102 Construction of Riesz kernels Analytic continuation: Let g D(Ω) and start w/ R λ (x, ξ) g = D + (ξ) Υ(x, ξ) λ n+1 2 H n+1 (λ) g(x)dx, for and R[λ] > n 1 2 H n+1 (λ) def = 2π n 1 2 Γ(λ)Γ(λ n 1 2 ).

103 Then, extend for all λ C the above definition by analytic continuation, using iterates of the (classical) formula which is valid for R[λ] > (n + 3)/2. R λ = R λ 1, (40) Fact: The distribution R λ (x, ξ) satisfies supp[r λ (x, ξ)] D + (ξ),

104 Then, extend for all λ C the above definition by analytic continuation, using iterates of the (classical) formula which is valid for R[λ] > (n + 3)/2. R λ = R λ 1, (40) Fact: The distribution R λ (x, ξ) satisfies supp[r λ (x, ξ)] D + (ξ), Exercise: R λ R µ = R λ+µ. (41) (x ξ x )R λ = (2λ n + 1)R λ, (42)

105 R 0 (x, ξ) = δ ξ (x), (43) Proposition: For an odd n λ {1, 2,..., (n 1)/2} we have n+1 R λ (x, ξ) = δ( 2 λ) + (Υ), (44) β n,λ where δ + (Υ) stands for the the Dirac s delta measure concentrated on C + (ξ) {Υ = 0}, and β n,λ a numerical constant. Now, look for an expansion of the fundamental solution E(x, ξ) U ν (x, ξ)r λ+ν (x, ξ). (45) ν=0

106 Substitute expression (45) into LE = δ ξ obtain a set of recursive transport relations (x ξ x )U ν (x, ξ) + νu ν (x, ξ) = 1 4 L[U ν 1(x, ξ)]. (46) Facts: The above recursive system has a unique solution provided one normalizes U 0 (x, ξ) = 1, (47)

107 and requires for r 1 U r (x, ξ) = O(1), x ξ. (48) If the operator L has analytic coefficients, then the series (46) is uniformly convergent in a sufficiently small neighborhood of Υ = 0. Riesz kernel for the operator L can be expanded as Φ Ω λ(x, ξ) = ν=0 4 νγ(λ + ν) U ν (x, ξ)r λ+ν (x, ξ). Γ(λ) Main argument for the proof of Hadamard s result:

108 If n is even then for ν = 0, 1, 2,... we have supp[r ν+1 (x, ξ)] = D + (ξ), and so Huygens principle does not hold. If n is odd then for ν = 0, 1, 2,..., (n 3)/2 we have supp[r ν+1 (x, ξ)] = C + (ξ). Notice: Using equation (46), and p = (n 1)/2 one gets Hadamard s classical formula for n odd E + (x, ξ) = 1 2πp(V (x, ξ)δ(p 1) + (Υ) + W (x, ξ)h + (Υ)),

109 where H + (γ) is the the Heaviside step distribution on the forward region D + (ξ). H + (γ) ϕ = ϕ(x)dx. D + (ξ) Furthermore, V (x, ξ) and W (x, ξ) are analytic in a neighborhood of x = ξ with expansions V (x, ξ) = p 1 ν=0 s ν U ν (x, ξ)υ ν, where s ν = [(1 p)... (ν p)] 1 and W (x, ξ) = ν=p 1 (ν p)! U ν(x, ξ)υ ν p.

110 i.e.: strict Huygens principle iff vanishing of the term W. Lemma: The term W (x, ξ) vanishes iff U p (x, ξ) = 0 for x on the surface of the forward light cone C + (ξ). Idea: The lemma follows from the fact that W (x, ξ) is a solution of the characteristic Goursat problem L[W (x, ξ)] = 0, with a bdry value given on the cone surface C + (ξ). This Goursat problem has a unique solution. Hence, W (x, ξ) 0 iff W (x, ξ) = 0 for x C + (ξ).

111 DEFINITION: The Hadamard series is said to be truncated (or terminated) at level ν 0 if the sequence defined by the recursions (46), (47), and (48) is zero for ν ν 0. Corollary of previous Lemma THEOREM: Let L be a real-analytic formally self-adjoint hyperbolic on a causal domain Ω R n+1 with n > 1. Then, L is strictly Huygens iff n is odd and the Hadamard series for L is truncated at level p = (n 1)/2.

112 References [1] J. J. Duistermaat and F. A. Grünbaum. Differential equations in the spectral parameter. Comm. Math. Phys., 103(2): , [2] J. P. Zubelli and F. Magri. Differential equations in the spectral parameter, Darboux transformations, and a hierarchy of master symmetries for KdV. Commun. Math. Phys., 141(2): , [3] Jorge P. Zubelli. Rational solutions of nonlinear evolution

113 equations, vertex operators, and bispectrality. Equations, 97(1):71 98, J. Differential [4] Jorge P. Zubelli and D.S. Valerio Silva. Rational solutions of the master symmetries of the KdV equation. Commun. Math. Phys., 211(1):85 109, [5] George Wilson. Collisions of Calogero-Moser particles and an adelic Grassmannian (with an appendix by I. G. Macdonald). Invent. Math., 133(1):1 41, 1998.