INVERSE PROBLEMS for NONLINEAR HYPERBOLIC EQUATIONS Gunther Uhlmann Jian Zhai 1. Introduction. Many Physical Phenomena Are Model

DISCRETE AND CONTINUOUS doi:10.3934/dcds.2020380 DYNAMICAL SYSTEMS Volume 41, Number 1, January 2021 pp. 455–469

INVERSE PROBLEMS FOR NONLINEAR HYPERBOLIC EQUATIONS

Gunther Uhlmann∗ Department of Mathematics, University of Washington, Seattle, WA 98195, USA Institute for Advanced Study, The Hong Kong University of Science and Technology Kowloon, Hong Kong, China Jian Zhai Institute for Advanced Study, The Hong Kong University of Science and Technology Kowloon, Hong Kong, China

Abstract. There has been considerable progress in recent years in solving inverse problems for nonlinear hyperbolic equations. One of the striking aspects of these developments is the use of nonlinearity to get new information, which is not possible for the corresponding linear equations. We illustrate this for several examples including Einstein equations and the equations of nonlinear elasticity among others.

1. Introduction. Many physical phenomena are modeled by nonlinear hyperbolic equations (see for example [40]). In recent years, starting with the paper [20] several inverse problems for nonlinear hyperbolic equations have been solved, ranging from semilinear wave equations to nonlinear elastic equations and Einstein equations. The nonlinearity of the equation is used in an essential way. The interaction of waves, caused by the nonlinearity of the equation, produces new waves giving information that is not available for the corresponding linear equation. We will survey in this paper some recent progress on inverse problems for nonlinear hyperbolic equations. The most exciting breakthrough is the possibility of recovering time-dependent coefficients for nonlinear evolution equations. The unique recovery of a time-dependent coefficient in a linear hyperbolic equation is in general still not known. To solve inverse problems for linear hyperbolic equations, a powerful method, called the boundary control (BC) method, was developed by Belishev starting with the acoustic wave type equation ([2]). The BC method relies on a sharp unique continuation result for the wave equation that was proven later by Tataru [33]. However, this result is only valid if the coefficients of the equation depend ana- lytically on time. One way to avoid using Tataru’s result is to reduce the inverse problems to some geometrical inverse problems. We refer to [29, 27, 31, 30] for

2020 Mathematics Subject Classiﬁcation. Primary: 35R30; Secondary: 35A27, 35L70. Key words and phrases. Inverse problems, nonlinear hyperbolic equations, Lorentzian manifolds. The ﬁrst author was partially supported by NSF, a Walker Professorship at UW and a Si-Yuan Professorship at IAS, HKUST. ∗ Corresponding author: [email protected].

455 456 GUNTHER UHLMANN AND JIAN ZHAI this direction of work. Microlocal analysis is then used as a powerful tool to analyze the propagation of singularities of solutions. Nonlinear waves have even richer geometric behaviors, which can be utilized to solve inverse problems. For an inverse problem for a nonlinear equation, usually the nonlinear equation is linearized, and the results for the linearized equation are applied when available. For example, a semilinear parabolic equation is treated in [15]; the nonlinear Calderón problem in the isotropic case is reduced to the linear one [32]; the inverse problem for the nonlinear Navier-Stokes equation is reduced to the problem for the linear Stokes equation [24]. Since the paper [20], the nonlinearity has been used to solve some inverse problems, whereas the results for the associated linearized equations are not available. What is used is the interaction of several waves to produce new waves that give information not available for the linearized equation. The technique of higher order linearization is a necessary part in the treatment. We shall briefly explain this. The higher order linearization can be explained as follows. Assume Λ(f) is the measured response of a medium with an input f. For example, Λ can be the Dirichlet-to- Neumann map associated to some partial differential equation. One can take an PN input of the form f = i=1 ifi, where i, i = 1, ··· ,N are small parameters. If the response of the medium to the input is nonlinear, then Λ is a nonlinear map, and PN consequently Λ( i=1 ifi) contains extra information other than what is contained in {Λ(fi)}i=1,··· ,N . Therefore useful information can possibly be extracted from the higher order linearization of the map Λ: N ∂N X Λ( ifi). ∂ ··· ∂ =···= =0 1 N 1 N i=1 For hyperbolic equations modeling waves, one usually chooses input sources of the PN form f = i=1 ifi such that fi, i = 1, 2, ··· N produce distorted plane waves or Gaussian beams. The outline of this paper is as follows. In Section 2 we consider the inverse problem of determining a globally hyperbolic Lorentzian metric by measuring the source-to-solution map in a neighborhood of a time-like geodesic for the wave equation plus a quadratic term. The result of [17] is reviewed. The concept of earliest light observation set is defined and the result that the earliest light observation sets determine the conformal class of a globally hyperbolic Lorentzian metric in 3+1 dimensions is reviewed. In Section 3 more general semilinear equations are considered. In Section 4, several results for other equations like Einstein equations, a simpler analog of Yang-Mills equations and nonlinear elasticity are formulated.

2. Quadratic non-linearities. We ﬁrst introduce some notations and deﬁnitions. Let (M, g) be a C∞-smooth (1 + 3)-dimensional globally hyperbolic Lorentzian manifold, where the metric g is of signature (−, +, +, +). By [3], (M, g) is isometric to a smooth manifold (R × N, h), where N is a 3-dimensional manifold and the metric h has the form h = −β(t, y)dt2 + κ(t, y). (1) Here β : R × N → (0, +∞) is a smooth function and κ(t, ·) is a Riemannian metric on N depending smoothly on t ∈ R. A smooth curve µ :(a, b) → M is called causal if g(µ ˙ (s), µ˙ (s)) ≤ 0 andµ ˙ (s) 6= 0 for all s ∈ (a, b). Given p, q ∈ M, we denote p ≤ q if p = q or p can be joined to q by a future-pointing causal curve. We say p < q if p ≤ q + and p 6= q. We denote the causal future of p ∈ M to be Jg (p) = {q ∈ M : p ≤ q} NONLINEAR HYPERBOLIC EQUATIONS 457

− and the causal past of q ∈ M to be Jg (q) = {p ∈ M : p ≤ q}. The curve µ is called time-like if g(µ ˙ (s), µ˙ (s)) < 0 for all s ∈ (a, b). We denote p q if p 6= q and there is a future-pointing time-like path from p to q. Then the chronological + future of p ∈ M is the set Ig (p) = {q ∈ M; p q} and the chronological past of − + − q ∈ M is Ig (q) = {p ∈ M; p q}. We also denote Jg(p, q) := Jg (p) ∩ Jg (q) and + − Ig(p, q) := Ig (p) ∩ Ig (q). Take t0 > 0 to be a ﬁxed number and denote

M(t0) = (−∞, t0) × N. Consider the semilinear wave equations with source terms 2 gu(x) + a(x)u(x) = f(x), on M(t0), + (2) u = 0 in M(t0) \ Jg (supp(f)), where g is the Laplacian-Beltrami operator on (M, g) and a(x) is nowhere vanishing. In local coordinates,

1 p jk g := ∂j( | det g|g ∂k). p| det g| For p ∈ M, denote the set of light-like vectors at p by

LpM = {ζ ∈ TpM \{0} : g(ζ, ζ) = 0}. ∗ Also, the set of light-like covectors at p is denoted by LpM. The future and past + − ∗,+ ∗,− light-like vectors (covectors) are denoted by Lp M and Lp M (Lp M and Lp M). Assume V ⊂ M(t0) is an open, relatively compact, connected neighborhood of a future-pointing time-like pathµ ˆ =µ ˆ([−1, 1]) ⊂ M(t0).Take −1 < s− < s+ < 1, ± and denote p =µ ˆ(s±). 6 Deﬁnition 2.1. Let W = {f ∈ C0 (V ); kfkC6(V ) < ε}, where ε > 0 is so small 2 that the equation (2) has a unique solution u ∈ C (M(t0)) for all f ∈ W. The source-to-solution map LV : W → C(V ) is the nonlinear operator deﬁned as

LV : f 7→ u|V , f ∈ W, (3) where u satisﬁes the wave equation (2) on (M(t0), g). Kurylev, Lassas and Uhlmann [20] ﬁrst studied the inverse problem of determining the Lorentzian metric g from the source-to-solution operator LV . The main result is: Theorem 2.2. Let (M (j), g(j)), j = 1, 2 be two smooth, globally hyperbolic Lorentzian manifolds of dimension (1 + 3) that are represented in the form M (j) = R × N (j) withe a metric of the form (1). (j) + − Let µˆj :[−1, 1] → M (t0) be smooth time-like paths, pj =µ ˆj(s+), pj =µ ˆj(s−), (j) where −1 < s− < s+ < 1, and Vj ⊂ M be neighborhoods of µˆj([−1, 1]).

Let LVj , j = 1, 2 be the source-to-solution maps for wave equations (2) on man- (j) (j) (j) ifolds (M , g ) with a = aj, where aj : M → R \{0} are nowhere vanishing C∞-smooth functions, j = 1, 2. − − Assume that there is a diﬀeomorphism Φ: V1 → V2 such that Φ(p1 ) = p2 , + + Φ(p1 ) = p2 and the source-to-solution maps satisfy −1 ∗ ∗ ((Φ ) ◦ LV1 ◦ Φ )f = LV2 f 6 for f ∈ W, where W is a neighborhood of the zero function in C0 (V2). 458 GUNTHER UHLMANN AND JIAN ZHAI

− + − + Then there is a diﬀeomorphism Ψ: Ig(1) (p1 , p1 ) → Ig(2) (p2 , p2 ) and the metric ∗ (2) (1) − + (1) (1) Ψ g is conformal to g in Ig(1) (p1 , p1 ) ⊂ M , that is, there is b : M → R+ (1) ∗ (2) − + such that g (x) = b(x)(Ψ g )(x) in Ig(1) (p1 , p1 ). Moreover, b(x) = 1 for x ∈ V1. The corresponding inverse problem for linear equations is still open. The proof the above theorem uses the nonlinear nature of the equation. The key is to produce artiﬁcial point sources by nonlinear interactions of distorted plane waves that we will introduce next. 2.1. Distorted plane waves. We will follow closely the notations used in [23]. Distorted plane waves have singularities conormal to a submanifold of M and can be viewed as Lagrangian distributions. Recall that T ∗M is a symplectic man- P4 j ifold with canonical 2-form, given in local coordinates, ω = j=1 dξj ∧ dx .A submanifold Λ of T ∗M is called Lagrangian if dim Λ = 4 and ω vanishes on Λ. For K a submanifold of M, its conormal bundle ∗ ∗ N K = {(x, ζ) ∈ T M \{0} : x ∈ K, hζ, θi = 0, θ ∈ TxK} is a Lagrangian submanifold of T ∗M. Let Λ be a smooth conic Lagrangian submanifold of T ∗M \0. We denote by Iµ(Λ) the space of Lagrangian distribution of order µ associated with Λ. Let Λ0, Λ1 ⊂ T ∗M \ 0 be two Lagrangian submanifolds intersecting cleanly, i.e.,

TpΛ0 ∩ TpΛ1 = Tp(Λ0 ∩ Λ1), ∀p ∈ Λ0 ∩ Λ1. p,l We denote the paired Lagrangian distribution associated with (Λ0, Λ1) by I (Λ0, Λ1). For more details, we refer to [25, 12]. −3/2,−1/2 ∗ The wave operator g has a parametrix Qg ∈ I (N Diag, Λg). We de- −1 note g to be the unique causal inverse of the wave operator g. For convenience, −1 we just take Qg = g . Next we construct distorted plane waves. Let x ∈ U, ζ ∈ L+ M. Take a small number s > 0. Denote 0 0 x0 0

+ + + Vx0,ζ0,s0 = {η ∈ Tx0 M : kη − ζ0kg < s0, kηkg = kζ0kg } + be a neighborhood of ζ0 on a sphere. Here g is a Riemannain metric on M. Denote W ∈ L+ M ∩ V and let x0,ζ0,s0 x0 x0,ζ0,s0

Kx0,ζ0,s0 = {γx0,η(t) ∈ M(t0); η ∈ Wx0,ζ0,s0 , t ∈ (0, ∞)}. Let [ ∗ Σx0,ζ0,s0 = {(x0, rη ) ∈ T M; η ∈ Vx0,ζ0,s0 , r ∈ R \{0}}, [ ∗ Λx0,ζ0,s0 = {(γx0,η(t), rγ˙ x0,η(t) ) ∈ T M; η ∈ Wx0,ζ0,s0 , t ∈ (0, ∞), r ∈ R \{0}}. jk Denote p(x, ξ) = g (x)ξjξk, and ∗ Char(g) = {(x, ξ) ∈ T M \ 0; p(x, ξ) = 0}.

Note that Λx0,ζ0,s0 is the Lagrangian manifold that is the ﬂowout from Char(g) ∩

Σx0,ζ0,s0 by the Hamiltonian vector ﬁeld associated to p(x, ξ) in the future direction. ∗ If (M, g) has no conjugate point, then Λx0,ζ0,s0 = N Kx0,ζ0,s0 . Assume that f ∈ n+1 I (M, Σx0,ζ0,s0 ) is a compactly supported classical conormal distribution, then −1 the solution to the linear wave equation w = g f is a distorted plane wave whose singular support is concentrated near the geodesic γx0,ζ0 ([0, ∞)) and is singular on the surface Kx0,ζ0,s0 , n−1/2 u|M(t0)\{x0} ∈ I (M(t0) \{x0};Λx0,ζ0,s0 ).

When s → 0, the singular support of u tends to the set γx0,ζ0 ([0, ∞)). NONLINEAR HYPERBOLIC EQUATIONS 459

2.2. Determination of the earliest light observation sets. We can take the source of the form 4 X f = εjfj. j=1 Here n+1 fj ∈ I (M;Σxj ,ζj ,s0 ), where (xj, ζj) are light-like vectors with xj ∈ V . We assume that the sources satisfy + supp(fj) ∩ Jg (supp(fk)) = ∅, for all j 6= k, 4 + − [ Jg (W ) ∩ Jg (W ) ⊂ V, where W = supp(fj). j=1

This implies that the supports of the sources are causally independent. Assume uj is the solution of the linearized wave equation with source fj, j = 1, 2, 3, 4. That is, −1 µ uj := ∂εj u|εj =0 = g (fj) ∈ I (M(t0) \{xj}, Λxj ,ζj ,s0 ).

Then uj, j = 1, 2, 3, 4 represent distorted plane waves. We can extract useful information from the behavior of (4) −1 U = ∂ε1 ∂ε2 ∂ε3 ∂ε4 u|ε1=ε2=ε3=ε4=0 = g S, where X S = Sσ σ∈Σ(4) and −1 −1 −1 −1 Sσ = −ag (auσ(4)uσ(3))g (auσ(2)uσ(1)) − 4auσ(4)g (auσ(3)g (auσ(2)uσ(1))), where Σ(4) is the permutation group of the set {1, 2, 3, 4}. Notice (4) U |V = ∂ε1 ∂ε2 ∂ε3 ∂ε4 LV (f)|ε1=ε2=ε3=ε4=0 (4) The singularity of U is carefully analyzed in [20]. If γxj ,ζj , j = 1, 2, 3, 4 intersect at some point q ∈ I+(p−) ∩ I−(p+), then U (4) is a conormal distribution associated + + + to L (q). Here L (q) = {γq,ζ (t) ∈ M; ζ ∈ Lq M, t ≥ 0} ⊂ M is the future directed light-cone emanating from q. This means that the four distorted plane waves uj, j = 1, 2, 3, 4 generate a point source at q, which produce new waves propagating along L+(q). One can detect the singularities of U (4) in V and use them to recover the metric g on I+(p−) ∩ I−(p+). One diﬃculty is that the metric is unknown outside V , and we need to infer whether those distorted plane waves indeed intersect at some point from the observation. This can be done by very careful analysis of the singularities of U (4) in [20]. In particular, one needs to distinguish the singularities generated by interactions of four waves from those generated by three waves. Even we know when the four distorted plane waves intersect, the location of the intersection point is still unknown. However, such intersection can happen at any point in I+(p−) ∩ I−(p+) if we take a rich collection of four waves, and each point corresponds to one admissible observation in V . More precisely, by Lemma 4.3 and Theorem 4.5 in [20], the source-to-solution operator LV uniquely determines the + − − + family {EV (q); q ∈ I (p ) ∩ I (p )} of the earliest light observation sets. Note that the location of q is not known for each observation EV (q). Therefore, Theorem 2.2 follows from the results on the inverse problem with passive observations which we will summarize in Theorem 2.4 below. 460 GUNTHER UHLMANN AND JIAN ZHAI

2.3. Reconstruction from the earliest light observation sets. Deﬁnition 2.3. (i) The light observation set of q ∈ M in the observation set V is + V PV = L (q) ∩ V ∈ 2 . (ii) The earliest light observation set of q ∈ M in V is

EV (q) = {x ∈ PV (q) : there are no y ∈ PV (q) and future-pointing time-like path α : [0, 1] → V such that α(0) = y and α(1) = x} ⊂ V. (iii) Let W ⊂ M be open. The family of the earliest light observation sets with source points at W is V EV (W ) = {EV (q); q ∈ W } ⊂ 2 .

We emphasize here that EV (W ) is an unindexed set, that is, for an element EV (q) ∈ EV (W ), the corresponding point q ∈ W is not known. V1 V2 When Φ : V1 → V2 is a map, then the power set extension of Φ is Φe : 2 → 2 is given by 0 0 0 Φ(e V ) = {Φ(z); z ∈ V }, for V ⊂ V1. (4) The following result for the inverse problem with the passive observations is given in [20]. Theorem 2.4. Let (M (j), g(j)), j = 1, 2 be two open, C∞-smooth, globally hy- (j) perbolic Lorentzian manifolds of dimension n ≥ 3, µˆj :[−1, 1] → M be smooth ± (j) time-like paths, and pj =µ ˆj(s±). Let the observation sets Vj ⊂ M be neigh- (j) borhoods of µˆj([−1, 1]) and Wj ⊂ M be relatively compact sets such that W j ⊂ − + − − Jg(j) (pj ) \ Ig(j) (pj ). Let EVj (Wj) be the families of the earliest light observation sets with source points at Wj. Assume that there is a conformal diﬀeomorphism Φ: V1 → V2 such that Φ(ˆµ1(s)) =µ ˆ2(s), s ∈ [−1, 1] and

Φ(e EV1 (W1)) = EV2 (W2), where Φe is the power set extension of Φ, see (4). ∗ (2) Then there is a diﬀeomorphism Ψ: W1 → W2 such that the metric Ψ g is (1) conformal to g and Ψ|W1∩V1 = Φ|W1∩V1 .

When Mj, j = 1, 2 are Ricci-ﬂat. The conformal factor has to be 1.

Corollary 1. Assume that (Mj, gj) and Vj,Wj, j = 1, 2 satisfy the conditions of Theorem 2.4 with the resulting conformal map Ψ: W1 → W2. Morever, assume that Wj are Ricci-flat and that Φ: V1 → V2 is an isometry. Also, assume that all topological components of Wj intersect Vj, j = 1, 2. Then the map Ψ is an isometry. In order to reconstruct the metric g on W , one needs to construct another copy of the manifold, and determine its topological, differentiable and metric structures. See [1, 16, 18] for the reconstruction of Riemannian manifolds from the boundary distance functions. The reconstruction of a Lorentzian manifold is carried out in [20]. Here we sketch the outline of the procedure. One can take a subdomain U ⊂ V that is a union of time-like paths. Assume there is a family of future pointing, C∞-smooth, time-like NONLINEAR HYPERBOLIC EQUATIONS 461 paths µa :[−1, 1] → V , indexed by a ∈ A, where A is a connected metric space. The A denots the completion of A and is compact. Define [ U = µα([−1, 1]). α∈A For x < y, we define the time separation function τ(x, y) ∈ [0, +∞) to be the R 1 p supremum of the lengths L(α) = 0 −g(α ˙ (s), α˙ (s))ds of the piecewise smooth causal paths α : [0, 1] → M from x to y. If x < y, we define τ(x, y) = 0. This function satisfies the reverse triangle inequality τ(x, y) + τ(y, z) ≤ τ(x, z), for x ≤ y ≤ z. If (M, g) is globally hyperbolic, the time separation function (x, y) 7→ τ(x, y) is continuous on M × M. Let a ∈ A, and for x ∈ M define + fa (x) = inf({s ∈ (−1, 1); τ(x, µa(s)) > 0} ∪ {1}), − fa (x) = sup({s ∈ (−1, 1); τ(µa(s), x) > 0} ∪ {−1}). + Then fa (x) is the earliest observation time from the point x on the path µa. The + functions fa : M → R, a ∈ A are called the earliest observation time functions on the path µa. Then the map − + − − + F : Jg (p ) \ Ig (p ) → C(A), F(q): a 7→ fa (q) is continuous. By [20, Proposition 2.2], the map F : W → F(W ) ⊂ C(A) is a homeomorphism. Here F(W ) has the topology induced by C(A). Thus we have already constructed a homeomorphic image of the Lorentzian manifold W in the space C(A). One can then construct the differentiable structure on F(W ) such that F : W → F(W ) becomes a diffeormorphism. Finally, one can show that the conformal class of g|U determines a metric G on F(W ) such that the map

F :(W, g|W ) → (F(W ),G) is a diﬀeomorphism and a conformal map. This determines the conformal class of the metric g on W . Hintz and Uhlmann [14] also considered an inverse problem with passive observations on the boundary of a Lorentzian manifold with time-like boundary. The conformal class of the metric can be reconstructed from the boundary light observation sets.

3. More general nonlinear terms. 3.1. General semilinear equations. Lassas, Uhlmann and Wang [23] consider the equation with more general nonlinear terms

gu(x) + H(x, u(x)) = f(x). Here H(x, z) ∈ C∞(U × I) be real-valued, where U is open in M and I is a small neighborhood of 0. Assume that H is genuinely nonlinear in the sense that k H(x, 0) = ∂zH(x, 0) = 0, ∂z H(x, 0) 6= 0 for some k ≥ 2. One can determine the metric and the nonlinearity (up to conformal diﬀeomor- phism) simultaneously by analyzing more carefully the singularities and principal symbols of the waves generated by nonlinear interactions of distorted plane waves: 462 GUNTHER UHLMANN AND JIAN ZHAI

Theorem 3.1. Let (M (j), g(j)), j = 1, 2 be two smooth, globally hyperbolic Lorentzian manifolds of dimension (1 + 3). (j) + − Let µˆj :[−1, 1] → M (t0) be smooth time-like paths, pj =µ ˆj(s+), pj =µ ˆj(s−), (j) where −1 < s− < s+ < 1, and Vj ⊂ M be open, relatively compact neighborhoods of µˆj([−1, 1]).

Let LVj , j = 1, 2 be the source-to-solution maps for wave equations (j) (j) g(j) u(x) + H (x, u(x)) = f(x) on M (t0). (j) + u = 0 in M (t0) \ Jg(j) (supp(f)). Assume H(j), j = 1, 2 are genuinely nonlinear. Assume that there is a diffeomor- − − + + phism Φ: V1 → V2 such that Φ(p1 ) = p2 and Φ(p1 ) = p2 and the source-to- solution maps satisfy −1 ∗ ∗ ((Φ ) ◦ LV1 ◦ Φ )f = LV2 f 6 for f ∈ W, where W is a neighborhood of the zero function in C0 (V2). − + − + Then there is a diffeormorphism Ψ: Ig(1) (p1 , p1 ) → Ig(2) (p2 , p2 ) and γ ∈ ∞ − + − + C (Ig(1) (p1 , p1 )) such that for x ∈ Ig(1) (p1 , p1 ) we have g(1) = e2γ Ψ∗g(2), k (1) (k−3)γ(x) k (2) ∂z H (x, 0) = e ∂z H (Ψ(x), 0), ∀k ≥ 4. − + Also, for k = 2, 3 we have the for x ∈ Ig(1) (p1 , p1 ) 2 (1) 3 (1) −γ(x) 2 (2) 3 (2) ∂z H (x, 0) · ∂z H (x, 0) = e ∂z H (x, 0) · ∂z H (x, 0); 2 (1) −γ(x) 2 (2) 3 (1) ∂z H (x, 0) = e ∂z H (Ψ(x), 0) if ∂z H (x, 0) = 0. k A reconstruction method of the coefficients ∂z H(x, 0) in the case of the Minkowski metric using convolution neural networks has been proposed in [36]. 3.2. Quadratic derivative nonlinearity. Wang and Zhou [39] considered the inverse problem for the equation of the form

gu(x) + H(x, u(x), ∇gu(x)) = f(x), where the Taylor expansion of H(x, z, w) in (z, w) has the form 2 2 2 H(x, z, w) = N0(x, w) + zN1(x, w) + z M(x, w) + o(|z| · |w| ), 0 where N0, N1, M are quadratic forms in w. A quadratic form Q(x, w, w ) satisﬁes the null condition if

Q(x, w, w) = 0 for any w ∈ TxM such that |w|g(x) = 0. Assumption 3.2. Assume

N0, N1 are null forms and M is not null. Theorem 3.3. Let g(j), j = 1, 2 be two globally hyperbolic Lorentzian metrics on 1+3 (j) M = R . Let µˆj(t) be time-like geodesics on (M, g ), where t ∈ [−1, 1]. Assume µˆj([−1, 1]) ⊂ M(t0). Let Vj ⊂ M(t0) be an open relatively compact neighborhood of µˆj(s−, s+), where −1 < s− < s+ < 1. Consider the nonlinear wave equations with source terms (j) g(j) u(x) + H (x, u(x), ∇g(j) u(x)) = f(x), on M(t0), + u = 0, in M(t0) \ Jg(j) (supp(f)), NONLINEAR HYPERBOLIC EQUATIONS 463

(j) (j) (j) where supp(f) ⊂ Vj. Suppose that H (x, z, w) ∼ N0 (x, w) + zN1 (x, w) + 2 (j) z M (x, w) satisfy the Assumption 3.2. If there is a diffeormorphism Φ: V1 → V2 ± ± such that Φ(p1 ) = p2 and the source-to-solution maps LVj satisfy −1 ∗ ∗ (Φ ) ◦ LV1 ◦ Φ (f) = LV2 (f) 4 (2) for all f in a small neighborhood of the zero function in C0 (V ), then − + − + (1) 1. There is a diffeormorphism Ψ: Ig(1) (p1 , p1 ) → Ig(2) (p2 , p2 ) such that g is (2) ∗ (2) 2γ (1) − + conformal to g , that is, Ψ g = e g in Ig(1) (p1 , p1 ) with γ smooth. 2. In addition, if M(1) = M(2) are independent of x or g(1) and g(2) are Ricci flat, the diffeomorphism Ψ is an isometry.

4. Other nonlinear equations. In this section, we introduce the results for var- ious nonlinear hyperbolic equations. 4.1. Einstein equations. The papers [19, 17] consider inverse problems for Ein- stein equations coupled with scalar fields. The paper [22] considers an inverse problem for Einstein equations coupled with Maxwell’s equations in vacuum. Uhlmann and Wang [37] provide a general approach, which can deal with Einstein equations coupled with scalar fields and coupled with Maxwell’s equations in a general background medium. To explain this result in [37], we first introduce some notations used therein. We denote Ric(g) to be the Ricci curvature tensor of the Lorentzian metric g. Let Sym2 be the vector bundle of symmetric covariant 2-tensors on M. For any T ∈ Sym2, αβ the trace of T with respect to g is Trg(T ) = g Tαβ. The scalar curvature of g is S(g) = Trg(Ric(g)). The Einstein equations are as follows. Ein(g) = T, where T ∈ Sym2 and 1 Ein(g) = Ric(g) − S(g)g. 2 We assumeg ˆ is the background metric satisfying Einstein equations coupled with fields ψˆ, Ein(ˆg) = Tb(ˆg, ψˆ), (5) ˆ ˆ gˆψl + Vl(ψ) = 0, t > 0. Here Tˆ = Tˆ(g, ψ) is a smooth function of g ∈ Sym2 and ψ ∈ BL, where BL is a vector bundle on M with fiber dimension L ≥ 1. We regard ψˆ as the background field that produces the background metricg ˆ. We assume (M, gˆ) has no conjugate points. Now consider a freely falling observer that is represented by a time-like geodesic µgˆ(t), t ∈ [−1, 1] on (M, gˆ). Let V be an open relatively compact neighborhood 4 2 Ψ of µgˆ([−1, 1]) and V ⊂ M(t0) for some t0 > 0. Let F ∈ C (M; Sym ), F ∈ C4(M; BL) be compactly supported in V and sufficiently small. Let g be the perturbed metric satisfying the Einstein equation

Ein(g) = Tb(g, ψ) + F , on M(t0), Ψ gψl + Vl(ψ) = Fl , l = 1, 2, ··· , L, (6) ˆ + g =g, ˆ ψ = ψ, in M(t0) \ Jg (supp(F )), 464 GUNTHER UHLMANN AND JIAN ZHAI

Ψ where F = (F , F ). The source Tsour = Tb(g, ψ) + F is subject conservation law

divgTsour = 0. We can consider (ψ, F ), instead of (F , F Ψ) as the source ﬁelds, and consider the following Einstein equations

Ein(g) = Tb(g, ψ) + F , on M(t0), (7) + g =g, ˆ in M(t0) \ Jg (supp(F )), Then the corresponding source-to-solution map is

LV :(ψ, F ) 7→ g|V , where g satisfies the equation (7), and F ∈ C4(V ; Sym2) is compactly supported, 4 L 4 2 ˆ 4 ψ ∈ C (M; B ), and kF kC (M;Sym ) < δ, kψ − ψkC (M(t0)) < δ. The inverse problem of interest is to determine the background metricg ˆ from the source-to-solution operator LV . It is motivated by the problem of determining the structure of the Universe by observing gravitational waves. The paper [37] considered a very general class of Einstein equations coupled with different fields as long as the microlocal linearization condition is satisfied. This condition is used to construct distorted plane waves for the system (7). The main result of [37] reads Theorem 4.1. Let (M, gˆ(i)), i = 1, 2 be two 4-dimensional time oriented, globally hyperbolic smooth Lorentzian manifolds and Tb(i) = Tb(i)(g(i), ψ(i)), ψ(i) ∈ BL. As- sume (M, gˆ(i)), i = 1, 2 have no conjugate points, the equation (5) holds for gˆ(i) and (i) Tb (g, ψ) are smooth in g and ψ. Let µgˆ(i) (t), t ∈ [−1, 1] be time-like geodesics on (i) (M, gˆ ). Let V be open relatively compact neighborhoods of both µgˆ(i) ([−1, 1]) and (i) (i) V ⊂ M (t0) for some t0. Let −1 < s− < s+ < 1 and p± = µgˆ(i) (s±) such that (1) (2) (i) p± = p± . Let L , i = 1, 2 be the source-to-solution operator for the equation (i) (i) (i) (i) (i) (i) Ein(g ) = Tb (g , ψ ) + F , in M (t0), (8) (i) (i) (i) + (i) g =g ˆ , in M (t0) \ Jg(i) (supp(F )). Suppose the microlocal linearization stability condition holds. If L(1) = L(2), then (1) (1) (2) (2) ∗ (2) there is a diffeormorphism Ψ: Igˆ(1) (p− , p+ ) → Igˆ(2) (p− , p+ ) such that Ψ gˆ = (1) (1) (1) gˆ in Igˆ(1) (p− , p+ ). It is worth mentioning that compared with the result in Theorem 2.2, the metric is determined up to isometries rather than its conformal class. This is done by analyzing the principal symbols of waves generated by nonlinear interactions. 4.2. Nonlinear elastic wave equation. de Hoop, Wang and Uhlmann [7] considered the initial boundary value problem for a quasilinear elastic wave equation ∂2u ρ − ∇ · S(x, u) = 0, (t, x) ∈ (0,T ) × Ω, ∂t2 u(t, x) = f(t, x), (t, x) ∈ (0,T ) × ∂Ω, (9) ∂ u(0, x) = u(0, x) = 0, x ∈ Ω. ∂t Here Ω ⊂ R3 is a bounded domain with smooth boundary ∂Ω. We denote x = (x1, x2, x3) to be the Cartesian coordinates. Then we can write the displacement NONLINEAR HYPERBOLIC EQUATIONS 465 vector as u = (u1, u2, u3) under the Cartesian coordinates. The stress tensor S has the form ∂ui ∂ui Sij =λεmmδij + λεemm + 2µ εij + εejn ∂xj ∂xn (10) 3 + A εeinεejn + B(2εennεeij + εemnεemnδij) + C εemmεennδij + O(u ), where ε is the strain tensor defined as 1 ∂ui ∂uj ∂uk ∂uk εij(u) = + + , 2 ∂xj ∂xi ∂xi ∂xj and εe is the linearized strain tensor 1 ∂ui ∂uj εeij(u) = + . 2 ∂xj ∂xi By using the notation O(u3) we are considering the small displacement asymptotics. The functions λ(x), µ(x), ρ(x), A (x), B(x), C (x) are all smooth on Ω. The parameters λ and µ are called Lamémoduli and ρ is the density. This model is widely used and can be found in [21,8,7]. The inverse problem is to recover the parameters λ, µ, ρ, A , B, C from the displacement-to- traction map Λ: f → ν · S(x, u)|(0,T )×∂Ω, where ν is the exterior normal unit vector to ∂Ω. Denote S = SL + SN , where SL is the linearized stress L Sij(x, u) = λεemmδij + µεeij, and

N λ + B ∂um ∂um ∂um ∂un B ∂um ∂un ∂um ∂uj Sij (x, u) = δij + C δij + δij + B 2 ∂xn ∂xn ∂xm ∂xn 2 ∂xn ∂xm ∂xm ∂xi A ∂uj ∂um ∂um ∂ui + + (λ + B) 4 ∂xm ∂xi ∂xm ∂xj A ∂um ∂um ∂ui ∂uj ∂ui ∂um + µ + + + + O(u3). 4 ∂xi ∂xj ∂xm ∂xm ∂xm ∂xj The linear elastic wave equation reads ∂2u ρ − ∇ · SL(x, u) = 0, (t, x) ∈ (0,T ) × Ω, ∂t2 u(t, x) = f(t, x), (t, x) ∈ (0,T ) × ∂Ω, (11) ∂ u(0, x) = u(0, x) = 0, x ∈ Ω. ∂t We denote the Dirichlet-to-Neumann map for the above linear elastic wave equation as lin L Λ : f → ν · S (x, u)|(0,T )×∂Ω.

Let diamP/S(Ω) be the diameter of Ω with respect to gP/S, and assume T > 2diamS(Ω). One can ﬁrst recover the Dirchlet-to-Neumann map Λlin for the linear elastic wave equation (11) by ﬁrst order linearization of Λ (cf. [7]) ∂ Λ(f)| = Λlin(f). ∂ =0 466 GUNTHER UHLMANN AND JIAN ZHAI

In an isotropic elastic medium, there are two types of waves traveling in diﬀerent q λ+2µ wave speeds. The longitudinal waves (P- waves) travel with speed cP = ρ , q µ and the transversal waves (S-waves) travel with speed cS = ρ . Denote the Riemannian metrics associated with P/S- wave speeds to be −2 2 gP/S = cP/Sds , where ds2 is the Euclidean metric. It was shown in [13] that from Λlin one can recover the scattering relation associated to the metrics gP and gS. Thus, one can recover the two wavespeeds cP and cS if either (1) (Ω, gP/S) have no conjugate points [27] or (2) they satisfy the foliation condition [30]. Recall that a Riemannian manifold (M, g) satisﬁes the foliation condition if it can be foliated by strictly convex hypersurfaces [35]. Simply assume ρ = 1, then λ, µ can be recovered, and we will focus on the recovery of the nonlinear parameters A , B, C .

Remark 1. Assume cP 6= 2cS except at isolated points in Ω, then the density ρ can be determined separately under the foliation condition [4] and the no conjugate points condition with extra restriction on curvatures [28]. In [7], the authors use the nonlinear interactions of one distorted plane P- wave and one distorted plane S- wave to recover the parameters A and B. It is observed that those nonlinear interactions of waves do not depend on the parameter C . The authors of this survey [38] gave an alternative approach, based on the con- struction of Gaussian beam solutions, to prove the uniqueness of A and B, and further showed the uniqueness of C . Next we introduce this approach. A Gaussian beam solution to the linear elastic wave equation ∂2u ρ − ∇ · SL(x, u) = 0 ∂t2 is of the form i%ϕ(t,x) u(t, x) = e a(t, x) + R%(t, x), with a large parameter %. The phase function ϕ is complex-valued. The principal i%ϕ(t,x) 2 term e a(t, x) is concentrated near a null geodesic ϑ in ((0,T )×Ω, −dt +gP/S). For more discussions about Gaussian beam solution, we refer to [9, 10]. Assume p is an arbitray point in Ω. Take ϑ(0), ϑ(2) to be the null geodesics in the 2 (1) Lorentzian manifold ((0,T )×Ω, −dt +gS), and ϑ the null geodesic in Lorentzian 2 (0) (1) (2) T manifold ((0,T ) × Ω, −dt + gP ), such that ϑ , ϑ , ϑ intersect at ( 2 , p). We can construct solutions: • u(1),P : the Gaussian beam solution representing P-waves, concentrated near the null geodesic ϑ(1); • u(2),S: the Gaussian beam solution representing S-waves, concentrated near the null geodesic ϑ(2); • u(0),S: the Gaussian beam solution representing S-waves, concentrated near the null geodesic ϑ(0). From the second order linearization of the the displacement-to-traction map Λ, one can recover Z T Z G(∇u(1), ∇u(2), ∇u(0)) dxdt, 0 Ω NONLINEAR HYPERBOLIC EQUATIONS 467 where G is a cubic form explicitly given by

G(∇u(1), ∇u(2), ∇u(0)) =(λ + B)(∇u(1) : ∇u(2))(∇ · u(0)) + 2C (∇ · u(1))(∇ · u(2))(∇ · u(0)) + B (∇ · u(1))(∇u(2) : ∇T u(0)) + (∇ · u(2))(∇u(1) : ∇T u(0)) + (∇u(1) : ∇T u(2))(∇ · u(0)   ∂u(1) (2) ∂u(2) (1) (0) (1) T (2) (0) A j ∂um j ∂um ∂ui + B(∇u : ∇ u )(∇ · u ) +  +  4 ∂xm ∂xi ∂xm ∂xi ∂xj + (λ + B) (∇ · u(1))(∇u(2) : ∇u(0)) + (∇ · u(2))(∇u(1) : ∇u(0)) (1) (2) (2) (1) (1) (2) (2) (1) A ∂u ∂u ∂u ∂u ∂u ∂uj ∂u ∂uj + µ + m m + m m + i + i 4 ∂xi ∂xj ∂xi ∂xj ∂xm ∂xm ∂xm ∂xm (1) (2) ! (0) ∂u ∂u(2) ∂u ∂u(1) ∂u + i m + i m i . ∂xm ∂xj ∂xm ∂xj ∂xj (12) Note that the principal term of G(∇u(1), ∇u(2), ∇u(0)) is supported only in a neighborhood of p. By proper choices of the null geodesics ϑ(0), ϑ(1), ϑ(2) and the Gaussian beam solutions, one can recover the parameters A and B at the point p. Then one can use Gaussian beam solutions: • u(1),P , u(2),P , u(0),P : the Gaussian beam solutions representing P-waves, concentrated near the same null geodesic ϑ(1); Since A and B have been recovered, one now recover Z T Z C (∇ · u(1))(∇ · u(2))(∇ · u(0)) dxdt. 0 Ω Now C (∇·u(1))(∇·u(2))(∇·u(0)) is concentrated near the null-geodesic ϑ(1). Assume ϑ(1)(t) = (t, γ(1)(t)). From the above integral, one can recover the Jacobi weighted −9/2 −3/2 (1) ray transform of the ﬁrst kind (cf. [11]) of C cP ρ along the geodesic γ (in the Riemannian manifold (Ω, gP )). By the invertibility of this weighted ray transform under no conjugate points condition [11] or the foliation condition [26], one can recover C . The results the nonlinear elastic wave equations (9) are summarized as follows.

Theorem 4.2. Assume ρ ≡ 1, T > 2diamS(Ω), and ∂Ω is strictly convex with respect to gP/S. Assume further that either of the following conditions holds

1. (Ω, gP/S) has no conjugate points; 2. (Ω, gP/S) satisﬁes the foliation condition. Then the parameters λ, µ, A , B, C are uniquely determined in Ω by Λ. 4.3. Recovery of an Hermitian connection. Chen et. al. [5] considered an inverse problem of recovering a Hermitian connection A on E = M × Cn in the equation 2 Aφ + κ|φ| φ = 0, (13) ∗ from the source-to-solution map. Here φ is a section of E, A = (d + A) (d + A). Assume g is the Minkowski metric for simplicity. Denote LA to be the source-to- solution map corresponding to measurements gathered in

f = (0, 1) × B(0). 468 GUNTHER UHLMANN AND JIAN ZHAI

Denote [ + − D = (J (x) ∩ J (z)). x,z∈f Then 1+3 Theorem 4.3. Let A and B be two connections in R such that LA = LB. Suppose that κ 6= 0. Then there exists a smooth u : D → U(n) such that u|f = Id and B = u−1du + u−1Au. The equation (13) can be regarded as a component of Yang-Mills-Higgs equations (cf. [34]) with the Mexican hat type potential.

4.4. Determination of an interface of discontinuity. Assume (R3, g) be a Riemannian manifold. Consider the equation 2 2 3 (∂t − ∆g)u(t, x) + a(t, x)u (t, x) = f, in (−∞, t0) × R , 3 (14) u(t, x) = 0, in (−∞, 0) × R . In [6], de Hoop, Uhlmann and Wang analyze the interaction of two distorted plane waves at an interface S0, where the coeﬃcient a is discontinuous. In fact, a is a conormal distribution. It is noted in [23] that the nonlinear interaction of two distorted plane waves does not produce new propagating singularities if a is smooth. Associated inverse problems are also discussed in [6].

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