1.1 The semi-linear model

Our main aim is to study quasi-linear equations, but to describe how the method works, we start our considerations with semi-linear equations. We consider the model setup

(1.2) $$ \begin{align} \begin{aligned} \begin{cases} \Box_{g}u+u^m=f\,\quad \text{on}\ (-\infty,T)\times M_0, \\ u=0 \quad \text{on}\ (-\infty,-T)\times M_0. \end{cases} \end{aligned} \end{align} $$

Here, $m\geqslant 3$ is an integer and $T>0$ is a parameter to be fixed later and the source f is real-valued and compactly supported in the set $(-T,T)\times M_0$ . The wave operator, $\Box _g$ , is defined in local coordinates $(x^0,\ldots ,x^n)$ by the expression

$$ \begin{align*}\Box_{g} u = -\sum_{j,k=0}^n\left|\det g\right|{}^{-\frac{1}{2}} \frac{\partial}{\partial x^j}( \left|\det g\right|{}^{\frac{1}{2}}g^{jk} \frac{\partial u}{\partial x^k}),\end{align*} $$

where $g^{jk}$ stands for the elements of the inverse of g. Note that we are using the $(x^0,x')$ -coordinate system on M that is given by (1.1).

1.2 The quasi-linear model

For the quasi-linear wave equation, we first consider a family of smooth real-valued symmetric tensors $G_z(x)=G(x,z){=(G_{jk}(x,z))_{j,k=0}^n}$ with $x \in M$ and $z \in \mathbb R$ satisfying

1. (i) $G(x,0)=g(x)$ and $\partial _z G(x,0)=0$ for all $x \in M$ .

2. (ii) The tensor $h(x)=\frac {1}{2}\partial ^2_z G(x,0)$ satisfies $\langle v,v\rangle _h\neq 0$ for all nonzero $v\in LM$ .

Here, $LM$ denotes the bundle of light-like vectors on M with respect to the metric g. We subsequently consider the equation

(1.3) $$ \begin{align} \begin{aligned} \begin{cases} \Box_{G_u}u=f\,\quad \text{on}\ (-\infty,T)\times M_0, \\ u=0 \quad \text{on}\ (-\infty,-T)\times M_0. \end{cases} \end{aligned} \end{align} $$

Here, the quasi-linear wave operator $\Box _{G_u}$ is defined in local coordinates $(x^0,\ldots ,x^n)$ through

$$ \begin{align*}\Box_{G_u}u = -\sum_{j,k=0}^n\left|\det G(x,{ u(x)})\right|{}^{-\frac{1}{2}} \frac{\partial}{\partial x^j} \bigg( \left|\det G(x,{ u(x)})\right|{}^{\frac{1}{2}}G^{jk}(x,{ u(x)}) \frac{\partial u}{\partial x^k}{ (x)}\bigg),\end{align*} $$

where $G^{jk}$ stands for elements of the inverse of G. We will assume that the source f in (1.3) is real-valued and compactly supported in $(-T,T)\times M_0$ .

In Subsection 2.1 we show that each of the Cauchy problems (1.2)–(1.3) above admits a unique solution

$$ \begin{align*}u \in \mathcal C^2((-\infty,T)\times M_0),\quad \forall\, f \in \mathscr C_O,\end{align*} $$

where given any relatively compact open set $\mathcal O \subset (-T,T)\times M_0$ , we define

(1.4) $$ \begin{align} \mathscr C_{\mathcal O}=\{h \in H^{n+1}(\mathbb R\times M_0;\mathbb R)\,:\,\operatorname{\mathrm{supp}} h \subset \mathcal O,\quad \|h\|_{H^{n+1}(\mathbb R\times M_0)}\leqslant r_{\mathcal O}\}\end{align} $$

and $r_{\mathcal O}$ is a sufficiently small constant depending on $(M,g)$ , $\mathcal O$ and T.

1.3 Source-to-solution map and the remote sensing inverse problem

Our primary interest lies in the setting that the sources can be actively placed near a world line $\mu _{\mathrm{in}}$ and the corresponding unique small solution u will be be measured near another disjoint world line $\mu _{\mathrm{out}}$ corresponding to some observer. The main question is whether such experiments corresponding to the separated source and observation regions determine the structure of the background unperturbed media; that is, $(M,g)$ .

To state the inverse problem precisely, let us consider two disjoint time-like future-pointing smooth paths

$$ \begin{align*}\mu_{\text{in}}:[t_0^-,t_0^+]\to M \quad \text{and}\quad \mu_{\text{out}}:[s_0^-,s_0^+]\to M\end{align*} $$

and impose the conditions that

(1.5) $$ \begin{align} \mu_{\text{out}}(s_0^+)\notin I^+(\mu_{\text{in}}(t_0^+))\quad \text{and}\quad \mu_{\text{in}}(t_0^-) \notin J^-(\mu_{\text{out}}(s_0^-)). \end{align} $$

Next, let us consider the source and observation regions $\Omega _{\mathrm{in}}$ and $\Omega _{\mathrm{out}}$ as small neighbourhoods of $\mu _{\mathrm{in}}([t_0^-,t_0^+])$ and $\mu _{\mathrm{out}}([s_0^-,s_0^+])$ in M, respectively. These two open neighbourhoods will be precisely defined in Subsection 1.4. We will also make the standing assumption that

$$ \begin{align*}(\Omega_{\text{in}}, g_{|_{\Omega_{\text{in}}}}) \quad \text{and}\quad (\Omega_{\text{out}},g_{|_{\Omega_{\text{out}}}})\end{align*} $$

are a priori known as Lorentzian manifolds; that is, we are given local coordinates, the transition functions between the local charts and the metric tensors on these coordinate charts.

The partial data inverse problem with separated sources and observations (or the remote sensing problem) can now be formulated as follows. Is it possible to uniquely determine the unperturbed manifold $(M,g)$ (recall that $G(x,0)=g(x)$ ) by observing solutions to the nonlinear wave equations (1.2) or (1.3) in $\Omega _{\mathrm{out}}$ that arise from small sources placed in $\Omega _{\mathrm{in}}$ ? The inverse problems with partial data are widely encountered in applications. The partial data problems where the sources and observations are made only on a part of boundary have been a focus of research for inverse problems for linear elliptic equations [Reference Dos Santos Ferreira, Kenig, Salo and Uhlmann22, Reference Alinhac and Baouendi2 Reference Ambartsoumian, Felea, Nolan and Quinto3, Reference Guillarmou and Tzou37, Reference Ambartsoumian, Felea, Quinto and Nolan4 Reference Barreto, Uhlmann and Wang7, Reference Ambartsoumian, Felea, Quinto and Nolan4 Reference Belishev8, Reference Imanuvilov, Uhlmann and Yamamoto49, Reference Anderson, Katsuda, Kurylev, Lassas and Taylor5 Reference Alinhac1, Reference Anderson, Katsuda, Kurylev, Lassas and Taylor5 Reference Babich and Ulin6, Reference Anderson, Katsuda, Kurylev, Lassas and Taylor5 Reference Barreto, Uhlmann and Wang7, Reference Krupchyk and Uhlmann59, Reference Krupchyk and Uhlmann60, Reference Babich and Ulin6 Reference Alinhac and Baouendi2, Reference Lassas, Liimatainen, Lin and Salo68, Reference Barreto, Uhlmann and Wang7 Reference Alinhac and Baouendi2]. However, in most of these results it is assumed that the sets where the sources are supported and where the solutions are observed do intersect, with the notable exceptions in [Reference Ambartsoumian, Felea, Quinto and Nolan4 Reference Belishev8, Reference Imanuvilov, Uhlmann and Yamamoto49, Reference Anderson, Katsuda, Kurylev, Lassas and Taylor5 Reference Babich and Ulin6]. In particular, in inverse problems for hyperbolic equations it is essential that the sources, which are physically implemented by acoustic transducers, or by air guns or explosives in seismic imaging, are located far away from the sensors that observe the wave fields with a high precision. This is why the partial data problems are essential in applications. The partial data problems with separated sources and observations have been studied for linear hyperbolic equations, but the present results require convexity or other geometrical restrictions that guarantee the exact controllability of the system [Reference Anderson, Katsuda, Kurylev, Lassas and Taylor5 Reference Belishev8, Reference Lassas and Saksala70]. Let us remark that we can also apply the results in this article in the case when $\Omega _{\mathrm{in}}$ and $\Omega _{\mathrm{out}}$ intersect.

To formulate the inverse problem precisely, we define the source-to-solution map $\mathscr L$ associated to the semi-linear Cauchy equation (1.2) through the expression

(1.6) $$ \begin{align} \mathscr L f= u\,|_{\Omega_{\text{out}}}, \quad \forall\, f \in \mathscr{C}_{\Omega_{\text{in}}}, \end{align} $$

where u is the unique small solution to (1.2) subject to the source f and the set $\mathscr C_{\Omega _{\mathrm{in}}}$ is as defined by (1.4). Analogously, we define the source-to-solution map $\mathscr N$ for the quasi-linear Cauchy equation (1.3) through the expression

(1.7) $$ \begin{align} \mathscr N f= u\,|_{\Omega_{\text{out}}}, \quad \forall \,f\in \mathscr{C}_{\Omega_{\text{in}}}, \end{align} $$

where u is the unique small solution to (1.3) subject to the source f.

Our inverse problem can now be restated as whether the manifold $(M,g)$ can be uniquely recovered given the source-to-solution map $\mathscr L$ or $\mathscr N$ . Recall that $g(x)=G(x,0)$ in the quasi-linear model.

Due to finite speed of propagation for the wave equation, the optimal region where one can recover the geometry is the causal diamond generated by the source region $\Omega _{\mathrm{in}}$ and $\Omega _{\mathrm{out}}$ that is defined through

(1.8) $$ \begin{align} \mathbb D_e=\left(\bigcup_{q \in \Omega_{\text{in}}} I^{+}(q)\right) \cap \left(\bigcup_{q \in \Omega_{\text{out}}} {I}^{-}(q)\right),\end{align} $$

given the knowledge of the source-to-solution map $\mathscr L$ or $\mathscr N$ . As we will see, we are able to recover the geometry in the slightly smaller set; that is, a causal diamond determined by the points $\mu _{\mathrm{in}}(t_0^-)$ and $\mu _{\mathrm{out}}(s_0^+)$ ,

(1.9) $$ \begin{align} \mathbb D= I^{+}(\mu_{\text{in}}(t_0^-)) \cap {I}^{-}(\mu_{\text{out}}(s_0^+)).\end{align} $$

1.4 Main results

Before stating the main results, let us define, in detail, the source and observation neighbourhoods of the two future-pointing time-like curves

$$ \begin{align*}\mu_{\text{in}}:[t_0^-,t_0^+]\to M \quad \text{and}\quad \mu_{\text{out}}:[s_0^-,s_0^+]\to M\end{align*} $$

satisfying (1.5). We begin by extending the time-like paths to slightly larger intervals

$$ \begin{align*}\mu_{\text{in}}:[t_1^-,t_1^+]\to M \quad \text{and}\quad \mu_{\text{out}}:[s_1^-,s_1^+]\to M\end{align*} $$

and proceed to define the source and observation regions as a foliation of time-like future-pointing paths near the paths $\mu _{\mathrm{in}}((t_1^-,t_1^+))$ and $\mu _{\mathrm{out}}((s_1^-,s_1^+))$ . To make this precise, we use Fermi coordinates near these paths.

Let $\{\alpha _{i}\}_{i=1}^n$ be an orthonormal basis for $\dot {\mu }_{\mathrm{in}}(t_1^-)^\perp $ and subsequently consider $\{e_{i}(t)\}_{i=1}^{n}$ to denote the parallel transport of $\{\alpha _i\}_{i=1}^n$ along $\mu _{\mathrm{in}}$ to the point $\mu _{\mathrm{in}}(t)$ . Let

$$ \begin{align*}F_{\text{in}}: (t_1^-,t_1^+) \times B(0,\delta) \to M\end{align*} $$

be defined through

$$ \begin{align*}F_{\text{in}}(t,y) = \exp_{\mu_{\text{in}}(t)}(\sum_{i=1}^n y^i e_{i}(t)).\end{align*} $$

Here, $B(0,\delta )$ is the ball of radius $\delta $ in $\mathbb R^n$ . For $\delta $ sufficiently small, the map $F_{\mathrm{in}}$ is a smooth diffeomorphism and the paths

$$ \begin{align*}\mu_a(t) = F_{\text{in}}(t,a) \quad a \in B(0,\delta)\end{align*} $$

are smooth time-like paths. We define $F_{\mathrm{out}}: (s_1^-,s_1^+)\times B(0,\delta )\to M$ analogously as above with $\mu _{\mathrm{in}}$ replaced by $\mu _{\mathrm{out}}$ . Finally, we define the source and observation regions through the expression

(1.10) $$ \begin{align} \begin{aligned} \Omega_{\text{in}}&= \{ F_{\text{in}}(t,y)\,:\, t \in (t_1^-,t_1^+),\quad y\in B(0,\delta)\}\\ \Omega_{\text{out}}&= \{ F_{\text{out}}(t,y)\,:\, t \in (s_1^-,s_1^+),\quad y \in B(0,\delta)\}. \end{aligned} \end{align} $$

We will impose that $\delta $ is small enough so that the following condition is satisfied. This can always be guaranteed in view of (1.5).

(1.11) $$ \begin{align} \Omega_{\text{in}} \cap J^{-}(F_{\text{out}}(\{s_1^-\}\times B(0,\delta))=\emptyset \quad \text{and}\quad \overline{\Omega_{\text{out}}}\cap J^+(\mu_{\text{in}}(t_1^+))=\emptyset. \end{align} $$

Our main result regarding the inverse problems for the semi-linear and quasi-linear models above can be stated as follows.

Theorem 1.1. Let $m \geqslant 3$ be an integer and $(M^{(1)},g^{(1)})$ , $(M^{(2)},g^{(2)})$ be smooth globally hyperbolic Lorentzian manifolds of dimension $1+n\geqslant 3$ . Let $G^{(j)}_z$ , $j=1,2$ , be a symmetric tensor on $M^{(j)}$ that satisfies conditions (i)–(ii) and recall that $g^{(j)}(x)=G^{(j)}(x,0)$ for all $x \in M^{(j)}$ . Let $\mu _{\mathrm{in}}^{(j)}:[t_0^-,t_0^+]\to M^{(j)}$ and $\mu _{\mathrm{out}}^{(j)}:[s_0^-,s_0^+]\to M^{(j)}$ be smooth time-like paths satisfying (1.5). For $j=1,2$ , let the source region $\Omega _{\mathrm{in}}^{(j)}$ and the observation region $\Omega ^{(j)}_{\mathrm{out}}$ be defined by (1.10). We assume that these neighbourhoods are sufficiently small so that (1.11) holds. Let $T>0$ be sufficiently large so that

$$ \begin{align*}\mathbb D_e^{(j)}\subset (-T,T)\times M^{(j)}_0\quad \text{for}\ j=1,2,\end{align*} $$

and also that there exists isometric diffeomorphisms

$$ \begin{align*}\Psi^k:(\Omega_{k}^{(1)},g^{(1)}|_{\Omega_{k}^{(1)}})\to (\Omega_{k}^{(2)},g^{(2)}|_{\Omega_{k}^{(2)}})\quad k \in \{\textit{in},{\textit{out}}\}.\end{align*} $$

Next, and for $j=1,2$ , we consider the source-to-solution maps $\mathscr L^{(j)}$ and $\mathscr N^{(j)}$ associated to (1.2)–(1.3) respectively and assume that one of the following statements hold:

1. (i) $\Psi ^{\text {in}}\circ (\mathscr L^{(1)}(f))= \mathscr L^{(2)}(f\circ (\Psi ^{\text {in}})^{-1})$ for all sources $f\in \mathscr C^{(1)}_{\Omega _{\mathrm{in}}^{(1)}}$ ,

or

1. (ii) $\Psi ^{\text {in}}\circ (\mathscr N^{(1)}(f))= \mathscr N^{(2)}(f\circ (\Psi ^{\text {in}})^{-1})$ for all sources $f\in \mathscr C^{(1)}_{\Omega _{\mathrm{in}}^{(1)}}$ ,

where the set $\mathscr C^{(1)}_{\Omega ^{(1)}_{\mathrm{in}}}$ is defined by (1.4) associated to $T>0$ and the manifold $(M^{(1)},g^{(1)})$ .

Then, under the hypotheses above, there exists a smooth diffeomorphism $\Psi :\mathbb D^{(1)} \to \mathbb D^{(2)}$ that is equal to $\Psi ^{\text {in}}$ on the set $\Omega ^{(1)}_{\mathrm{in}}\cap \mathbb D^{(1)}$ and equals $\Psi ^{\text {out}}$ on the set $\Omega ^{(1)}_{\mathrm{out}}\cap \mathbb D^{(1)}$ and such that

$$ \begin{align*}\Psi^*g^{(2)}=c\,g^{(1)}\quad \text{on}\ \mathbb D^{(1)},\end{align*} $$

for some smooth real-valued function $c=c(x)$ , $c:\mathbb D^{(1)}\to \mathbb R_+$ .

Moreover, in the case that statement (i) holds and if $(n,m)\neq (3,3)$ , we have $c\equiv 1$ on the causal diamond $\mathbb D^{(1)}$ .

1.5 Recovery of geometry from the three-to-one scattering relation

The proof of Theorem 1.1 will be divided into an analytical and a geometrical part, with Sections 2–5 covering the analytical part and Sections 6–7 covering the geometric part of the analysis. In the analysis part, we use the idea of multiple-fold linearisation of the wave equation first used in [Reference Kurylev, Lassas and Uhlmann64] together with the principle of propagation of singularities for the wave equation, resulting in a geometrical data on the set $\mathbb D$ , the three-to-one scattering relation, that we will next define.

Before formulation of the definition, we set some notations. We say that geodesics $\gamma _{v_{1}}$ and $\gamma _{v_2}$ are distinct if the maximal geodesics that are extensions of $\gamma _{v_{1}}$ and $\gamma _{v_{2}}$ do not coincide as subsets of M. Also, for $v=(x,\eta )\in L^+M$ , let $s(v)=\sup \{s>0:\ \gamma _{v}([0,s))\subset M\}$ and $\rho (v)=\sup \{s\in [0,s(v)]:\ \gamma _{v}(s)\not \in I^+(x)\}$ . As discussed in Subsection 6.3, $\gamma _v(\rho (v))$ is called the first cut point of $\gamma _v$ .

For $v=(x,\eta )\in L^+M$ , let $\overrightarrow {\;\gamma _{v}}=\gamma _{x,\eta }([0,s(x,\eta )))$ be the light-like geodesics that are maximally extended in M to the future from v. Also, let $\overleftarrow {\;\gamma _{v}}=\gamma _{x,-\eta }([0,s(x,-\eta )))$ be the geodesic that emanates from v to the past.

Next we consider a set $\operatorname {\mathrm {R}}\subset (L^+M)^4$ of 4-tuples of vectors $(v_0,v_1, v_2,v_3)$ and say that these vectors satisfy relation $\operatorname {\mathrm {R}}$ if $(v_0,v_1, v_2,v_3)\in \operatorname {\mathrm {R}}$ .

In other words, (R1) means that if $(v_0,v_1, v_2,v_3) \in \operatorname {\mathrm {R}}$ , then it is necessary that the future-pointing geodesics $\gamma _{v_1},\gamma _{v_2}$ and $\gamma _{v_3}$ must intersect at some point y and some future-pointing geodesic emanating from y arrives to $v_0$ . The condition (R2) means for $(v_0,v_1, v_2,v_3) \in \operatorname {\mathrm {R}}$ it is sufficient that the future-pointing geodesics $\gamma _{v_1},\gamma _{v_2}$ and $\gamma _{v_3}$ intersect at some point y before their first cut points and that the past-pointing null geodesic $\gamma _{v_0}$ arrives to the point y in the direction $\xi _0$ that is in the span of the velocity vectors of $\gamma _{v_1},\gamma _{v_2}$ and $\gamma _{v_3}$ at the point y and, finally, that the geodesic $\gamma _{v_0}([s_0,0])$ has no cut points. The relation $\operatorname {\mathrm {R}}$ is visualised in Figure 3.

Figure 1

Nonlinear interaction of three waves when $m=3$ and g is the Minkowski metric. The four figures are snapshots of the waves in the space $\mathbb R^3$ at different times, with time progressing from top to bottom. First figure from the top: Plane waves before interactions. Second figure: The 2-wave interactions (black line segments) appear but do not cause singularities that propagate in new directions. Third figure: The 3-wave interactions generate a new wave (blue cone). Fourth figure: The new wave propagates near the $x^1, x^2$ plane. By varying the directions of the incoming plane waves, a wave propagating near any plane can be generated. Note that, for general g, the picture may be more complicated due to caustics.

Figure 2

Schematic of the geometric setting. The time-like paths $\mu _{\mathrm{in}}$ and $\mu _{\mathrm{out}}$ are in green and their neighbourhoods $\Omega _{\mathrm{in}}$ and $\Omega _{\mathrm{\rm out}}$ are in orange. The set $\mathbb D$ is enclosed by the black rectangle. The set $I^+({ \mu _{\mathrm{\rm in}}(t_1^+)})$ is in light blue, and the set $J^-({ \mu _{\mathrm{out}}(s_1^+)})$ is in light red; cf. (1.5).

Figure 3

Three-to-one scattering relation $\operatorname {\mathrm {R}}$ in the $1+2$ -dimensional Minkowski space. The red vectors $v_1$ , $v_2$ and $v_3$ and the green vector $v_0$ satisfy $(v_0,v_1,v_2,v_3) \in \operatorname {\mathrm {R}}$ . The vector $\xi _0$ in Definition 1.2 is in blue and the vectors $\xi _1$ , $\xi _2$ and $\xi _3$ are in black. The latter three have the same base point with $\xi _0$ and are not labelled.

We emphasise that in Definition 1.2 there are no requirements for the 4-tuples $(v_0,v_1, v_2,v_3)$ for which the conditions in (R1) or (R2) are not valid. This has the important consequence that to verify that the source-to-solution map $\mathscr L$ or $\mathscr N$ determines a three-to-one scattering relation, we need to consider only those 4-tuples $(v_0,v_1, v_2,v_3)$ for which the geodesics $\gamma _{v_j}$ , $j=0,1,2,3$ do not intersect at the conjugate points and the structure of the multiple pairwise intersections of the geodesics is simple. This makes it possible to avoid technical geometrical difficulties encountered in several studies on inverse problems for linear and nonlinear differential equations; see, for example, [Reference Dos Santos Ferreira, Kurylev, Lassas, Liimatainen and Salo23, Reference Dos Santos Ferreira, Kurylev, Lassas and Salo24, Reference Hintz, Uhlmann and Zhai40, Reference Kurylev, Lassas and Uhlmann64].

The following theorem states that the three-to-one scattering relation determines uniquely the topological, differential and conformal structure of the set $\mathbb D$ .

Theorem 1.3. Let $(M^{(1)},g^{(1)})$ , $(M^{(2)},g^{(2)})$ be smooth globally hyperbolic Lorentzian manifolds of dimension $1+n\geqslant 3$ . Let $\mu _{\mathrm{in}}^{(j)}:[t_0^-,t_0^+]\to M^{(j)}$ and $\mu _{\mathrm{out}}^{(j)}:[s_0^-,s_0^+]\to M^{(j)}$ be smooth time-like paths satisfying (1.5). For $j=1,2$ , let the source region $\Omega _{\mathrm{in}}^{(j)}$ and the observation region $\Omega ^{(j)}_{\mathrm{out}}$ be defined by (1.10). We assume that these neighbourhoods are sufficiently small so that (1.11) holds. Moreover, we assume that there are isometric diffeomorphisms

$$ \begin{align*}\Psi^k:(\Omega_{k}^{(1)},g^{(1)}|_{\Omega_{k}^{(1)}})\to (\Omega_{k}^{(2)},g^{(2)}|_{\Omega_{k}^{(2)}})\quad k \in \{{\textit{in}},{\textit{out}}\}.\end{align*} $$

Suppose next that there are relations $\operatorname {\mathrm {R}}^{(j)}\subset L^+\Omega _{out}^{(j)}\times (L^+\Omega _{in}^{(j)})^3$ , $j=1,2$ , that satisfy conditions (R1) and (R2) in Definition 1.2 for manifolds $(M^{(j)},g^{(j)})$ and that

(1.12) $$ \begin{align} \operatorname{\mathrm{R}}^{(2)}=\bigg\{(\Psi^{in}_*v_0,\Psi^{out}_*v_1,\Psi^{out}_*v_2,\Psi^{out}_*v_3)\ \bigg|\ (v_0,v_1,v_2,v_3)\in \operatorname{\mathrm{R}}^{(1)}\bigg\}. \end{align} $$

Then there exists a smooth diffeomorphism $\Psi :\mathbb D^{(1)} \to \mathbb D^{(2)}$ that is equal to $\Psi ^{\text {in}}$ on the set $\Omega ^{(1)}_{\mathrm{in}}\cap \mathbb D^{(1)}$ and equals $\Psi ^{\text {out}}$ on the set $\Omega ^{(1)}_{\mathrm{out}}\cap \mathbb D^{(1)}$ and such that

$$ \begin{align*}\Psi^*g^{(2)}=c\,g^{(1)}\quad \text{on}\ \mathbb D^{(1)},\end{align*} $$

for some smooth real-valued function $c=c(x)$ , $c:\mathbb D^{(1)}\to \mathbb R_+$ .

The motivation of Definition 1.2 and Theorem 1.3 is to provide a general framework that allows the results of this article to be applicable for other nonlinear hyperbolic equations similar to those studied in this article. Indeed, to consider some different kind of nonlinear hyperbolic equation (for example, such as $\square _g u+\|\nabla u\|_g^3=f$ in $\mathbb R^{1+n}$ ; see [Reference Wang and Zhou92] for the case $n=3$ ) one can define that $(v_0,v_1, v_2,v_3)$ satisfies the relation $\operatorname {\mathrm {R}}_{sim}$ if three singular waves sent to directions $v_1,v_2$ and $v_3$ interact so that the interaction produces a wave whose wave front contains the covector corresponding to $v_0$ . Then to apply Theorem 1.3 one has to show that $\operatorname {\mathrm {R}}_{sim}$ satisfies conditions (R1) and (R2). We note that condition (R1) is natural as the second-order interaction of waves does not produce singularities that propagate to new directions; see [Reference Greenleaf and Uhlmann36, Reference Kurylev, Lassas and Uhlmann64]. Condition (R2) is motivated by the general results for the interaction of three waves; see [Reference Barreto, Uhlmann and Wang7, Reference Barreto, Uhlmann and Wang7 Reference Barreto, Uhlmann and Wang7, Reference Sa Barreto and Wang83] and references therein. We emphasise that to verify condition (R2) one has to consider only geodesics that have no conjugate points and thus this condition can be verified without analysing interaction of waves near caustics.

1.6 Previous literature

The study of nonlinear wave equations is a fascinating topic in analysis with a rich literature. In contrast with the study of linear wave equations, there are numerous challenges in studying the existence, uniqueness and stability of solutions to such equations. These equations physically arise in the study of general relativity, such as the Einstein field equations. They also appear in the study of vibrating systems or the detection of perturbations arising in electronics, such as the telegraph equation or the study of semi-conductors; see, for instance, [Reference Cazenave and Haraux15]. We mention in particular that the quasi-linear model (1.3) studied in this article is a model for studying Einstein’s equations in wave coordinates [Reference Barreto, Uhlmann and Wang7 Reference Anderson, Katsuda, Kurylev, Lassas and Taylor5].

This article uses extensively the nonlinear interaction of three waves to solve the inverse problems. To analyse this, we use Gaussian beams. An alternative way to consider the nonlinear interaction is to use microlocal analysis and conormal singularities; see [Reference Greenleaf and Uhlmann36, Reference Guillemin and Uhlmann38, Reference Barreto, Uhlmann and Wang7 Reference Belishev8]. There are many results on such nonlinear interaction, starting with the studies of Bony [Reference Alinhac1 Reference Ambartsoumian, Felea, Quinto and Nolan4], Melrose and Ritter [Reference Melrose and Ritter77] and Rauch and Reed [Reference Belishev8 Reference Alinhac and Baouendi2]. However, these studies concern the direct problem and differ from the setting of this article in that they assumed that the geometrical setting of the interacting singularities and, in particular, the locations and types of caustics, is known a priori. In inverse problems we need to study waves on an unknown manifold, so we do not know the underlying geometry and, therefore, the location of the singularities of the waves. For example, the waves can have caustics that may even be of an unstable type.

Earlier, inverse problems for nonlinear hyperbolic equations with unknown metric have been studied using interaction of waves only in the (1+3)-dimensional case using interaction of four or more waves. Inverse problems for nonlinear scalar wave equations with a quadratic nonlinearity were studied in [Reference Kurylev, Lassas and Uhlmann64] using multiple-fold linearisation. Together with the phenomenon of propagation of singularities for the wave equation, the authors reduced the inverse problem for the wave equation to the study of light observation sets. This approach was extended in [Reference de Hoop, Uhlmann and Wang19, Reference Hintz and Uhlmann39, Reference Anderson, Katsuda, Kurylev, Lassas and Taylor5 Reference Alinhac and Baouendi2, Reference Barreto, Uhlmann and Wang7 Reference Alinhac1]. In [Reference Kurylev, Lassas, Oksanen and Uhlmann63], the coupled Einstein and scalar field equations were studied. The result has been more recently strengthened in [Reference Barreto, Uhlmann and Wang7 Reference Ambartsoumian, Felea, Nolan and Quinto3, Reference Uhlmann and Wang91] for the Einstein scalar field equations with general sources and for the Einstein–Maxwell equations. In particular, a technique to determine the conformal factor using the microlocal symbols of the observed waves was developed in [Reference Uhlmann and Wang91].

Aside from the works mentioned above, the majority of works have been on inverse problems for semi-linear wave equations, with quadratic nonlinearities studied in [Reference Kurylev, Lassas and Uhlmann64], a general semi-linear term studied in [Reference Hintz, Uhlmann and Zhai40, Reference Barreto, Uhlmann and Wang7 Reference Ambartsoumian, Felea, Nolan and Quinto3] and quadratic derivatives studied in [Reference Wang and Zhou92]; see also [Reference Babich and Ulin6 Reference Barreto, Uhlmann and Wang7] and references therein. All these works concern the $(1+3)$ -dimensional case. In recent works [Reference Chen, Lassas, Oksanen and Paternain16, Reference Chen, Lassas, Oksanen and Paternain17, Reference Feizmohammadi and Oksanen29], the authors have also studied problems of recovering zeroth- and first-order terms for semi-linear wave equations with Minkowski metric. We note that three wave interactions were used in [Reference Chen, Lassas, Oksanen and Paternain16, Reference Chen, Lassas, Oksanen and Paternain17] to determine the lower order terms in the equations and in modelling nonlinear elastic scattering from discontinuities [Reference de Hoop, Uhlmann and Wang20, Reference Alinhac and Baouendi2 Reference Alinhac1]. In [Reference Feizmohammadi and Oksanen28, Reference Babich and Ulin6 Reference Belishev and Katchalov9] similar multiple-fold linearisation methods were introduced to study inverse problems for elliptic nonlinear equations; see also [Reference Krupchyk and Uhlmann61].

All of the aforementioned works consider inverse problems for various types of nonlinear wave equations subject to small sources. The presence of a nonlinear term in the partial differential equation (PDE) is a strong tool in obtaining the uniqueness results. To discuss this feature in some detail, we note that the analogous inverse problem for the linear wave equation (see (2.1)) is still a major open problem. For this problem, much is known in the special setting that the coefficients of the metric are time-independent. We refer the reader to the work of Belishev and Kurylev in [Reference Alinhac1 Reference Alinhac and Baouendi2] that uses the boundary control method introduced in [Reference Belishev and Kurylev10] to solve this problem and to [Reference Anderson, Katsuda, Kurylev, Lassas and Taylor5, Reference Isozaki and Kurylev50, Reference Anderson, Katsuda, Kurylev, Lassas and Taylor5 Reference Ambartsoumian, Felea, Nolan and Quinto3, Reference Babich and Ulin6 Reference Anderson, Katsuda, Kurylev, Lassas and Taylor5] for a state-of-the-art result in the application of the boundary control method and finally to [Reference Hörmander41] for the related scattering control method. The boundary control method is known to fail in the case of general time-dependent coefficients, since it uses the unique continuation principle of Tataru [Reference Tataru89]. This principle is known to be false in the cases that the time dependence of coefficients is not real-analytic [Reference Alinhac1, Reference Alinhac and Baouendi2]. We refer the reader to [Reference Eskin25] for recovery of coefficients of a general linear wave equation under an analyticity assumption with respect to the time coordinate.

In the more challenging framework of general time-dependent coefficients and by using the alternative technique of studying the propagation of singularities for the wave equation, the inverse problem for the linear wave equation (see (2.1)) is reduced to the injectivity of the scattering relation on M; see the definition (5.1). The injectivity of the scattering relation is open unless the geometry of the manifold is static and an additional convex foliation codition is satisfied [Reference Stefanov, Uhlmann and Vasy86] on the spatial part of the manifold. In the studies of recovery of sub-principal coefficients for the linear wave equation, we refer the reader to the recent works [Reference Feizmohammadi, Ilmavirta, Kian and Oksanen26, Reference Feizmohammadi, Ilmavirta and Oksanen27, Reference Stefanov85] for recovery of zeroth- and first-order coefficients and to [Reference Stefanov and Yang87] for a reduction from the boundary data for the inverse problem associated to (2.1) to the study of geometrical transforms on M. This latter approach has been recently extended to general real principal type differential operators [Reference Ralston80].

The main underlying principle in the presence of a nonlinearity is that linearisation of the equation near the trivial solution results in a nonlinear interaction of solutions to the linear wave equation producing much richer dynamics for propagation of singularities. Owing to this richer dynamics, and somewhat paradoxically, inverse problems for nonlinear wave equations have been solved in much more general geometrical contexts than their counterparts for the linear wave equations.

Let us now discuss some of the main novelties of the present work. Firstly, we consider threefold linearisation of the nonlinear equations (1.2)–(1.3) and can therefore analyse inverse problems using interaction of three waves instead of earlier works relying on interaction of four waves. This makes it possible to consider more general equations with simpler techniques. Due to the new techniques, we can consider inverse problems on Lorenzian manifolds with any dimension $n+1\ge 3$ . As a second novelty, we introduce a new concept, the three-to-one scattering relation that can be applied for many kinds of nonlinear equations and which we hope to be useful for other researchers in the field of inverse problems. Also, this makes it possible to consider inverse problems in the remote sensing setting that includes both forward- and back-scattering problems. Finally, we mention our quite general quasi-linear model problem (1.3) with an unknown nonlinearity in the leading order term. We successfully study this complicated model with the use of Gaussian beams and show that the source-to-solution map determines the three-to-one scattering relation.

1.7 Outline of the article

We begin with some preliminaries in Section 2, starting with Proposition 2.1 that shows that the forward problems (1.2)–(1.3) are well-posed. We also recall the technique of multiple-fold linearisation and apply it to the semi-linear and quasi-linear equations separately. This will relate the source-to-solution maps, $\mathscr L$ and $\mathscr N$ , to the study of products of solutions to the linear wave equation; see (2.6) and (2.10). In Section 3, we briefly recall the construction of the classical Gaussian beams for the linear wave equation. We also show that it is possible to explicitly construct real-valued sources supported in the source and observation regions that generate exact solutions to the linear wave equations that are close in a suitable sense to the real parts of Gaussian beams. In Section 5 we prove the main analytical theorems, showing that the source-to-solution maps lead to a three-to-one scattering relation; see Theorem 5.1–5.2. Combined with Theorem 1.3, this proves the first half of Theorem 1.1 on the recovery of the topological, differential and conformal structure of the casual diamond $\mathbb D$ .

The geometrical sections of the article are concerned with the study of a general three-to-one scattering relation $\operatorname {\mathrm {R}}$ on M and the proof of Theorem 1.3. In Section 6, we recall some technical lemmas on globally hyperbolic Lorentzian geometries. In Section 7, we prove Theorem 7.10, showing that it is possible to use the three-to-one scattering relation to construct the earliest arrivals on $\mathbb D$ . Combining this with the results of [Reference Kurylev, Lassas and Uhlmann64] leads to unique recovery of the topological, differential and conformal structure of $(\mathbb D, g|_{\mathbb D})$ that completes the proof of Theorem 1.3.

Finally, Section 8 is concerned with the proof of Theorem 1.1. The first half of the proof – that is, the recovery of the topological, differential and conformal structure of the manifold – follows immediately from combining Theorems 5.1–5.2 together with Theorem 1.3. The remainder of this section deals with the recovery of the conformal factor c on $\mathbb D$ .