1 Introduction↩︎

1.1 Mathematical formulation and problem of interest↩︎

The present manuscript is concerned with an inverse problem associated with an initial boundary value problem (IBVP) for semilinear dynamical Schrödinger equation. More precisely, for \(m\geq 2\) an integer, we consider the following IBVP associated with the aforementioned equation by \[\begin{align} \label{govern95eqn} \begin{aligned} \begin{cases} \left(\mathrm{i}\partial_t +\sum_{j=1}^{n}\left(\partial_j+\mathrm{i}A_j(t,x)\right)^2 + q(t,x)\right)u(t,x)=B(t,x,u,\overline{u}),~(t,x)\in Q, \\ u(0,x) = 0,\quad x\in\Omega,\\ u(t,x) = f(t,x), \quad (t,x)\in \Sigma:=(0,T)\times\partial\Omega, \end{cases} \end{aligned} \end{align}\tag{1}\] where \(T>0\) be fixed, \(Q:=(0,T)\times\Omega\) with \(\Omega\subset \mathbb{R}^n~(n\geq 2)\) is a bounded domain with smooth boundary \(\partial \Omega\) and magnetic potential \(A(t,x):=(A_1(t,x),A_2(t,x),\cdots, A_n(t,x))\) be a real-vector field. Here we denoted by \(\partial_t\) and \(\partial_j\) (\(1\leq j\leq n)\), the partial derivatives with respect to time-variable \(t\in (0,T)\) and \(x_j\), the \(j\)th component of spacial variable \(x:=(x_1,x_2,\cdots,x_n)\in\Omega\), respectively. The non linear term \(B(t,x,u,\overline{u})\) is given by \[\begin{align} \label{non32linear32term32in32govern32equation} B(t,x,u,\overline{u}):=\sum_{a=1}^{m}r_a(t,x)u^a(t,x)(\overline{u})^{m-a}(t,x). \end{align}\tag{2}\] Throughout this manuscript, we assume that \(A \in C_c^{\infty}(Q,\mathbb{R}^n)\) and \(q, r_a \in C_c^\infty(Q,\mathbb{R})\) for each integer \(1\leq a\leq m\). Also, we denote the linear dynamical Schrödinger operator associated to 1 by \(\mathcal{P}_{A,q}\) and is given by \[\begin{align} \label{linear32operator32P} \begin{aligned} \mathcal{P}_{A,q}&:= \mathrm{i}\partial_t +\sum_{j=1}^{n}\left(\partial_j+\mathrm{i}A_j(t,x)\right)^2 + q(t,x)\\ &=i\partial_t +\Delta+2iA\cdot \nabla+i(\nabla\cdot A)-\lvert A\rvert^2+q. \end{aligned} \end{align}\tag{3}\] To discuss the well-posedness of the IBVP 1 , we first introduce some function spaces. For \(N=\Omega\), or \(\partial\Omega\), and \(s,t\geq 0\), we define the Hilbert space \(H^{s,t}((0,T)\times N)\) by \[H^{s,t}((0,T)\times N):= H^s(0,T;L^2(N))\cap L^2(0,T;H^t(N))\] with the norm \[\lVert u \rVert_{H^{s,t}((0,T)\times N)} :=\lVert u \rVert_{H^s(0,T;L^2(N))}+\lVert u \rVert_{L^2(0,T;H^t(N))}.\] Now if \(s=t\), then from (Proposition 2.3, [1]), we can identify \(H^{s,s}(Q)\) and \(H^{s,s}(\Sigma)\) by \(H^{s}(Q)\) and \(H^{s}(\Sigma)\), respectively and their associated norms are equivalent. The well-posedness of the forward problem 1 follows from (Proposition \(2.3\), [2]) for the small Dirichlet data \(f \in \mathcal{E}_{\delta}(\Sigma)\) where \(\mathcal{E}_{\delta}(\Sigma)\) is given by \[\begin{align} \label{definition32of32d32delta} \begin{aligned}\mathcal{E}_{\delta}(\Sigma):=\Big\{ f \in H^{2 \kappa+\frac{3}{2}}(\Sigma): \partial_t^\ell f(0,\cdot)=0on\partial \Omega,for\ell \leq 2 \kappa +1,~l\in \mathbb{N}and\lVert f \rVert_{H^{2 \kappa+(3/2)}(\Sigma)} \leq \delta\Big\} \end{aligned} \end{align}\tag{4}\] for \(\delta>0\) sufficiently small. That is, there exists a unique solution \[u \in \mathcal{H}_0^{2 \kappa}(Q):=\{h\in H^{2 \kappa}(Q): \partial_t^\ell f(0,\cdot)=0in\Omegafor\ell < 2 \kappa -1,l\in \mathbb{N} \}\] to the IBVP 1 which satisfies the following estimate \[\lVert u \rVert_{H^{2\kappa}(Q)}\leq C \lVert f \rVert_{H^{2\kappa+(3/2)}(\Sigma)},\] where \(\kappa> \frac{n+1}{2}\) is an integer. Also, we denote \[\begin{align} \widetilde{\mathcal{H}}^{2\kappa+\frac{3}{2}}_0(\Sigma):=\{g \in H^{2 \kappa+\frac{3}{2}}(\Sigma): \partial_t^\ell g(0,\cdot)=0on\partial \Omega,for\ell \leq 2 \kappa +1,~l\in \mathbb{N} \}. \end{align}\] Based on the aforementioned well-posedness of 1 , we define the Dirichlet-to-Neumann (DN) map \(\Lambda_{A,q,B}:\mathcal{E}_{\delta}(\Sigma) \rightarrow H^{2\kappa -\frac{3}{2}}(\Sigma)\), associated to 1 by \[\begin{align} \label{DN32map32for32gover32eqn} \begin{aligned} \Lambda_{A,q,B}(f):=\partial_{\nu} u_{f} \Big|_{ \Sigma},\;\; f\in \mathcal{E}_{\delta}(\Sigma) \end{aligned} \end{align}\tag{5}\] where \(u_f\) is a solution to IBVP 1 and \(\partial_\nu:=\nu\cdot\nabla_x\), normal derivative with respect to the outward unit normal \(\nu\) on \(\partial\Omega\). In the current manuscript, we are interested in the following inverse problem associated with the IBVP 1 .
Problem of interest: To recover the coefficients \(A\), \(q\), and \(r_a\) appearing in 1 from the knowledge of the aforementioned DN map \(\Lambda_{A,q,B}(f)\), measured for all \(f\in \mathcal{E}_{\delta}(\Sigma)\). More precisely, we prove Theorem 1, stated below as a main result of this manuscript.

1.2 Physical significance↩︎

The Schrödinger equation, named after Erwin Schrödinger is a building block of Quantum mechanics. It has a variety of applications, including quantum tunnelling, wave propagation in fibre optics, and more. The semilinear equation 1 considered in this manuscript has numerous applications in science and engineering, for example it is used to study physical processes such as plasma physics (like Langmuir wave dynamics [3]), nonlinear optics (like, optical pulse propagation [4], [5]), water waves (like, gravity waves on water layer [6], [7]), ocean surface waves (like, rogue wave/Killer waves [8]) and others. Also, we refer [9] for the connection between the semilinear equation 1 and mean-field theories. Moreover, if we we choose \(B = r_2 u^2 \overline{u}\) and \(A = \mathbf{0}\), in 1 , then it reduces to a well-known “the Gross–Pitaevskii equation (GPE)”(see for example [10] and references therein), which is employed to describe the dynamics of dilute Bose–Einstein condensates (BEC). Thus, the semilinear equation 1 can be seen as a generalization of GPE.

1.3 Related works↩︎

The inverse problems related to identification of coefficients appearing in Schrödinger equations have been a key object of study due to its appearance in various physical phenomena (see, for example [3], [4], [6]). For instance, Sun in [11] considered an inverse problem associated to unique determination of the time-independent magnetic field and electrical potential in a linear Schrödinger equation from the DN map. Bellassoud and Ferreira [12] investigated a dynamical anisotropic Schrödinger equation with \(A=0\) and derived a Hölder-type stability for the determination of the potential in a manifold setup. In a periodic quantum waveguide, Choulli et al [13] considered a linear Schrödinger equation with \(A=0\) and proved a stable estimate for the scalar potential from the boundary measurement. For the dynamical Schrödinger equation, Aı̈cha [14] established a stability estimate for the recovery of the magnetic field and electrical potential from the DN map. Kian and Soccorsi [15], established the Hölder-type stability in determining the electromagnetic potential \((A,q)\) in the Schrödinger equation and also showed that the unique recovery is achievable only if \(\nabla\cdot A=0\) and due to a natural gauge invariance, recovery of divergence free magnetic potential is optimal. For inverse problems related to uniqueness and stability in context of linear dynamical Schrödinger equation, we refer to [15]–[19] and references therein. In a manifold setup, for more works related to the determination of electrical and magnetic potentials, we refer to [15], [20]–[22].

So far, we have mentioned the prior works related to recovery of coefficients appearing in linear static and dynamical Schrödinger equations. Next we mention, the existing results related to coefficients identification inverse problems for nonlinear PDEs. Motivated by the pioneer work of Isakov [23], which introduced first time a linearization technique to deal with the inverse problem for nonlinear PDEs, we are concerned with coefficient determination inverse problem related to a semilinear dynamical Schrödinger equation. In this approach, we linearize the DN map and use the existing results for linear PDEs to solve the inverse problems related to nonlinear PDEs. In [24], the inverse problem for a semilinear elliptic equation with power-type nonlinearities is analyzed using a higher-order linearization method. In the context of nonlinear Schrödinger equation, Lai et al. [25] investigated a polynomial type nonlinearity in the absence of a magnetic potential. By constructing geometric optics solutions that are based on Gaussian beam quasimodes, they achieved a global uniqueness from partial boundary measurements. Moreover, they establish a logarithmic stability estimate for the recovery of the nonlinear coefficient using the unique continuation principle. For the corresponding problem in a Riemannian manifold setting, Lasses et al. [10] established the unique recovery of the coefficients from the source-to-solution map. Later, for a more general nonlinear Schrödinger equation, Lai et al. in [2] studied an inverse problem related to unique coefficient identification from the partial DN map. In the context of reconstruction of the coefficients, Carstea et al. [26] considered a quasilinear elliptic operator of the divergence form and derived the reconstruction formulae for determining the coefficients from the DN map. Recently, Bhardwaj et al. [27] established the reconstruction formulae for recovering the potential and damping coefficients in a semilinear wave equation with a power-type nonlinearity. In addition, for the nonlinear fourth-order Schrödinger equation, we refer the reader to [28], where the uniqueness of the associated coefficients is established in both Euclidean and Riemannian manifold settings. For more works on inverse problems related to nonlinear PDEs, which are closely related to the inverse problem considered in this manuscript, we refer the reader to [24], [26], [29]–[34], to [35]–[37] and to [27], [38]–[42] for recovering the coefficients appearing in nonlinear elliptic, parabolic and hyperbolic PDEs, respectively.

1.4 Statement of the main result↩︎

As mentioned earlier, we observe that from an inverse problem point of view, substantial progress has been made toward unique identification of the coefficients and stability aspects for both linear and nonlinear PDEs; however, few results are available for reconstructing the coefficients from boundary measurement data. Our main result stated below, provides the point-wise reconstruction formulae for determining the time-dependent coefficients of linear and nonlinear terms, appearing in semilinear dynamical Schrödinger equation 1 from the knowledge DN map measured on \(\Sigma\).

Recently, in [2], authors established a partial data unique determination result related to recovery of the coefficients appearing in nonlinear magnetic Schrödinger equation, in which they proved the uniqueness of inverse problems when the coefficients are known in a small enough open neighborhood of the boundary of the spatial domain. Inspired by [2], we consider the inverse problem related to finding reconstruction formulae for determining the time-dependent coefficients appearing in 1 from the boundary measurements of solution. Our result can be viewed as a continuation of the analysis undertaken in [2]. To the best of our knowledge, our work is the first result in the direction of reconstruction of time-dependent coefficients in semilinear dynamical Schrödinger using the knowledge of DN map \(\Lambda_{A,q,B}\).
The rest of the manuscript is organized as follows. 2 is devoted to proving the main result of this manuscript, where in Subsection 2.1 we provide the reconstruction formulae for the vector potential \(A\) and \(q\) while in Subsection 2.2 we provide the reconstruction formulas for the nonlinear coefficients.

2 Proof of Theorem 1↩︎

This section is devoted to proving the main result of this manuscript. First, we provide the reconstruction formulae for the vector and scalar potentials \(A\) and \(q\), respectively. Inspired by prior works (see for example [23], [24], [27], [38] and references therein) related to coefficients identification inverse problems for nonlinear PDEs, we use the first-order linearization technique to recover the linear coefficients \(A\) and \(q\) and the reconstruction of coefficients \(r_i\) (\(1\leq i\leq m\)) of nonlinear terms, are established via the higher-order linearization techniques along with asymptotic solutions to the linearized equation. We divide the present section into two subsections, in which first one consists of the proof related to reconstruction of the coefficients of linear terms and the second subsection consists of reconstruction of the coefficients of nonlinear terms.

2.1 Reconstruction of coefficients associated with linear terms↩︎

In this subsection, we derive reconstruction formulae for finding vector and scalar potentials \(A\) and \(q\) respectively. As mentioned above, following [23], [24], [27], [38], we use the first-order linearization of solution to the nonlinear problem to recover these coefficients. Given \(f_1, f_2,\cdots,f_m \in \widetilde{\mathcal{H}}^{2\kappa+\frac{3}{2}}_0(\Sigma)\), choose \(\epsilon:=(\epsilon_1,\epsilon_2,\cdots,\epsilon_m)\) with \(\epsilon_i\geq 0\) (\(1\leq i\leq m\)), such that \(\epsilon f:=\sum_{i=1}^m \epsilon_if_i \in \mathcal{E}_{\delta}(\Sigma)\) and denote \(u:=u_{\epsilon f}\), a solution to the following IBVP \[\begin{align} \label{govern32pde32with32boundary32data32ef} \begin{cases} \mathcal{P}_{A,q}u(t,x)=B(t,x,u,\overline{u}),~(t,x)\in Q, \\ u(0,x) = 0,\quad x\in\Omega,\\ u(t,x) = \epsilon f(t,x), \quad (t,x)\in \Sigma. \end{cases} \end{align}\tag{6}\] Also, the Dirichlet-to-Neumann map is given by \[\begin{align} \label{DN32map32for32gover32eqn321} \begin{aligned} \Lambda_{A,q,B}(\epsilon f):=\partial_{\nu} u |_{ \Sigma}=\text{Known}, \;for any \epsilon f\in \mathcal{E}_{\delta}(\Sigma) \end{aligned} \end{align}\tag{7}\] is well-defined, whenever \(u\) is a solution to 6 .

2.1.1 First-order linearization↩︎

The first order linearization is done via differentiating 6 with respect to \(\epsilon_i\) (\(1\leq i\leq m\)) and substituting \(\epsilon=0\). Now after differentiating 6 with respect to \(\epsilon_i\) for \(1\leq i \leq m\), we get \[\begin{align} \begin{cases} \mathrm{i}\partial_t (\partial_{\epsilon_i}u) +\sum_{j=1}^{n}\left(\partial_j+\mathrm{i}A_j\right)^2 \partial_{\epsilon_i}u + q(t,x)\partial_{\epsilon_i}u=\partial_{\epsilon_i}B(t,x,u,\overline{u}),~(t,x)\in Q,\\ \partial_{\epsilon_i}u(0,x) = 0,\quad x\in\Omega,\\\partial_{\epsilon_i}u(t,x) = f_i(t,x), \quad (t,x)\in \Sigma, \end{cases} \end{align}\] where \[\partial_{\epsilon_i}B(t,x,u,\overline{u})=\sum_{a=1}^mr_a\left( a\partial_{\epsilon_i}u^{a-1}(\overline{u})^{m-a}+(m-a)u^a (\overline{u})^{m-a-1}\right).\] Now due the well-posedness of the forward problem, we observe that \(\displaystyle \partial_{\epsilon_i}B(t,x,u,\overline{u})|_{\epsilon=0}=0\). Hence, after evaluating the above equations at \(\epsilon=0\), we obtain \[\begin{align} \begin{cases} \mathrm{i}\partial_t (\partial_{\epsilon_i}u\lvert_{\epsilon=0}) +\sum_{j=1}^{n}\left(\partial_j+\mathrm{i}A_j\right)^2 \partial_{\epsilon_i}u\lvert_{\epsilon=0} + q(t,x)\partial_{\epsilon_i}u\lvert_{\epsilon=0}=0,~(t,x)\in Q,\\ \partial_{\epsilon_i}u\lvert_{\epsilon=0}(0,x) = 0,\quad x\in\Omega,\\\partial_{\epsilon_i}u\lvert_{\epsilon=0}(t,x) = f_i(t,x), \quad (t,x)\in \Sigma. \end{cases} \end{align}\] Now if we denote \(v_i(t,x):=\partial_{\epsilon_i}u\lvert_{\epsilon=0}\) (\(1\leq i\leq m\)), then the above IBVP reduces to \[\begin{align} \label{IBVP32when32k611} \begin{cases} \mathrm{i}\partial_t v_i(t,x) +\sum_{j=1}^{n}\left(\partial_j+\mathrm{i}A_j\right)^2 v_i(t,x) + q(t,x)v_i(t,x)=0,~(t,x)\in Q,\\ v_i(0,x) = 0,\quad x\in\Omega,\\v_i(t,x) = f_i(t,x), \quad (t,x)\in \Sigma. \end{cases} \end{align}\tag{8}\] From (Lemma 4.1, [2]), we have that \(v_i \in H^{2\kappa}(Q)\) for \(1\leq i\leq m\). The first order linearization of the DN map in 7 is given by \[\partial_{\epsilon_i}\Lambda_{A,q,B}(\epsilon f)\lvert_{\epsilon=0}= \left(\partial_{\epsilon_i} \partial_{\nu} u\lvert_{\Sigma}\right)\lvert_{\epsilon=0}= \partial_{\nu}v_i\lvert_{\Sigma},\;1\leq i\leq m.\] Thus, it follows from 7 that \[\begin{align} \label{DN32map32when32k611} \partial_{\nu}v_i\lvert_{\Sigma}=\text{Known},\;\;for any f_i\in \widetilde{\mathcal{H}}^{2\kappa+\frac{3}{2}}_0(\Sigma)\;and\;1\leq i\leq m. \end{align}\tag{9}\] Now since the Dirichlet data \(f_i\) (\(1\leq i\leq m\)) in 8 are part of measurement data, therefore we denote \(v_i:=v\) and \(f_i:=\mathfrak{f}\), in 8 and end up with the following IBVP for \(v\) \[\begin{align} \label{IBVP32v} \begin{cases} i\partial_t v(t,x) +\sum_{j=1}^{n}\left(\partial_j+\mathrm{i}A_j(t,x)\right)^2 v(t,x) + q(t,x)v(t,x)=0,~(t,x)\in Q, \\ v(0,x) = 0,\quad x\in\Omega,\\ v(t,x) = \mathfrak{f}, \quad (t,x)\in \Sigma, \end{cases} \end{align}\tag{10}\] and from first-order linearization of DN map \(\Lambda_{A,q,B}\), we have \[\begin{align} \label{DN32map32use32for32reconstructing32a32and32q} \Lambda_{A,q}(\mathfrak{f}):= \partial_{\nu} v \lvert_{\Sigma}=\text{Known}, \;for any \mathfrak{f}\in \widetilde{\mathcal{H}}^{2\kappa+\frac{3}{2}}_0(\Sigma). \end{align}\tag{11}\] The equation 11 represent the DN map for the linearized IBVP given by 10 .

2.1.2 Reconstruction of \(A\)↩︎

To reconstruct \(A\), we multiply 10 by \(\overline{w}(t,x)\), where \(w(t,x)\) satisfies the following backward problem \[\begin{align} \label{adjoint32equation321} \mathrm{i}\partial_t w(t,x)+ \Delta w(t,x)=0, \;(t,x)\in Q \;and\; w(T,x)=0, \;x\in\Omega \end{align}\tag{12}\] and integrate over \(Q\), to yield \[\begin{align} \int_{Q} \left(\mathrm{i}\partial_{t} v +\Delta v +2\mathrm{i}A \cdot \nabla v-\lvert A\rvert^2 v + q v\right)(t,x) \overline{w}(t,x) \;dxdt=0. \end{align}\] Next, the use of integration-by-parts formula leads to \[\begin{align} \int_{\Omega} \mathrm{i}[v(T,x)\overline{w(T,x)}- v(0,x)\overline{w(0,x)}]\;dx-\int_{Q} \mathrm{i}v\partial_t \overline{w} \;dx dt +\int_{\Sigma}\overline{w}\partial_{\nu} v\; dS_xdt-\int_{\Sigma}v\partial_{\nu} \overline{w}\; dS_xdt \\+ \int_{Q} v\Delta \overline{w} \;dxdt+\int_{\Sigma} 2 \mathrm{i}(\nu \cdot A ) v \overline{w} ~dS_x dt -2\mathrm{i}\int_{Q}(A\cdot \nabla \overline{w})v\;dx dt + \int_{Q} q_1 v \overline{w}\;dx dt=0 \end{align}\] where \(q_1:=q-\lvert A\rvert^2\). Using DN map given by 11 along with equations 10 and 12 , the above expression boils down to the following integral identity \[\begin{align} \label{integral32identity32in32case32of32linear} -2\mathrm{i}\int_{Q}(A\cdot \nabla \overline{w})v \;dx dt+ \int_{Q} q_1 v \overline{w}\;dx dt =\text{Known}, \end{align}\tag{13}\] for all \(v\) and \(w\) solutions to 10 and 12 , respectively. Next, to reconstruct the coefficient \(A\), we use Geometric optics (GO) solutions from (Section 3, [15]). For \(\varrho>0\), let \(\Xi \in C_c^{\infty}(\mathbb{R},[0,1])\) with \(\Xi=1\), in \([2\varrho,T-2\varrho]\) and \(\mathop{\mathrm{supp}}(\Xi)\subset (\varrho,T-\varrho)\) be smooth cut-off function satisfying \[\lVert \Xi \rVert_{W^{l,\infty}(\mathbb{R})} \lesssim \varrho^{-l}\] for each \(l\in \mathbb{N}\). Now for a fixed vector \(\omega\in \mathbb{S}^{n-1}\) and \(\lambda>0\) a large parameter, we define \[\begin{align} \label{definition32of32u032and32u00} \begin{aligned} &\phi(t,x):=\lambda(x \cdot \omega-\lambda \lvert \omega\rvert^2t),\;(t,x)\in Q,\\&T_v(t,x):=\Xi(t) \dfrac{\xi}{\lvert \xi\rvert}\cdot\nabla\left( e^{-\mathrm{i}(\tau t+ x \cdot \xi)} e^{-\mathrm{i}\int_{\mathbb{R}} \omega\cdot A(t,x+s\omega)~ds}\right)e^{\mathrm{i}\int_{0}^{\infty}\omega\cdot A(t,x+s\omega)ds}, \;(\tau,\xi) \in \mathbb{R}\times \omega^{\perp} \\& T_w(t,x):=\Xi(t) \end{aligned} \end{align}\tag{14}\] where \(\displaystyle \omega^{\perp}:=\left\{x\in\mathbb{R}^n:\;x\cdot \omega=0\right\}\). Now following (Section 3, [15]), we choose the GO solutions for \(v\) and \(w\) of 10 and 12 respectively, of the following form \[\begin{align} \label{vi32cgo321} v(t,x)=e^{\mathrm{i}\phi(t,x)} T_v(t,x)+R_v(t,x) \text{ and } w(t,x)=e^{\mathrm{i}\phi(t,x)}T_w(t,x)+R_w(t,x) \end{align}\tag{15}\] where \(v(0,\cdot)=w(T,\cdot)=0\), in \(\Omega\). Also, the correction terms, \(R_v\) and \(R_w\) satisfy the following estimates \[\begin{align} \label{reminder32terms32bound321} \lambda\lVert R_v\rVert_{L^2(Q)}+\lVert \nabla R_v\rVert_{L^2(Q)}\leq C\quad \text{ and }\quad \lambda\lVert R_w\rVert_{L^2(Q)}+\lVert \nabla R_w\rVert_{L^2(Q)}\leq C. \end{align}\tag{16}\] Substitute 15 into the integral identity 13 , to arrive at \[\begin{align} -2 \lambda \int_{Q}(\omega\cdot A)T_v \overline{T_w}~dx dt -2 \mathrm{i}\int_{Q} e^{\mathrm{i}\phi}T_v(A\cdot \nabla\overline{R_w}) ~ dxdt-2 \mathrm{i}\int_{Q} R_v(A\cdot \nabla \overline{R_w})~dxdt \\-2 \lambda \int_{Q}(\omega\cdot A)\overline{T_w}\left( e^{-\mathrm{i}\phi} R_v \right)~dxdt+\int_Q q_1 T_v\overline{T_w}(t,x)~dxdt\\+\int_Q q_1\left(e^{\mathrm{i}\phi} \overline{R_w}T_v+e^{-\mathrm{i}\phi} \overline{R_v}\overline{T_w}+R_v\overline{R_w} \right)~ dxdt=\text{ Known} \end{align}\] for any choice of \(T_v\) and \(T_w\) given by 14 . Multiply by \(\lambda^{-1}\) in above expression and letting \(\lambda \rightarrow\infty\) along with using equations 14 and 16 , we obtain \[\begin{align} \int_{Q} (\omega \cdot A)\Xi^2(t)\dfrac{\xi}{\lvert \xi\rvert}\cdot\nabla\left( e^{-\mathrm{i}(\tau t+ x \cdot \xi)} e^{-\mathrm{i}\int_{\mathbb{R}} \omega\cdot A(t,x+s\omega)~ds}\right)e^{\mathrm{i}\int_{0}^{\infty}\omega\cdot A(t,x+s\omega)ds}~dxdt= \text{ Known} \end{align}\] for any \(\Xi \in C_c^{\infty}(\mathbb{R},[0,1])\), \(\xi\in\omega^{\perp}\) and \(\tau\in \mathbb{R}\). Since \(\Xi \in C_c^{\infty}(\mathbb{R},[0,1])\) and \(A \in C_c^{\infty}(Q)\), therefore the above integral can be written as \[\begin{align} \int_{\mathbb{R}^{1+n}} (\omega \cdot A)\Xi^2(t)\dfrac{\xi}{\lvert \xi\rvert}\cdot\nabla\left( e^{-\mathrm{i}(\tau t+ x \cdot \xi)} e^{-\mathrm{i}\int_{\mathbb{R}} \omega\cdot A(t,x+s\omega)~ds}\right)e^{\mathrm{i}\int_{0}^{\infty}\omega\cdot A(t,x+s\omega)ds}~dxdt= \text{ Known} \end{align}\] for any \(\Xi \in C_c^{\infty}(\mathbb{R})\), \(\xi\in\omega^{\perp}\) and \(\tau\in \mathbb{R}\). The use of decomposition \(\mathbb{R}^n:= \mathbb{R}\omega \oplus \omega^{\perp}\) along with the fact that \(\Xi \in C_c^{\infty}(\mathbb{R},[0,1])\) and \(\tau\in \mathbb{R}\) are arbitrary, in the above integral, yields \[\begin{align} \int_{\omega^{\perp}} \int_{\mathbb{R}} \left(\omega\cdot A(t,l+\sigma \omega)\right)\dfrac{\xi}{\lvert \xi\rvert} \cdot \nabla(e^{-\mathrm{i} \xi\cdot l} e^{-\mathrm{i}\int_{\mathbb{R}} \omega\cdot A(t,l+s\omega)~ds} )e^{ \mathrm{i}\int_{0}^{\infty} \omega\cdot A(t,l+\sigma \omega+s\omega) ds})d\sigma dl = \text{Known} \end{align}\] for any \(t\in [0,T]\) and \(\xi\in \omega^{\perp}\), where \(dl\) denotes the Lebesgue measure on \(\omega^{\perp}\). Now using the change of variable \(\sigma+s:= r\), in the above equation, we obtain that \[\begin{align} \label{on32the32left32side} \int_{\omega^{\perp}} \int_{\mathbb{R}} e^{ \mathrm{i}\int_{\sigma}^{\infty} \omega\cdot A(t,l+r \omega) dr}\omega\cdot A(t,l+\sigma \omega)\dfrac{\xi}{\lvert \xi\rvert} \cdot \nabla e^{-\mathrm{i}( \xi\cdot l+\mathrm{i}\int_{\mathbb{R}} \omega\cdot A(t,l+s\omega)~ds}d\sigma dl= \text{Known} \end{align}\tag{17}\] for any \(t\in [0,T]\) and \(\xi\in \omega^{\perp}\). Now if we denote \(\Psi:=\exp{\bigg( \mathrm{i}\int_{\sigma}^{\infty} \omega\cdot A(t,l+r\omega) dr}\bigg)\), then \(\partial_{\sigma}\Psi= -\mathrm{i}\omega\cdot A(t,l+\sigma \omega) \Psi\). Integrating this over \(\mathbb{R}\) w.r.t. \(\sigma\), yields that \[\begin{align} -\mathrm{i}\int_{\mathbb{R}} \omega\cdot A(t,l+\sigma \omega) \Psi d\sigma=\int_{\mathbb{R}} \partial_{\sigma}\Psi \;d\sigma=1- \exp{\bigg( \mathrm{i}\int_{\mathbb{R}} \omega\cdot A(t,l+r\omega) dr}\bigg). \end{align}\] Substitute the above expression in 17 , we get \[\begin{align} \label{24612} \mathrm{i} \int_{\omega^{\perp}} \left(1- e^{\mathrm{i}\int_{\mathbb{R}} \omega\cdot A(t,l+r\omega) dr}\right) \dfrac{\xi}{\lvert \xi\rvert} \cdot \nabla\left(e^{-\mathrm{i} \xi\cdot l} e^{-\mathrm{i}\int_{\mathbb{R}} \omega\cdot A(t,l+s\omega)~ds} \right)dl = \text{Known} \end{align}\tag{18}\] for any \(t\in [0,T]\) and \(\xi\in \omega^{\perp}\). Now since \(A\) is compactly supported in \(\Omega\) for each \(t\in [0,T]\) and \(\Omega\subset\mathbb{R}^n\) is bounded, therefore there exist \(R>0\) such that \(\mathop{\mathrm{supp}}A(t,\cdot)\subset\Omega\subset B(0,R)\) and \[\begin{align} \label{support32condition} \mathop{\mathrm{supp}}\left(1- e^{ \mathrm{i}\int_{\mathbb{R}}\omega\cdot A(t,l+r\omega) dr}\right) \subset \omega^{\perp}\cap B(0,R). \end{align}\tag{19}\] The use of decomposition \(\nabla g:=\nabla_{\perp} g+\omega(\omega\cdot \nabla g)\) and the integration by parts formula along with support condition 19 in 18 leads to \[\begin{align} \text{Known}&= \mathrm{i} \int_{\omega^{\perp}} \left(1- e^{\mathrm{i}\int_{\mathbb{R}} \omega\cdot A(t,l+r\omega) dr}\right) \dfrac{\xi}{\lvert \xi\rvert} \cdot \nabla\left(e^{-\mathrm{i} \xi\cdot l} e^{-\mathrm{i}\int_{\mathbb{R}} \omega\cdot A(t,l+s\omega)~ds} \right)dl \\&=\mathrm{i} \int_{\omega^{\perp}} \left(1- e^{\mathrm{i}\int_{\mathbb{R}} \omega\cdot A(t,l+r\omega) dr}\right) \dfrac{\xi}{\lvert \xi\rvert} \cdot \nabla_{\perp}\left(e^{-\mathrm{i} \xi\cdot l} e^{-\mathrm{i}\int_{\mathbb{R}} \omega\cdot A(t,l+s\omega)~ds} \right)~dl\\& =\mathrm{i} \int_{\omega^{\perp}\cap B(0,R)} \left(1- e^{\mathrm{i}\int_{\mathbb{R}} \omega\cdot A(t,l+r\omega) dr}\right) \dfrac{1}{\lvert \xi\rvert} \nabla_{\perp}\cdot \left(\xi e^{-\mathrm{i} \xi\cdot l} e^{-\mathrm{i}\int_{\mathbb{R}} \omega\cdot A(t,l+s\omega)~ds} \right)~dl \\&=-\mathrm{i} \int_{\omega^{\perp}} \left(e^{-\mathrm{i} \xi\cdot l} e^{-\mathrm{i}\int_{\mathbb{R}} \omega\cdot A(t,l+s\omega)~ds} \right)\dfrac{\xi}{\lvert \xi\rvert} \cdot \nabla_{\perp} \left(1- e^{\mathrm{i}\int_{\mathbb{R}} \omega\cdot A(t,l+r\omega) dr}\right)~dl \\& =-\int_{\omega^{\perp}}e^{-\mathrm{i} \xi\cdot l}\dfrac{\xi}{\lvert \xi\rvert} \cdot \nabla_{\perp}\left(\int_{\mathbb{R}} \omega\cdot A(t,l+s\omega)~ds\right)~dl\\&=\int_{\omega^{\perp}}\int_{\mathbb{R}}e^{-\mathrm{i}\xi\cdot (l+s\omega)}\dfrac{\xi}{\lvert \xi\rvert}\cdot \nabla_{\perp}\left( \omega\cdot A(t,l+s\omega)\right)~ds~dl\\&=\int_{\omega^{\perp}}\int_{\mathbb{R}}e^{-\mathrm{i}\xi\cdot (l+s\omega)}\dfrac{\xi}{\lvert \xi\rvert}\cdot \nabla\left( \omega\cdot A(t,l+s\omega)\right)~ds~dl= \int_{\mathbb{R}^n}e^{-\mathrm{i}\xi\cdot x}\dfrac{\xi}{\lvert \xi\rvert}\cdot \nabla\left( \omega\cdot A(t,x)\right)~dx \end{align}\] for any \(t\in [0,T]\) and \(\xi\in \omega^{\perp}\). Using Fourier transform of the space variable, the above expression reduces to \[\begin{align} \label{omega32dot32A32known} \text{Known}=\int_{\mathbb{R}^{n}}e^{-\mathrm{i}\xi\cdot x}\dfrac{\xi}{\lvert \xi\rvert} \cdot \nabla \left( \omega\cdot A(t,x)\right)dx =\mathrm{i}\lvert \xi\rvert \omega\cdot \widehat{A}(t,\xi) \end{align}\tag{20}\] for any \(t\in [0,T]\) and \(\xi\in \omega^{\perp}\). Also, using the Fourier transform along with the given hypothesis \(\nabla \cdot A=0\), we have \(\xi \cdot \widehat{A}(t,\xi)=0\), for all \(\xi \in \mathbb{R}^n\setminus\{0\}\). Also for any \(\xi\in \mathbb{R}^n\), we choose \(\omega_i\in \mathbb{S}^{n-1}\) for \(1\leq i\leq n-1\) such that \(\left\{\omega_1,\cdots,\omega_{n-1},\frac{\xi}{\lvert \xi\rvert}\right\}\) form an orthonormal basis for \(\mathbb{R}^n\). Using these basis elements, we can write \(\widehat{A}(t,\xi)=\sum_{i=1}^{n-1}a_i \omega_i+a_n\frac{\xi}{\lvert \xi\rvert}\) where \(a_i:=\langle \widehat{A}(t,\xi),\omega_i\rangle\) for \(1\leq i\leq n\) and \(a_n=\langle \widehat{A}(t,\xi),\frac{\xi}{\lvert \xi\rvert}\rangle\). Since \(\xi \cdot \widehat{A}(t,\xi)=0\), invoking 20 , we have \[\widehat{A}(t,\xi)=\sum_{i=1}^{n-1} \langle \widehat{A}(t,\xi),\omega_i\rangle \omega_i=\text{ Known}, \;\text{for any t\in [0,T] and \xi \in \mathbb{R}^n.}\] Now, by after utilizing the Fourier inversion formula, we get \(A(t,x)=\text{Known}, \;\text{for all }(t,x) \in Q.\)0◻

2.1.3 Reconstruction of \(q\)↩︎

To reconstruct \(q\), we multiply equation 10 by \(\overline{w}(t,x)\), where \(w(t,x)\) satisfies the following backward problem \[\begin{align} \label{modified32adjoint32equation322} \begin{aligned} \mathrm{i}\partial_t w + \Delta w+2\mathrm{i} A\cdot \nabla w=0, \;(t,x)\in Q \;and\; w(T,x)=0, \;x\in\Omega \end{aligned} \end{align}\tag{21}\] and integrating over \(Q\), we get \[\begin{align} \int_{Q} \left(\mathrm{i}\partial_{t} v(t,x) +\Delta v(t,x) +2\mathrm{i}A \cdot \nabla v(t,x) +q_1(t,x) v(t,x)\right) \overline{w(t,x)} \;dxdt=0 \end{align}\] where \(q_1=q-\lvert A \rvert^2\). Next, the use of integration-by-parts formula in the above equation, leads to \[\begin{align} \int_{\Omega} \mathrm{i}\left[v(T,x)\overline{w}(T,x)- v(0,x)\overline{w}(0,x)\right]\;dx-\int_{Q} \mathrm{i}v\partial_t \overline{w} \;dx dt +\int_{\Sigma}\overline{w}\partial_{\nu} v\; dS_xdt-\int_{\Sigma}v\partial_{\nu} \overline{w}\; dS_xdt \\+ \int_{Q} v\Delta \overline{w} \;dxdt+\int_{\Sigma} 2 \mathrm{i}(\nu \cdot A ) v \overline{w} ~dS_x dt -2\mathrm{i}\int_{Q}(A\cdot \nabla \overline{w})v\;dx dt + \int_{Q} q_1 v \overline{w}\;dx dt=0. \end{align}\] Using DN map 11 along with equations 10 and 21 , the above expression simplifies to \[\begin{align} \label{q95132integral} \int_{Q}q_1(t,x) v(t,x) \overline{w(t,x)}\;dxdt= \text{Known} \end{align}\tag{22}\] for any \(v\) and \(w\) solutions to 10 and 21 , respectively. Now as before, we choose the GO solutions for \(v\) and \(w\), solutions to equations 10 and 21 respectively and taking the following form \[\begin{align} v(t,x)=e^{\mathrm{i}\phi(t,x)} \Xi(t)e^{-\mathrm{i}(\tau t+ x\cdot \xi)}e^{\mathrm{i}\int_{0}^{\infty}\omega\cdot A(t,x+s\omega)ds}+R_v(t,x) \end{align}\] and \[\begin{align} w(t,x)=e^{\mathrm{i}\phi(t,x)}\Xi(t)e^{\mathrm{i}\int_{0}^{\infty}\omega\cdot A(t,x+s\omega)ds}+R_w(t,x) \end{align}\] where \(v(0,\cdot)=w(T,\cdot)=0\) in \(\Omega\). Also, the correction terms, \(R_v\) and \(R_w\) satisfy the following estimates \[\begin{align} \label{estimate32on32remainder32terms} \lambda\lVert R_v\rVert_{L^2(Q)}+\lVert \nabla R_v\rVert_{L^2(Q)}\leq C\quad \text{ and }\quad \lambda\lVert R_w\rVert_{L^2(Q)}+\lVert \nabla R_w\rVert_{L^2(Q)}\leq C. \end{align}\tag{23}\] Using aforementioned \(v\) and \(w\) in 22 , we get \[\begin{align} \int_{Q} q_1 \Xi^2(t)& e^{-\mathrm{i}(\tau t + x \cdot \xi)}\;dx dt+\int_{Q} q_1 \Xi(t)R_v e^{-\mathrm{i}\phi}e^{-\mathrm{i}\int_{0}^{\infty}\omega\cdot A(t,x+s\omega)ds}\;dx dt\\&+\int_{Q} q_1 \Xi(t) \overline{R_w}e^{\mathrm{i}\phi}e^{-\mathrm{i}(\tau t + x \cdot \xi)}e^{\mathrm{i}\int_{0}^{\infty}\omega\cdot A(t,x+s\omega)ds}\;dx dt+ \int_{Q} q_1 R_v\overline{R_w}\;dx dt=\text{ Known}. \end{align}\] After taking \(\lambda \rightarrow\infty\) along with using equation 23 , we get \[\begin{align} \int_{Q}\Xi^2(t) q_1(t,x) e^{-\mathrm{i}(\tau t + x \cdot \xi)}\;dx dt= \text{ Known}, \;\text{for any \Xi \in C_c^{\infty}(\mathbb{R},[0,1]), \xi\in\omega^{\perp} and \tau\in \mathbb{R}}. \end{align}\] Next utilize the fact that \(\Xi \in C_c^{\infty}(\mathbb{R},[0,1])\) is arbitrary and \(q_1 \in C_c^{\infty}(Q)\), in the above integral, to obtain \[\int_{\mathbb{R}^{n}} q_1(t,x) e^{-\mathrm{i} x \cdot \xi}\;dx=\text{Known},\;\text{for any t\in [0,T] and \xi\in \omega^{\perp}}.\] Now using the fact that \(\omega\in \mathbb{S}^{n-1}\) is arbitrary and the fact that \(\displaystyle \int_{\mathbb{R}^{n}} q_1(t,x) e^{-\mathrm{i} x \cdot \xi}\;dx\) is known for any \(\xi\in \omega^{\perp}\) and \(t\in [0,T]\), we obtain that\[\int_{\mathbb{R}^{n}} q_1(t,x) e^{-\mathrm{i} x \cdot \xi}\;dx=\text{known}, \;\text{for any \xi\in \mathbb{R}^n and t\in [0,T]}.\] Finally, using the Fourier inversion formula, we get \[q_1(t,x)=\text{Known}, \; \text{ for all }(t,x) \in Q.\] Since \(A\) is already been reconstructed therefore we conclude that \(q(t,x)\) is known for all \((t,x)\in Q\).0◻

2.2 Reconstruction of nonlinearity coefficients↩︎

This subsection is devoted to deriving the reconstruction formulae for \(r_i\) \((1\leq i\leq m)\). As mentioned earlier, following [24], [27], [29], [34], [38], we use the higher-order linearization of solution to the nonlinear equation for establishing the reconstruction of these coefficients.

2.2.1 Higher order linearization↩︎

The \(k\)-th (\(2\leq k\leq m\)) order linearization is done by applying \(\partial_{\epsilon_{j_1}\cdots\epsilon_{j_k}}^k\) (\(j_1,\cdots,j_k\in\{1,2,\cdots,m\}\)) to 6 and substituting \(\epsilon=0\). Hence, we start by applying the partial differential operator \(\partial_{\epsilon_1\cdots \epsilon_k}^k\) \((2\leq k\leq m)\) to 6 and obtain that \[\begin{align} \label{upto32m32deivative32of32gover32eqn} \begin{aligned} \begin{cases} \mathrm{i}\partial_t\left( \partial^k_{\epsilon_1\cdots \epsilon_k}u\right) +\sum_{j=1}^{n}\left(\partial_j+\mathrm{i}A_j\right)^2 \left( \partial^k_{\epsilon_1\cdots \epsilon_k}u\right) + q\left( \partial^k_{\epsilon_1\cdots \epsilon_k}u\right)=\partial^k_{\epsilon_1\cdots \epsilon_k} B(t,x,u,\overline{u}),~(t,x)\in Q,\\ \partial^k_{\epsilon_1\cdots \epsilon_k}u(0,x) = 0,\quad x\in\Omega,\\ \partial^k_{\epsilon_1\cdots \epsilon_k}u(t,x) = 0, \quad (t,x)\in \Sigma. \end{cases} \end{aligned} \end{align}\tag{24}\] Now using well-posedness of the IBVP 6 , we get that \(\partial^k_{\epsilon_1\cdots \epsilon_k}\rvert_{\epsilon=0} \left(B(t,x,u,\overline{u})\right)=0,\) for any \(2\leq k\leq m-1\). Thus, after evaluating 24 at \(\epsilon=0\), along with denoting by \(z_k:=\partial^k_{\epsilon_1\cdots \epsilon_k}u\lvert_{\epsilon=0}\), for \(2\leq k\leq m-1\), we obtain the following IBVP for \(z_k\) (\(2\leq k\leq m-1\)) \[\begin{align} \label{intermediate32pde} \begin{cases} \mathrm{i}\partial_t z_k(t,x) +\sum_{j=1}^{n}\left(\partial_j+\mathrm{i}A_j(t,x)\right)^2 z_k(t,x) + q(t,x)z_k(t,x)=0,~(t,x)\in Q,\\ z_k(0,x) = 0,\quad x\in\Omega,\\z_k(t,x) = 0, \quad (t,x)\in \Sigma. \end{cases} \end{align}\tag{25}\] The well-posedness of the IBVP 25 implies that the \(z_k\equiv 0\), in \(Q\) for any \(2\leq k\leq m-1\). Finally, applying \(\displaystyle \partial^{m}_{ \epsilon_1\epsilon_2\cdots\epsilon_m}\) to 6 and evaluating at \(\epsilon = 0\), we obtain \[\begin{align} \label{IBVP32when32k61m} \begin{cases} \mathrm{i}\partial_t X +\sum_{j=1}^{n}\left(\partial_j+\mathrm{i}A_j\right)^2 X + qX= \partial^m_{\epsilon_1\cdots \epsilon_m}\rvert_{\epsilon=0} \left(B(t,x,u,\overline{u})\right):=B_1(t,x),~(t,x)\in Q, \\ X(0,x) = 0,\quad x\in\Omega,\\ X(t,x) = 0, \quad (t,x)\in \Sigma, \end{cases} \end{align}\tag{26}\] where \(X:=\partial^m_{\epsilon_1\cdots \epsilon_m}u\lvert_{\epsilon=0}\) and the function \(B_1(t,x)\) is given by \[\begin{align} B_1(t,x):&=\partial^m_{\epsilon_1\cdots \epsilon_m}\rvert_{\epsilon=0} \left(B(t,x,u,\overline{u})\right)=\partial^m_{\epsilon_1\cdots \epsilon_m}\rvert_{\epsilon=0}\left[\sum_{a=1}^{m}r_a(t,x)u^{a}(t,x)(\overline{u})^{m-a}(t,x)\right]\\&=m! r_m\prod_{k=1}^m v_k+\sum_{a=1}^{m-1}a!(m-a)!r_a\sum_{\substack{I\subset\{1,2,\cdots,m\} \\ \lvert I \rvert =a}}\left(\prod_{j \in I}v_j\right)\left(\prod_{j \notin I}\overline{v_j}\right) \end{align}\] where \(v_k\) for \(1\leq k\leq m\) are solutions to IBVP 8 . The \(m\)-th order linearization of the DN map 7 is given by \[\partial^m_{\epsilon_1\cdots \epsilon_m}\Lambda_{A,q,B}(\epsilon f)\lvert_{\epsilon=0}= \left(\partial^m_{\epsilon_1\cdots \epsilon_m} \partial_{\nu} u\lvert_{\Sigma}\right)\lvert_{\epsilon=0}= \partial_{\nu}X\lvert_{\Sigma} \; for any\;\epsilon f\in \mathcal{E}_{\delta}(\Sigma).\] Thus, it follows from 7 that \[\begin{align} \label{DN32map32when32k61m} \partial_{\nu}X\lvert_{\Sigma}=\text{Known}. \end{align}\tag{27}\] With this information, we are now in a position to establish the reconstruction formulae for the nonlinearity coefficients \(r_a\) for \(1\leq a\leq m\). We split the proof for reconstruction of these coefficients into two parts: the first focuses on reconstructing \(r_m\), while the second addresses the reconstruction of \(r_a\) for \(1\leq a\leq m-1\).

2.2.2 Reconstruction of \(r_m\)↩︎

To reconstruct \(r_m\), we multiply equation 26 by \(\overline{w}(t,x)\), where \(w(t,x)\) satisfies the following backward problem \[\label{adjoint32equation32v32power32m32plus32one} \mathcal{P}_{A,q} w(t,x)=0,~(t,x)\in Q, \;and\;w(T,x)=0, \;x \in \Omega\tag{28}\] and integrating over \(Q\), we have \[\int_{Q} \bigg(\mathrm{i}\partial_t X +\Delta X+2\mathrm{i}A\cdot \nabla X-\lvert A\rvert^2 X + qX -B_1 \bigg)(t,x) \overline{w}(t,x)~ dx dt =0.\] After using the integration-by-parts formula in above equation, we obtain \[\begin{align} &\int_{\Omega} \mathrm{i}\left[X(T,x)\overline{w}(T,x)- X(0,x)\overline{w}(0,x)\right]\;dx-\int_{Q} \mathrm{i}X(t,x)\partial_t \overline{w}(t,x) \;dx dt\\ & \quad +\int_{\Sigma}\left[\overline{w}(t,x)\partial_{\nu} X(t,x)-X(t,x)\partial_{\nu} \overline{w}(t,x)\right] dS_xdt + \int_{Q} X(t,x)\Delta \overline{w}(t,x) \;dxdt \\ & \quad +\int_{\Sigma} 2 \mathrm{i}(\nu \cdot A )(t,x) X(t,x) \overline{w}(t,x) ~dS_x dt -2\mathrm{i}\int_{Q}(A\cdot \nabla \overline{w})(t,x)X(t,x)\;dx dt \\ & \quad - \int_{Q}\left[ \lvert A\rvert^2 +q\right](t,x) X(t,x)\overline{w}(t,x)\;dx dt -\int_{Q} B_1(t,x) \overline{w}(t,x)~ dx dt=0. \end{align}\] Now utilizing equations 27 , 28 , and 26 , the foregoing expression simplifies to the following integral identity. \[\begin{align} \label{integral32identity} \begin{aligned} &\int_{Q} B_1(t,x)\overline{w}(t,x)\;dxdt=m!\int_{Q}r_m(t,x)\prod_{k=1}^m v_k(t,x)dxdt\\ & \quad + \sum_{a=1}^{m-1}a!(m-a)!\int_{Q}r_a(t,x)\sum_{\substack{I\subset\{1,2,\cdots,m\}\\ \lvert I \rvert =a}}\left(\prod_{j \in I}v_j\right)\left(\prod_{j \notin I}\overline{v_j}\right) \overline{w}(t,x) \;dxdt=\text{known} \end{aligned} \end{align}\tag{29}\] for any \(v_k\;(1\leq k\leq m)\) and \(w\) solutions to 8 and 28 , respectively. From the above integral identity, we reconstruct the coefficients \(r_a\) for \(1\leq a\leq m\). To this end, we use the GO solutions with concentrated amplitudes and we refer to [2] for construction of such solutions. We also refer the reader to [10], [15], [27], in which similar approaches have been utilized. We start with fixing \(\omega_l \in \mathbb{R}^n\setminus\{\boldsymbol{0}\}\) and denote \(\phi_l:=\lambda(x \cdot \omega_l-\lambda \lvert \omega_l\rvert^2t),\) for \(0\leq l\leq m\), where \(\lambda>0\) is a large parameter. Next for each \(\omega_{l}\) (\(1\leq l\leq m\)), we choose \(\xi_1^l, \cdots\xi_{n-1}^l\in \mathbb{R}^n\) such that \(\left\{\dfrac{\omega_l}{\lvert{\omega_l}\rvert},\xi_1^l, \cdots\xi_{n-1}^l\right\}\) forms an orthonormal basis for \(\mathbb{R}^n\). Also choose \(\mathcal{T} \in C^{\infty}_c(0,T)\) and \(\chi \in C^{\infty}_c(\mathbb{R},[0,1])\), the smooth cutoff functions such that \(\chi(x)=1\), for \(\lvert x\rvert\leq\frac{1}{2}\) and \(\mathop{\mathrm{supp}}(\chi) \subset B(0,1)\). Now for a fixed \(x_0\in \Omega\) and \(h>0\) sufficiently small, we define \[\begin{align} \label{concentrated32amplitude} \begin{aligned} U_0^l(t,x):=\mathcal{T}(t)\prod_{j=1}^{n-1} \chi\left( \dfrac{(x-x_0)\cdot \xi_j^l}{h} \right) \exp\left({\mathrm{i} \int_{0}^{\infty} \omega_l\cdot A(t,x+r \omega_l) ~dr}\right),\;(t,x)\in Q \end{aligned} \end{align}\tag{30}\] for \(0\leq l\leq m\). Also, for fixed \(\omega\in \mathbb{S}^{n-1}\) and \(x:=y+(x\cdot\omega)\omega\in \Omega\), we define \[\begin{align} \label{concentrated32amplitude321} U_j^l(t,x)&:=\mathrm{i} \int_{-x\cdot\omega}^0 \exp\left(-\mathrm{i}\int_{r}^0\omega\cdot A(t,x+r_1\omega)~dr_1\right)\mathcal{P}_{A,q} U_{j-1}^l (t,x+r\omega)~ dr \;(1\leq j\leq N) \end{align}\tag{31}\] with initial condition \(U_j^l(t,x)\big\lvert_{x\cdot\omega=0}=0\) for all \(t\in (0,T)\). In the expression above, the linear operator \(\mathcal{P}_{A,q}\) is as defined in 3 . Now following (Proposition \(3.2\), [2]), we choose the GO solutions for \(v_k\) and \(w\) solving 8 and 28 respectively, and taking the following form \[\begin{align} \label{vi32cgo} v_k(t,x)=e^{\mathrm{i}\phi_k(t,x)}\left( U_0^k(t,x)+\sum_{j=1}^{N} \lambda^{-j}U_j^k (t,x)\right)+R_{v_k}(t,x),\;\text{for 1\leq k\leq m} \end{align}\tag{32}\] and \[\begin{align} \label{w32cgo} w(t,x)=e^{\mathrm{i}\phi_0(t,x)}\left( U_0^0(t,x)+\sum_{j=1}^{N} \lambda^{-j}U_j^0 (t,x)\right)+R_{w}(t,x) \end{align}\tag{33}\] where \(v_k(0,\cdot)=0,\) for \(1\leq k \leq m\) and \(w(T,\cdot)=0\), in \(\Omega\). For any natural number \(s\) and \(0\leq l\leq m\), we have \[\begin{align} \label{reminder32terms32bound} \begin{aligned} \lVert U_j^l \rVert_{H^s(Q)}&=\mathcal{O}(1), \quad 0\leq j\leq N , \\ \lVert R_{v_k}\rVert_{H^s(Q)}=\mathcal{O}(\lambda^{-N+2s}) \;&\text{ and } \lVert R_{w}\rVert_{H^s(Q)}=\mathcal{O}(\lambda^{-N+2s}). \end{aligned} \end{align}\tag{34}\] Next, we substitute 32 and 33 into the integral equation 29 and use the Sobolev embedding theorem to arrive at \[\begin{align} \label{reduced32integral32identity} \begin{aligned} \text{Known}&= \displaystyle \int_{Q} m! r_me^{\mathrm{i}\left(\sum_{k=1}^m\phi_k-\phi_0\right)} \left(\prod_{k=1}^m U_0^k\right)\overline{U_0^0}~ dx dt\\&\quad+ \displaystyle \sum_{a=1}^{m-1}\sum_{\substack{I\subset\{1,2,\cdots,m\}\\ \lvert I \rvert =a}}\int_{Q} a!(m-a)!r_ae^{\mathrm{i}\left(\sum_{j\in I}\phi_j- \sum_{j\notin I}\phi_j-\phi_0\right)}\left(\prod_{j \in I}U_0^j\right)\left(\prod_{j \notin I}\overline{U_0^j}\right) \overline{U_0^0} \;dxdt\\&\quad+\mathcal{O}(\lambda^{-1}) \end{aligned} \end{align}\tag{35}\] for \(s\geq \kappa+1\) and \(N> 2s\), by virtue of 34 . In the next step, we select \(\phi_l\) for \(0\leq l\leq m\) so that the exponential factor in the first term of 35 vanishes while in the second term of 35 it must remain non-vanishing for every admissible set \(I\). To reconstruct \(r_m\), we choose \(\omega_k\) for \(0\leq k\leq m\) so that \[\begin{align} \label{omega32sum32in32reconstruction32of32rm} \omega_1+\cdots+\omega_{m}=\omega_0\;\text{ and }\lvert \omega_1\rvert^2+\cdots \lvert \omega_{m}\rvert^2=\lvert \omega_0\rvert^2. \end{align}\tag{36}\]

A particular choice of \(\omega_k\) (\(0\leq k\leq m\)) satisfying 36 is given in 1, below. Hence, building on this choice of \(\omega_k's\), we observe that \[\begin{align} \phi_1+\cdots+\phi_m-\phi_0= \lambda x\cdot(\omega_1+\cdots+\omega_{m}-\omega_0)- \lambda^2 t(\lvert \omega_1\rvert^2+\cdots \lvert \omega_{m}\rvert^2-\lvert \omega_0\rvert^2)=0. \end{align}\] Furthermore, for these choices of \(\omega_i\), \(\sum_{j\in I}\phi_j- \sum_{j\notin I}\phi_j-\phi_0\neq0\) for any \(I\subset \{1,2,\cdots,m\}\) with \(\lvert I \rvert =a \;\text{for}\;1\leq a\leq m-1\). If possible, let us suppose that \(\sum_{j\in I}\phi_j- \sum_{j\notin I}\phi_j-\phi_0=0\), where \(I=\{i_1,\cdots ,i_a\}\), then \[\begin{align} \label{permutation32of32phi32i} \Phi:=\phi_{i_1}+\cdots +\phi_{i_a}-(\phi_{j_1}+\cdots+\phi_{j_{m-a}}+\phi_0)=0 \end{align}\tag{37}\] where \(\{i_1,\cdots ,i_a,j_1,\cdots ,j_{m-a}\}\) is some permutation of \(\{1,2,\cdots,m\}\). From definition of \(\phi_l\) for \(0\leq l\leq m\), the foregoing expression reduces to \[\begin{align} \label{omega32sum32in32reconstruction32of32rm32for32124l12461a} \begin{aligned} \omega_{i_1}+\cdots+ \omega_{i_a}&=\omega_{j_1}+\cdots+\omega_{j_{m-a}}+\omega_0,\\ \lvert \omega_{i_1}\rvert^2+\cdots+\lvert \omega_{i_a}\rvert^2&=\lvert \omega_{j_1}\rvert^2+\cdots+\lvert \omega_{j_{m-a}}\rvert^2+\lvert \omega_{0}\rvert^2. \end{aligned} \end{align}\tag{38}\] Since \(\{i_1,\cdots ,i_a,j_1,\cdots ,j_{m-a}\}\) is some permutation of \(\{1,2,\cdots,m\}\), we can rewrite 36 as \[\begin{align} \label{rewritten32omega32sum32in32reconstruction32of32rm} \begin{aligned} \omega_{i_1}+\cdots+ \omega_{i_a}&+\omega_{j_1}+\cdots+\omega_{j_{m-a}}=\omega_0,\\ \lvert \omega_{i_1}\rvert^2+\cdots+\lvert \omega_{i_a}\rvert^2&+\lvert \omega_{j_1}\rvert^2+\cdots+\lvert \omega_{j_{m-a}}\rvert^2=\lvert \omega_{0}\rvert^2. \end{aligned} \end{align}\tag{39}\] The difference of 38 and 39 leads to \[\omega_{j_1}+\cdots+\omega_{j_{m-a}}=0 \;\text{ and }\;\lvert \omega_{j_1} \rvert^2+\cdots +\lvert \omega_{j_{m-a}} \rvert^2 =0.\] The aforementioned expression implies that \(\omega_{j_1}=\cdots=\omega_{j_{m-a}}=0\), which is a contradiction. Thus, the exponential factor in the second term of 35 is non-vanishing for every admissible set \(I\).

2.2.3 Reconstruction of \(r_l~(1\leq l\leq m-1)\)↩︎

To reconstruct \(r_l\) for \(1\leq l\leq m-1\), we start with rewriting 29 as below \[\begin{align} \label{r32term} \begin{aligned} \text{Known}&= \int_{Q}\left( m! r_m \prod_{k=1}^m v_k\right)\overline{w}~dxdt+\sum_{a=1}^{m-1}\sum_{\substack{I\subset\{1,2,\cdots,m\}\\ \lvert I \rvert =a}}\displaystyle \int_{Q} a!(m-a)!r_a\left(\prod_{j \in I}v_j\right)\left(\prod_{j \notin I}\overline{v_j}\right) \overline{w} \;dxdt\\& =\int_{Q}\left( m! r_m \prod_{k=1}^m v_k\right)\overline{w}~dxdt+\int_{Q} l!(m-l)!r_l \left(\prod_{j=1}^l v_j\right) \left(\prod_{j=l+1}^m \overline{v_j}\right) \overline{w}\;dxdt \\&\qquad+\sum_{\substack{I\subset\{1,2,\cdots,m\}\\ \lvert I \rvert =l,~I\neq \{1,\cdots,l\}}}\displaystyle \int_{Q} l!(m-l)!r_l\left(\prod_{j \in I}v_j\right)\left(\prod_{j \notin I}\overline{v_j}\right) \overline{w} \;dxdt\\&\qquad+ \sum_{\substack{a=1\\a\neq l}}^{m-1}\sum_{\substack{I\subset\{1,2,\cdots,m\}\\ \lvert I \rvert =a}}\displaystyle \int_{Q} a!(m-a)!r_a\left(\prod_{j \in I}v_j\right)\left(\prod_{j \notin I}\overline{v_j}\right) \overline{w} \;dxdt. \end{aligned} \end{align}\tag{42}\] As discussed earlier, we substitute 32 and 33 into the integral identity 42 to arrive at \[\begin{align} \label{integral32identity32to32recover32r95l} \begin{aligned} \text{ Known}&=\int_{Q} m! r_me^{\mathrm{i}(\sum_{k=1}^m\phi_k-\phi_0)} \left(\prod_{k=1}^m U_0^k\right)\overline{U_0^0}~ dx dt\\&\quad+\int_{Q} l!(m-l)!r_l e^{\mathrm{i}(\sum_{k=1}^l \phi_k-\sum_{k=l+1}^m\phi_i-\phi_0)}\left(\prod_{k=1}^l U_0^k\right) \left(\prod_{k=l+1}^m \overline{U_0^k}\right) \overline{U_0^0}\;dxdt \\&\quad+\sum_{\substack{I\subset\{1,2,\cdots,m\}\\ \lvert I \rvert =l,~I\neq \{1,\cdots,l\}}}\displaystyle \int_{Q} l!(m-l)!r_le^{\mathrm{i}\left(\sum_{j\in I}\phi_j- \sum_{j\notin I}\phi_j-\phi_0\right)}\left(\prod_{j \in I}U_0^j\right)\left(\prod_{j \notin I}\overline{U_0^j}\right) \overline{U_0^0} \;dxdt\\&\quad+ \sum_{\substack{a=1\\a\neq l}}^{m-1}\sum_{\substack{J\subset\{1,2,\cdots,m\}\\ \lvert J \rvert =a}}\displaystyle \int_{Q} a!(m-a)!r_ae^{\mathrm{i}\left(\sum_{j\in J}\phi_j- \sum_{j\notin J}\phi_j-\phi_0\right)}\left(\prod_{j \in J}U_0^j\right)\left(\prod_{j \notin J}\overline{U_0^j}\right) \overline{U_0^0} \;dxdt\\&\quad+\mathcal{O}(\lambda^{-1}). \end{aligned} \end{align}\tag{43}\] Now, we select \(\phi_l\) for \(0\leq l\leq m\) so that the exponential factor in the second term of 43 vanishes while in other terms of 43 , it must remain non-vanishing. To achieve this, we choose \(\omega_k~(0\leq k\leq m)\) so that \[\begin{align} \label{omega32i32relation321} \begin{aligned} \omega_{1}+\cdots+\omega_{l}&=\omega_0+\omega_{l+1}+\cdots+\omega_{{m}},\\ \lvert \omega_1\rvert^2+\cdots +\lvert \omega_{l}\rvert^2&=\lvert \omega_0\rvert^2+\lvert \omega_{l+1}\rvert^2+\cdots +\lvert \omega_{m}\rvert^2 \end{aligned} \end{align}\tag{44}\] where \(1\leq l \leq m-1\). We provide the existence of above mentioned \(\omega_k's\) in two different cases viz. \(l=1\) and \(2\leq l\leq m-1\).
Case(1): When \(l=1\)

In this case, equation 44 can be rewritten as

\[\begin{align} \label{omega32i32relation32l611} \begin{aligned} \omega_{1}&=\omega_0+\omega_{2}+\omega_3+\cdots+\omega_{{m}},\\ \lvert \omega_1\rvert^2&=\lvert \omega_0\rvert^2+\lvert \omega_{2}\rvert^2+\lvert \omega_3\rvert^2+\cdots +\lvert \omega_{m}\rvert^2 . \end{aligned} \end{align}\tag{45}\] A particular choice of \(\omega_k's\) satisfying 45 can be seen in ¿tbl:table2? and with this choice, it is easy to see that \[\begin{align} & \phi_1-(\phi_2+\cdots+\phi_m+\phi_0)\\&= \lambda x\cdot(\omega_1-(\omega_2+\cdots+\omega_{m}-\omega_0))- \lambda^2 t(\lvert \omega_1\rvert^2-(\lvert \omega_2\rvert^2+\cdots \lvert \omega_{m}\rvert^2+\lvert \omega_0\rvert^2)=0. \end{align}\] Furthermore, for these choices of \(\omega_k\), the exponential factors other than the second term in 43 will remain non-vanishing. To prove this, let us consider the following cases:

1. If \(\sum_{k=1}^m\phi_k-\phi_0=0\) then \[\begin{align} \label{omega32m32sum} \omega_1+\cdots+\omega_{m}=\omega_0\;\text{ and }\lvert \omega_1\rvert^2+\cdots \lvert \omega_{m}\rvert^2=\lvert \omega_0\rvert^2. \end{align}\tag{46}\] The difference of 45 and 46 results into \[\begin{align} \omega_{2}+\omega_3+\cdots+\omega_{{m}}=0\;\text{ and }\;\lvert \omega_{2}\rvert^2+\lvert \omega_3\rvert^2+\cdots +\lvert \omega_{m}\rvert^2=0, \end{align}\] which is a contradiction, as \(\omega_k \in \mathbb{R}^n\setminus\{0\}\) for each \(0\leq k\leq m\).

2. If \(\sum_{j\in I}\phi_j- \sum_{j\notin I}\phi_j-\phi_0=0\) for some \(I\subset\{1,2,\cdots,m\}\), \(\lvert I\rvert=1\) and \(I\neq\{1\}\).
   Without loss of generality, let us assume \(I=\{a\}\), where \(a\neq 1\) then we have \[\begin{align} \label{omega32a} \begin{aligned} \omega_a&=\omega_1+\omega_2+\cdots+\omega_{a-1}+\omega_{a+1}+\cdots+\omega_m+\omega_0,\\\lvert \omega_a\rvert^2&=\lvert \omega_1\rvert^2+\lvert \omega_2\rvert^2+\cdots+\lvert \omega_{a-1}\rvert^2+\lvert \omega_{a+1}\rvert^2+\cdots+\lvert \omega_m\rvert^2+\lvert \omega_0\rvert^2. \end{aligned} \end{align}\tag{47}\] After subtracting 45 and 47 , we get \[\begin{align} \omega_a=\omega_1 \;\text{ and }\;\lvert \omega_a\rvert^2=\lvert \omega_1\rvert^2. \end{align}\] The above expression, together with the second term in 47 leads to \[\begin{align} \omega_k=0 \text{ for 0\leq k\leq m,~k\neq\{1,a\}}, \end{align}\] which is a contradiction.

3. If \(\sum_{j\in I}\phi_j- \sum_{j\notin I}\phi_j-\phi_0=0\) for some \(I\subset \{1,2,\cdots,m\}, \lvert I\rvert=a\) and \(a\neq 1\).
   Without loss of generality, let \(I=\{i_1,\cdots,i_a\}\), then we have \[\begin{align} \label{omega32124l12461a} \begin{aligned} \omega_{i_1}+\cdots+ \omega_{i_a}&=\omega_{j_1}+\cdots+\omega_{j_{m-a}}+\omega_0,\\ \lvert \omega_{i_1}\rvert^2+\cdots+\lvert \omega_{i_a}\rvert^2&=\lvert \omega_{j_1}\rvert^2+\cdots+\lvert \omega_{j_{m-a}}\rvert^2+\lvert \omega_{0}\rvert^2. \end{aligned} \end{align}\tag{48}\] The difference of 45 and 48 leads to

   1. Either \[\begin{align} \omega_{i_1}+\cdots+\omega_{i_{s-1}}+\omega_{i_{s+1}}+\cdots+\omega_{i_a}=0,\\ \lvert \omega_{i_1}\rvert^2+\cdots+\lvert \omega_{i_{s-1}}\rvert^2+\lvert \omega_{i_{s+1}}\rvert^2+\cdots+\lvert \omega_{i_a}\rvert^2=0, \end{align}\] if \(i_s=1\) for some \(i_s\in I\), which is a contradiction.

   2. Or \[\begin{align} \omega_1= \omega_{i_1}+\cdots+ \omega_{i_a}\;\text{ and }\;\lvert \omega_1\rvert^2=\lvert\omega_{i_1}\rvert^2+\cdots+\lvert \omega_{i_a}\rvert^2, \end{align}\] if \(i_s\neq 1\) for any \(i_s \in I\). On comparing the above expression with 45 , we get \(\omega_{j_1}=\cdots=\omega_{j_{m-a}}=0\), which is a contradiction.

Thus, for these choices of \(\omega_k\), the exponential factors other than the second term in 43 will remain non-vanishing. Next, following a similar analysis as used in the establishing for reconstruction of \(r_m\) in Step 1, we arrive at \[\begin{align} \label{r95132integral32identity} \begin{aligned} \text{Known}&=\int_{B(x_0,h)}r_1(t,x)\left(\prod_{k=1}^m \left[\prod_{j=1}^{n-1} \chi\left( \dfrac{(x-x_0)\cdot \xi_j^k}{h} \right)\right] \prod_{j=1}^{n-1} \chi\left( \dfrac{(x-x_0)\cdot \xi_j^0}{h} \right)\right) \times \\&\qquad \bigg[ \exp\bigg(\mathrm{i}\int_{0}^{\infty} \omega_1\cdot A(t,x+r \omega_1) ~dr -\mathrm{i}\sum_{k=2}^m \int_{0}^{\infty} \omega_k\cdot A(t,x+r \omega_k) ~dr\\&\qquad \qquad -\mathrm{i} \int_{0}^{\infty} \omega_0\cdot A(t,x+r \omega_0) ~dr\bigg)\bigg]~dx \end{aligned} \end{align}\tag{49}\] for all \(t\in (0,T)\). Again, we multiply the above expression by \(h^{-n}\) and letting \(h \rightarrow 0\), to obtain that \[\begin{align} r_1(t,x_0)= \text{Known, for all t\in(0,T)}. \end{align}\] Now since \(x_0 \in \Omega\) is arbitrary therefore \(r_1(t,x)=\text{Known}\), for all \((t,x) \in Q\).
Case(2): When \(2\leq l \leq m-1\)
Let \(\omega_k\) for \(0\leq k\leq m\) are as in 44 . Clearly \[\begin{align} \sum_{k=1}^l\phi_k - \sum_{k=l+1}^m \phi_k-\phi_0=0. \end{align}\] Also, except for \(J=\{1,\cdots,l\}\), we seek for \[\begin{align} \label{sum32we32want} \begin{aligned} \sum_{j\in J}\phi_j&- \sum_{j\notin J}\phi_j-\phi_0\\&=\lambda x\cdot \left(\sum_{j\in J}\omega_j-\sum_{j\notin J}\omega_j-\omega_0\right)-\lambda^2 t \left( \sum_{j\in J}\lvert \omega_j\rvert^2-\sum_{j\notin J}\lvert \omega_j\rvert^2-\lvert \omega_0\rvert^2 \right)\neq 0 \end{aligned} \end{align}\tag{50}\] However, for the choice of \(\omega_k\) for \(0\leq k\leq m\) satisfying 44 , the above expression is not true in general.

Now, to ensure that both 44 and 50 hold, we have to impose some extra condition on \(\omega_k\). If possible, let us assume that \(\sum_{j\in J}\phi_j- \sum_{j\notin J}\phi_j-\phi_0=0\) for \((J\neq\{1,\cdots,l\}) \subset \{1,2,\cdots,m\}\), then \[\begin{align} \label{omega32i32relation322} \sum_{j\in J}\omega_j-\sum_{j\notin J}\omega_j-\omega_0=0 \quad \text{ and }\quad \sum_{j\in J}\lvert \omega_j\rvert^2-\sum_{j\notin J}\lvert \omega_j\rvert^2-\lvert \omega_0\rvert^2=0. \end{align}\tag{51}\] Subtract 44 and 51 , we get \[\begin{align} \label{where32the32contradiction32needed} \sum_{j \in K_1} \omega_j=\sum_{j \in K_2} \omega_j \end{align}\tag{52}\] where \(K_1\subsetneq \{1,\cdots,l\}\) and \(K_2 \subsetneq \{l+1,\cdots,m\}\). Next, we choose \(\omega_k\) for \(0\leq k\leq m\) which satisfy 44 in such a way that 52 does not holds for any proper subset \(K_1\) and \(K_2\), introduced earlier. In particular, we choose \(\omega_k\) for \(0\leq k\leq m\) as in 2.

Again, the foregoing calculations follows from Step 1, and we achieve

\[\begin{align} \label{r95232to32m32integral32identity} \begin{aligned} \text{Known}&=\int_{B(x_0,h)}r_l(t,x)\left(\prod_{k=1}^m \left[\prod_{j=1}^{n-1} \chi\left( \dfrac{(x-x_0)\cdot \xi_j^k}{h} \right)\right] \prod_{j=1}^{n-1} \chi\left( \dfrac{(x-x_0)\cdot \xi_j^0}{h} \right)\right) \times \\&\qquad \bigg[ \exp\bigg(\sum_{k=1}^l\mathrm{i}\int_{0}^{\infty} \omega_k\cdot A(t,x+r \omega_k) ~dr -\mathrm{i}\sum_{k=l+1}^m \int_{0}^{\infty} \omega_k\cdot A(t,x+r \omega_k) ~dr\\&\qquad \qquad -\mathrm{i} \int_{0}^{\infty} \omega_0\cdot A(t,x+r \omega_0) ~dr\bigg)\bigg]~dxdt \end{aligned} \end{align}\tag{53}\] for all \(t\in (0,T)\). Thus, on multiplying the above expression by \(h^{-n}\) and letting \(h \rightarrow 0\), we get \[\begin{align} r_l(t,x_0)= \text{Known, for all t\in(0,T)}. \end{align}\] Since \(x_0 \in \Omega\) is arbitrary therefore \(r_l(t,x)=\text{Known}\), for all \((t,x) \in Q\). As \(2\leq l \leq m-1\) is arbitrary, we have \(r_l(t,x)=\text{Known}\), in \(Q\) for every \(2\leq l \leq m-1\). This completes the proof.0◻

Acknowledgments↩︎

• Parveen Kumar acknowledges financial support from the Council of Scientific and Industrial Research (CSIR), India, through the fellowship 09/1005(19269)/2024-EMR I.

• Manmohan Vashisth’s work was supported by the ARG-MATRICS grant from the ANRF, Government of India (File No. ANRF/ARGM/2025/002368/MTR).

• This research also received partial support under the FIST program of the Department of Science and Technology, Government of India (Ref. No. SR/FST/MS-I/2018/22(C)).
  

Data availability statement. No datasets were generated or analyzed during the current study; therefore, data sharing is not applicable.

Conflict of interest. The authors declare that they have no conflicts of interest regarding the research, authorship, and/or publication of this manuscript.

References↩︎