Uniqueness for a Cauchy problem for the generalized Schrödinger equation

1.   Introduction

The generalized Schrödinger equation appears in many physical models in quantum kinetic theory ^[1], in the study of water wave problems and in ferromagnetism. Some of these models are Davey-Stewartson ^[2], Zakharov-Schulman ^[3] and Ishimori systems ^[4]. In these integrable systems, we frequently encounter the nonlinear Schrödinger equation and its generalized form, which provide a description of nonlinear waves such as propagation of a laser beam in a medium whose index of refraction is sensitive to the wave amplitude, water waves at the free surface of an ideal fluid and plasma waves, ^[5]. A detailed analysis of these types of equations was presented in ^[5]. Moreover, we refer to ^[6,7,8,9,10] for the new results related to the existence, uniqueness and asymptotic properties of solutions including radial cases for the nonlinear Schrödinger equation. We also refer to ^{[11,12,13,14]}, where some new integrable systems were introduced based on the non-semisimple Lie algebra.

In this article, we deal with the generalized Schrödinger equation

in the domain $\Omega = \{(t, x):t\in \mathbb{R},$ $x\in D^{n+m}\subset \mathbb{R} ^{n+m}\},$ where the bounded domain $D^{n+m}$ is defined as

and supported by $x_{1} = 0.$ Throughout the paper, we use the following notations:

$\partial _{t}u = \frac{\partial u}{\partial t},$ $\partial _{1}^{2}u = \frac{ \partial ^{2}u}{\partial x_{1}^{2}},$ $\partial _{j}u = \frac{\partial u}{ \partial x_{j}},$ $\partial _{s}\partial _{j}u = \frac{\partial ^{2}u}{ \partial x_{s}\partial x_{j}}$ $(1\leq s,$ $j\leq n+m)$ and $i = \sqrt{-1}.$ In (1.1), the coefficients $a_{sj}$ $(1\leq s,$ $j\leq n+m)$ are real-valued, and there exist constants $M$, $M_{1}$ such that $\left\Vert a_{sj}\right\Vert _{C^{3}(\overline{D^{n+m}}~~)}~ \leq M$ and $\left\Vert fe^{-\kappa t^{2}}\right\Vert _{H^{2}(\mathbb{R})}\leq M_{1},$ where $\kappa > 0$ is a constant. The coefficients $b_{j}\in L_{loc}^{\infty }(\Omega)$ $(j = 0, 1, ..., n+m)$ are complex-valued, and there exists a constant $M_{2}$ such that $\left\vert \partial _{t}^{\beta _{0}}b_{j}\right\vert \leq M_{2},$ where $0\leq \beta _{0}\leq 2,$ $\partial _{t}^{\beta _{0}} = \frac{\partial ^{\beta _{0}}}{\partial t^{\beta _{0}}}.$ We consider the following problem:

Problem 1. Find the function $u(t, x)$ in $\Omega$ which satisfies equation (1.1) and the following conditions

We set

where $\beta _{0} = 0, 1, 2, 3;$ $\beta _{1},$ $\beta _{2} = 0, 1, 2;$ $s,$ $j = 0, 1, 2, ..., n+m;$ and $C_{u},$ $\kappa _{u} > 0$ are constants depending on $u(t, x).$ In ^[15,16], the local well-posedness of some initial value problems for the generalized Schrödinger equation was considered in appropriate Sobolev spaces. In ^[17], a non-standard Cauchy problem for the classical Schrö dinger equation was studied, and solvability of the related inverse problem was investigated. A conditional stability estimate for the ultrahyperbolic Schrödinger equation was obtained in ^[18], where the ultrahyperbolic part of the equation consists of the term $\Delta _{y}u-\Delta _{x}u$. In ^[19], a Hölder stability estimate was established for the inverse problem. Problem 1 is a Cauchy problem for the generalized Schrödinger equation because the data are given on the $x_{1} = 0.$ We investigate the uniqueness of the solution of the problem, and the main tool is a pointwise Carleman estimate. To the authors' best knowledge, this is the first result devoted to the solvability of Problem 1.

The main result of this paper is given below:

Theorem 1. Assume that there exist two positive constants $\alpha _{1}, \alpha _{2}$ such that

for any $\zeta = (\zeta _{2}, ..., \zeta _{n})\in \mathbb{R} ^{n-1},$ $\eta = (\eta _{n+1}, ..., \eta _{n+m})\in \mathbb{R} ^{m},$ $x\in D^{n+m}.$ Then, there exists at most one solution $u\in \mathcal{U}$ satisfying (1.1) and (1.2).

Theorem 1 is proved in Section 3 by using a Carleman type inequality which we call "Proposition 1". The proof of Proposition 1 will be given in Section 2.

Here, we suppose that the metric $g$ is given in semigeodesic coordinates, that is, it has the property $g_{11} = 1,$ $g_{1j} = 0$, $j = 2, 3, ..., n+m;$ so (1.1) takes the form

Since we shall prove the uniqueness, we consider the homogeneous version of Problem 1. Now, we define a new function $z = ue^{-\kappa t^{2}}$, so we can write

where $F(t, x) = f(t, x)e^{-\kappa t^{2}}$.

If we apply the Fourier transform to (1.6) and conditions (1.7) with respect to $t,$ we get

where $\widehat{z},$ $\widehat{F},$ $\widehat{b_{j}\partial _{j}z},$ $\widehat{b_{0}z}$ are the Fourier transforms with respect to $t$ of the functions $z,$ $F,$ $b_{j}\partial _{j}z,$ $b_{0}z$ $(j = 0, 1, ..., n+m),$ respectively. Here, $\xi$ is the parameter of the Fourier transforms and $\partial _{\xi }z = \frac{\partial z}{\partial \xi }$.

We write $\widehat{z} = w_{1}(\xi, x)+iw_{2}(\xi, x),$ $\widehat{b_{j}\partial _{j}z} = b_{1j}+ib_{2j}$ and $\widehat{b_{0}z} = b_{10}+ib_{20}$ in (1.8), and so we obtain the following system of equations:

where $k = 1, 2;$

By (1.9), we have

Thus, we shall show that this homogeneous problem has only zero solution.

2.   Pointwise Carleman estimate

T. Carleman ^[20] established the first Carleman estimate in 1939 for proving the unique continuation for a two-dimensional elliptic equation. A Carleman estimate is an $L^{2}$-weighted estimate with large parameters for a solution to a partial differential equation. These parameters play an essential role and are important how to choose a weight function in order to adjust given geometric configurations, ^[21]. This estimate is used as a powerful tool to establish the uniqueness and stability results for ill-posed Cauchy problems, ^[22]. In 1954, C. Müller extended Carleman's result to $\mathbb{R} ^{n}$, ^[23]. After that, A. P. Calderón ^[24] and L. Hörmander ^[25] improved these results based on the concept of pseudo-convexity. In ^[26,27,28], uniqueness and stability of various problems for the classical Schr ödinger equation were studied using Carleman estimates.

The Carleman estimate used in this paper asserts a pointwise inequality, while the conventional Carleman estimates are proved in terms of weighted $L^{2}$-integrals of solutions and boundary data. This type of pointwise Carleman estimate also was used in ^[22,29,30] for various equations including hyperbolic, parabolic and ultrahyperbolic equations.

We first reduce (1.10) to a form which is more suitable for a Carleman estimate. For this aim, we define a new variable $\widetilde{x}_{1} = \sqrt{ 2x_{1}}-\mu _{0},$ $\mu _{0} > 0,$ and then we define

Replacing the notations $\widetilde{w}_{k}, \widetilde{a}_{sj}, \widetilde{x} _{1}, \widetilde{b}_{r}, \widetilde{f}$ with $w_{k}, a_{sj}, x_{1}, b_{r}, f$ for the sake of simplicity, we have

where $l_{k}^{\prime } = (x_{1}+\mu _{0})l_{k},$ $k = 1, 2,$ and

Next, we define a subdomain of $D^{n+m}$

where $0 < \gamma < 1$, $2\mu _{0}\leq \min \left\{ \alpha _{0}, \gamma \right\},$ and a Carleman weight function

where $\alpha _{0} > 0$, the parameters $\delta,$ $\lambda,$ $v$ are positive numbers, and

Proposition 1. There exist $\delta _{\ast },$ $\nu _{\ast } = \nu _{\ast }(\delta)$ and $\lambda _{\ast } = \lambda _{\ast }(\nu)$ such that the inequality

holds for all $\delta > \delta _{\ast },$ $\nu > \nu _{\ast },$ $\lambda > \lambda _{\ast }$ and $w\in C^{2}(\overline{D_{\gamma }^{n+m}}).$ Here, the constant $\delta _{\ast }$ depends on $\alpha _{1}, \alpha _{2}, M, \psi, \gamma,$ and

Lemma 1. Assume that the conditions of Proposition 1 are satisfied. Then, there exist positive constants $\delta _{0},$ $\nu _{0} = \nu _{0}(\delta)$ and $\lambda _{0} = \lambda _{0}(\nu)$ such that

for all $\delta > \delta _{0},$ $\nu > \nu _{0},$ $\lambda > \lambda _{0}$ and $w\in C^{2}(\overline{D_{\gamma }^{n+m}}).$ Here, the constant $\delta _{0}$ depends on $\alpha _{1}, \alpha _{2}, M, \psi, \gamma,$ and $d_{j},$ $1\leq j\leq 3,$ represent some divergence terms which will be given explicitly in the proof below.

Proof. If we define a new function $\vartheta = \varphi w,$ then we write

where

and

Next, we calculate the terms in (2.5). Here, we note that $\partial _{1}\psi = \delta,$ $\partial _{1}^{2}\psi = 0,$ $\partial _{s}\partial _{j}\psi = \delta _{sj},$ where

First, we can write

where

On the other hand, by the inequality $2pq\geq -p^{2}-q^{2},$ we estimate the first, third, fourth and fifth terms in (2.5), so we have

Then, we can write

In (2.6), we used the inequality $2pq\geq -\tau _{0}p^{2}-\frac{q^{2}}{ \tau _{0}},$ for $\tau _{0} = (\nu +1)\psi ^{-1} > 0.$ Next, we continue to estimate the other terms in (2.5):

Since $\partial _{1}^{2}\psi = 0,$ we have

and

The last term has the following form:

Then, from (2.6)-(2.10), we can write

where

Here, we note that $||a_{sj}||_{C^{1}(\overline{D_{\gamma }^{n+m}})}\leq M,$ and $\partial _{j}\psi = (x_{j}-x_{j}^{0}), |\partial _{j}\psi | = |x_{j}-x_{j}^{0}|\leq \sqrt{2\gamma }, (2\leq j\leq n+m)$ in $D_{\gamma }^{n+m}$. Since $-2pq\geq -\frac{1}{\varepsilon _{0}(x_{1}+\mu _{0})} p^{2}-\varepsilon _{0}(x_{1}+\mu _{0})q^{2},$ for all $\varepsilon _{0} > 0$ and $p, q,$ we see that

The other terms can be estimated similarly. Hence, we can rewrite (2.11) in the following form:

where

First, we shall estimate the coefficient $E_{1}:$

where

If we take $\gamma$ as $0 < \gamma < \frac{2\sqrt{2\varepsilon _{0}}}{3M\sqrt{ (n-1)^{2}+m^{2}}},$ then $1-((n-1)^{2}+m^{2})\frac{M^{2}}{\varepsilon _{0}}(\frac{3}{4}\gamma)^{2} > \frac{1}{2},$ and so we obtain

in $D_{\gamma }^{n+m}.$ Setting

for $\delta \geq \delta _{1} = 2E_{12}^{^{\prime }}$, and then $\frac{1}{2} \delta -E_{12}^{^{\prime }}\geq 0,$ which implies that

As for the coefficient $E_{2},$ if we take $\varepsilon _{0}$ such that $0 < \varepsilon _{0} < \frac{\alpha _{1}}{4(n-1)},$ then we have

Here, we note that

Then, for $\delta \geq \delta _{2}$, from (2.20) it follows that

Next, we estimate the coefficient $E_{3}$. We choose $\varepsilon _{0}$ such that $0 < \varepsilon _{0} < \frac{\alpha _{2}}{4m},$

Since $\mu _{0} < \frac{1}{2}\gamma,$ $\delta \geq 4$ and $0 < \delta x_{1} < \gamma$ in $D_{\gamma }^{n+m},$ we get $(x_{1}+\mu _{0}) < \frac{3}{4 }\gamma.$ Here, we set

Then, if we take $\delta \geq \delta _{3}$, from (2.22) it follows that

Now, let us estimate the coefficient $E_{4}$. By the definition of the function $K$, we can write $E_{4}$ in the following form

where

We now estimate each of terms in $E_{41}$, respectively:

Then, from these relations we can write

Remembering that $(x_{1}+\mu _{0}) < \frac{3}{4}\gamma$ in $D_{\gamma }^{n+m},$ we have $\delta ^{4}(x_{1}+\mu _{0})^{-2} > \delta ^{4}.$ We set

then, we see that

for $\delta \geq \delta _{4}$ and $\lambda > \lambda _{0},$ which yields

Finally, if we write $\vartheta = \varphi w$ in (2.30), then we obtain

Choosing $\nu \geq \nu _{0} = 2\gamma ^{-3}\delta ^{-1}+\frac{3}{2}\gamma ^{2}\delta ^{-3}(\alpha _{1}+\alpha _{2})+1$, we have

On the other hand, if $\lambda \geq \lambda _{0} = 2\gamma ^{-3}(\nu +1)+\frac{ 3}{2}\gamma ^{2}(\nu +1)(\alpha _{1}+\alpha _{2})$, then we get

Therefore, for $\delta > \delta _{0},$ $\nu \geq \nu _{0}$ and $\lambda \geq \lambda _{0}$, from (2.30)-(2.33) we have

where $\delta _{0} = \max \{4,$ $\delta _{1},$ $\delta _{2},$ $\delta _{3},$ $\delta _{4}\},$ and

From relations (2.30) and (2.34), the proof of Lemma 1 is completed.

Lemma 2. If $\lambda$ exceeds some constant, then for all $w\in C^{2}(D_{\gamma }^{n+m}),$ we have

where $C > 0$ is a constant depending on $a_{sj}$ and dimension $n+m,$

Proof. By using the equality

and taking into account $\partial _{s}p\partial _{j}q = \partial _{j}((\partial _{s}p)q)-\partial _{j}(\partial _{s}p)q,$ we have

where

Remembering that $|S|\leq S^{0}$ and $||a_{sj}||_{C^{1}(D_{\gamma }^{n+m})}\leq M$ $(2\leq s, j\leq n+m),$ we see that the first term in (2.36) can be written as

Next, as for the coefficient in the second term, we write

Since

we see that

If we choose that

and $v\geq 1,$ then we have

where

Here, we note that $\left\vert a_{sj}\right\vert \leq M$ and $\left\vert \partial _{j}\psi \right\vert = \sqrt{2\gamma }, (2\leq s, j\leq n+m)$. We see that

Then, by relations (2.36) and (2.41), we write

Proof of Proposition 1.

We sum inequalities (2.4) and (2.35) and then we obtain

By choosing $\delta _{\ast } = \max \{\delta _{0},$ $\delta _{5}\},$ $\delta _{5} = \max \{(1+S^{0})\gamma ^{3},$ $(1+2S^{0}\gamma M(n-1))/\alpha _{1},$ $(1+2S^{0}\gamma Mm)/\alpha _{2}\},$ $\lambda _{\ast }\geq \max \{\lambda _{0},$ $\lambda _{1},$ $C\},$ $\nu _{\ast }\geq \max \{\nu _{0},$ $\frac{4}{ \delta ^{4}}(1+\psi S^{0}(\delta ^{2}+2M\gamma ((n-1)^{2}+m^{2})+\delta ^{2})\},$ we have (2.3) for $\delta > \delta _{\ast },$ $\nu > \nu _{\ast },$ $\lambda > \lambda _{\ast }.$ This completes the proof.

3.   Proof of Theorem 1

In order to prove Theorem 1, by (1.11) and the equality $l_{k}^{\prime } = (x_{1}+\mu _{0})l_{k}$, we write

Using (3.1), we have

Next, we apply the Carleman estimate in (3.2):

By (3.2) and (3.3), we write

If we multiply (3.4) by $(1+\xi ^{2})^{2}$ and integrate with respect to $\xi \ $over the interval $(-\infty, \infty)$, we have

In (3.5), we used the following relation by the Plancherel's theorem:

where $M_{3}$ is a constant depending on $M_{2}.$ Then, the inequality (3.5) takes the form

where

We estimate the coefficients $E_{5}, E_{6}, E_{7}:$ For $0 < \psi < \eta < 1,$ and $\lambda > 1,$ we have

for $\nu \geq \nu _{1} = \frac{1}{2}+8(n+m+1)M_{3},$ and

for $\nu \geq \nu _{2} = 1+16(n+m+1)M_{3}+C.$ If $\lambda > \max \left\{ \lambda _{2}, \lambda _{3}\right\},$ $\lambda _{2} = (S^{0})^{2},$ $\lambda _{3} = 16(n+m+1)M_{3}+8M_{1},$ and then we have

Hence, for $\lambda > 1$ and $v > \max \left\{ v_{1}, v_{2}\right\},$ (3.6)-(3.9) yield

Since $2(1+\xi ^{2})^{2} > 1$ and $\varphi ^{2} > 1$ on $D_{\gamma }^{n+m},$ we write

Integrating inequality (3.11) over the domain $D_{\gamma }^{n+m}$ and passing to the limit as $\lambda \rightarrow \infty,$ we get $w_{1} = w_{2} = 0,$, that is, $\widehat{z} = 0$ which implies $w(x) = 0$ in $D_{\gamma }^{n+m}$. Thus, Theorem 1 is proved.

4.   Conclusions

In this study, we consider a Cauchy problem for the generalized Schrö dinger equation. We prove the uniqueness of the solution of the problem by using the Carleman estimate. By similar arguments, a stability estimate can be obtained. Due to its multidimensional structure, the equation has an important place in some modern physics theories such as M-theory and twistor theory. In the future, we are planning to investigate the existence of the solution of the problem.

Acknowledgments

This research received no external funding.

Conflict of interest

The authors declare no conflict of interest.