**1. Introduction**

We deal with the generalized nonlinear Schrödinger equation:

$$i\frac{\partial}{\partial t}E\left(\mathbf{x},t\right) = -\Delta E\left(\mathbf{x},t\right) + h\left(\mathbf{x},|E|\right)E\left(\mathbf{x},t\right),$$

where *E* denotes the electromagnetic field in two-dimensional case, ∆ is the two-dimensional Laplacian and *h* describes in a general form the nonlinear contribution to the index of refrac‐ tion. Considering harmonic time dependence *E*(*x*, *t*) = *e*<sup>−</sup> *iωtu*(*x*) with frequency *ω* > 0, we obtain the steady-state nonlinear equation with fixed energy:

$$-\Delta u(\mathbf{x}) + h(\mathbf{x}, |u|)u(\mathbf{x}) = k^2 u(\mathbf{x}),\tag{1}$$

where *k*<sup>2</sup> = *ω* and fixed, and *u* denotes the complex amplitude of the field. Concerning the nonlinearity *h*(*x*, *s*), we pose the basic assumptions.


where *cρ* and *c<sup>ρ</sup>* ′ are constants and

$$
\sigma > 2 - 2/p, \ 1 < p \le \infty. \tag{2}
$$

Here, *L <sup>σ</sup> <sup>p</sup>*(ℝ<sup>2</sup> ) denotes the weighted Lebesgue space with the norm.

$$||f||\_{p,\sigma} = \left(\int\_{\mathbb{R}^2} (1+|\mathfrak{x}|)^{p\sigma} |f(\mathfrak{x})|^p d\mathfrak{x}\right)^{1/p}, \qquad ||f||\_p = ||f||\_{p,0}.$$

The main practical example (it can be considered as the motivation of this research) of such type equations (1) is the equation of the form:

$$-\Delta u(\mathbf{x}) + q\_1(\mathbf{x})u(\mathbf{x}) + q\_2(\mathbf{x})\frac{|u(\mathbf{x})|^2}{1+r|u(\mathbf{x})|^2}u(\mathbf{x}) = k^2u(\mathbf{x}),\tag{3}$$

with real number *k*, complex valued function *q*1(*x*) ∈ *L*<sup>2</sup> and real-valued function *q*2(*x*) ∈ *L*<sup>2</sup> , and parameter *r* ≥ 0. A particular nonlinearity in (3) of cubic type (*r* = 0) can be met in the context of a Kerr-like nonlinear dielectric film, while the case when *r* > 0 corresponds to the saturation model (see [1–4]).

We consider the inverse scattering problems for (1). For these purposes, we are interested of the scattering solutions to (1), i.e. solutions of the form

$$u(\mathbf{x},k,\theta) = u\_0(\mathbf{x},k,\theta) + u\_{\text{sc}}(\mathbf{x},k,\theta),\tag{4}$$

where , the unit sphere in ℝ<sup>2</sup> , *u*0(*x*, *k*, *θ*) = *eik*(*x*,*θ*) is the incident wave and *usc*(*x*, *k*, *θ*) is the scattered wave. The scattered wave must satisfy the Sommerfeld radiation condition at infinity:

$$\lim\_{r \to \infty} \sqrt{r} \left( \frac{\partial u\_{\rm sc}(\mathbf{x}, k, \theta)}{\partial r} - ik u\_{\rm sc}(\mathbf{x}, k, \theta) \right) = 0, \qquad r = |\mathbf{x}|\_{\mathbf{x}}$$

for fixed *k* > 0 and uniformly in . In that case, these solutions are the unique solutions of the Lippmann-Schwinger equation.

$$u(\mathbf{x},k,\theta) = u\_0(\mathbf{x},k,\theta) - \int\_{\mathbb{R}^2} G\_k^+(|\mathbf{x}-\mathbf{y}|)h(\mathbf{y},|u(\mathbf{y},k,\theta)|)u(\mathbf{y},k,\theta)d\mathbf{y}\,,\tag{5}$$

where *Gk* + is the outgoing fundamental solution of the operator − ∆ − *k*<sup>2</sup> , i.e. the kernel of the integral operator (−∆ − *k*<sup>2</sup> − *i*0)− 1. It is equal to

$$G\_k^+(|x|) = \frac{i}{4} H\_0^{(1)}(|kx|).$$

where *H*<sup>0</sup> (1) denotes the Hankel function of order zero and first kind.

The following main results concerning the direct scattering problem were proved in [5].

Under the basic assumptions and (2) for *h*, it is proved that for any *ρ* > 1 there is *k*0 > 0 such that for any *k* ≥ *k*0 in the ball *B<sup>ρ</sup>* ={*<sup>u</sup>* <sup>∈</sup> *<sup>L</sup>* <sup>∞</sup>(ℝ<sup>2</sup> ): *<sup>u</sup>* <sup>∞</sup> <sup>≤</sup>*ρ*}, there is a unique scattering solution (or the solutions of the form in (4) to (5) which satisfies the condition:

$$||u\_{\infty}||\_{\infty} \to 0, k \to \infty \tag{6}$$

uniformly in . What is more, the solution is obtained as the limit

$$u(\boldsymbol{x},k,\theta) = \lim\_{j \to \infty} u\_j(\boldsymbol{x},k,\theta)\_\*$$

where

**1. Introduction**

116 Applied Linear Algebra in Action

where *k*<sup>2</sup>

**1.** <sup>|</sup>*<sup>h</sup>* (*x*, *<sup>s</sup>*)|≤*cρα*(*x*), *<sup>α</sup>* <sup>∈</sup> *<sup>L</sup> <sup>σ</sup>*

′

saturation model (see [1–4]).

**2.** <sup>|</sup>*<sup>h</sup>* (*x*, *<sup>s</sup>*1) - *<sup>h</sup>* (*x*, *<sup>s</sup>*2)|≤*c<sup>ρ</sup>*

where *cρ* and *c<sup>ρ</sup>*

Here, *L <sup>σ</sup>*

We deal with the generalized nonlinear Schrödinger equation:

¶

the steady-state nonlinear equation with fixed energy:

nonlinearity *h*(*x*, *s*), we pose the basic assumptions.

′

are constants and

type equations (1) is the equation of the form:

*<sup>p</sup>*(ℝ<sup>2</sup> ), 0≤*s* ≤*ρ*,

*β*(*x*)|*s*<sup>1</sup> - *s*2|, *β* ∈ *L <sup>σ</sup>*

*<sup>p</sup>*(ℝ<sup>2</sup> ) denotes the weighted Lebesgue space with the norm.

The main practical example (it can be considered as the motivation of this research) of such

with real number *k*, complex valued function *q*1(*x*) ∈ *L*<sup>2</sup> and real-valued function *q*2(*x*) ∈ *L*<sup>2</sup>

and parameter *r* ≥ 0. A particular nonlinearity in (3) of cubic type (*r* = 0) can be met in the context of a Kerr-like nonlinear dielectric film, while the case when *r* > 0 corresponds to the

*i E xt E xt h x E E xt* () () , , , ,, ( ) ( ) *<sup>t</sup>*

where *E* denotes the electromagnetic field in two-dimensional case, ∆ is the two-dimensional Laplacian and *h* describes in a general form the nonlinear contribution to the index of refrac‐ tion. Considering harmonic time dependence *E*(*x*, *t*) = *e*<sup>−</sup> *iωtu*(*x*) with frequency *ω* > 0, we obtain

= *ω* and fixed, and *u* denotes the complex amplitude of the field. Concerning the

*<sup>p</sup>*(ℝ<sup>2</sup> ), 0≤*s*1, *<sup>s</sup>*<sup>2</sup> <sup>≤</sup>*ρ*,

(1)

(2)

(3)

,

¶ = -D +

$$u\_{f+1}(\mathbf{x},k,\theta) = u\_0(\mathbf{x},k,\theta) - \int\_{\mathbb{R}^2} G\_k^+(|\mathbf{x}-\mathbf{y}|)h(\mathbf{y},[u\_f(\mathbf{y},k,\theta)])u\_f(\mathbf{y},k,\theta)d\mathbf{y}$$

for *j* = 0, 1, … with *u*<sup>0</sup> as above. Let the function *h* have the same properties as above, but now with

$$
\sigma > \begin{cases} 2 - 2/p, 4/3 \le p \le \infty \\ 1/2, 1 < p < 4/3. \end{cases} \tag{7}
$$

Then for fixed *k* ≥ *k*0, the solution *u*(*x*, *k*, *θ*) admits the representation

$$u(\mathbf{x},k,\theta) = e^{ik\cdot(\mathbf{x},\theta)} - \frac{1+i}{4\sqrt{\pi k|\mathbf{x}|}} e^{ik|\mathbf{x}|} A(k,\theta',\theta) + o(|\mathbf{x}|^{-1/2}), \qquad |\mathbf{x}| \to \infty.$$

The function *A*(*k*, *θ*′, *θ*) is called the scattering amplitude and it is defined as

$$A(k, \theta', \theta) = \int\_{\mathbb{R}^2} e^{-ik(\theta', \circ)} h(\chi, |u(\circ, k, \theta)|) u(\circ, k, \theta) d\chi'$$

where is the direction of observation. This function *A* gives us the scattering data for inverse problem. More precisely, the inverse problem that is considered here is to extract some information about the function *h* by the knowledge of the scattering amplitude *A* for different sets of scattering data. There are four well-known inverse scattering data sets: (i) the full (scattering) data:

$$D = \{ A(k, \theta', \theta) \colon k > 0, \theta', \theta \in \mathbb{S}^1 \},$$

(ii) the backscattering data:

$$D\_{\mathcal{B}} = \{ A(k, \theta', \theta) \colon k > 0, \theta' = -\theta \in \mathbb{S}^1 \},$$

(iii) the fixed angle data:

$$D\_A = \{ A(k, \theta', \theta) \colon k > 0, \theta = \theta\_0 \text{ } f \text{ixed } \theta' \in \mathbb{S}^1 \} $$

and (iv) the fixed energy data:

$$D\_E = \{ A(k, \theta', \theta) \colon k = k\_0 > 0 \text{ if } \text{i} \text{ed}, \theta', \theta \in \mathbb{S}^1 \}.$$

We use the following notations for the direct and inverse Fourier transforms:

$$F(f)(\xi) = \int\_{\mathbb{R}^2} e^{i(\xi, y)} f(y) dy \tag{8}$$

$$F^{-1}(f)(\boldsymbol{x}) = \frac{1}{(2\pi)^2} \int\_{\mathbb{R}^2} e^{-i\langle \boldsymbol{\xi}, \boldsymbol{x} \rangle} f(\boldsymbol{\xi}) d\boldsymbol{\xi},\tag{9}$$

where (*ξ*, *x*) denotes the inner product in ℝ<sup>2</sup> , i.e. (*ξ*, *x*) = *ξ*1*x*1 + *ξ*2*x*2. By *C* > 0, we denote a generic constant that may change from one step to another. By *H <sup>t</sup>* (ℝ<sup>2</sup> )=*W*<sup>2</sup> *t* (ℝ<sup>2</sup> ), *t* ∈ ℝ we denote the standard *L*<sup>2</sup> based Sobolev space. A weighted Sobolev space *W <sup>p</sup>*,*<sup>σ</sup>* <sup>1</sup> (ℝ<sup>2</sup> ) is defined here by

$$\mathcal{W}^1\_{p,\sigma}(\mathbb{R}^2) = \{ f \in L^p\_{\sigma}(\mathbb{R}^2) \colon \nabla f \in L^p\_{\sigma}(\mathbb{R}^2) \}.$$

The following notation for the characteristic function is used:

$$\chi\_A(\mathbf{x}) = \begin{cases} 1, \mathbf{x} \in A \\ 0, \mathbf{x} \notin A. \end{cases}$$

## **2. Inverse scattering problems**

for *j* = 0, 1, … with *u*<sup>0</sup> as above. Let the function *h* have the same properties as above, but now

where is the direction of observation. This function *A* gives us the scattering data for inverse problem. More precisely, the inverse problem that is considered here is to extract some information about the function *h* by the knowledge of the scattering amplitude *A* for different sets of scattering data. There are four well-known inverse scattering data sets: (i) the

Then for fixed *k* ≥ *k*0, the solution *u*(*x*, *k*, *θ*) admits the representation

The function *A*(*k*, *θ*′, *θ*) is called the scattering amplitude and it is defined as

(7)

with

118 Applied Linear Algebra in Action

full (scattering) data:

(ii) the backscattering data:

(iii) the fixed angle data:

and (iv) the fixed energy data:

The direct scattering theory described above can also be reversed. The inverse scattering theory treats the function *h* as unknown and attempts to recover it from the knowledge of the scattering amplitude *A* for different data. Usually, the model in (1) is probed with one or more incident plane waves *u*0 and the resulting scattered waves are measured at a distance. This gives rise to several different scattering data sets which can be used to recover the unknowns.

The inverse backscattering problem for (1) was treated in [6]. Also for (1), the recovery of unknown function *h* is possible from the full scattering data. In addition to the two-dimensional studies mentioned above, certain particular nonlinear cases of (1) have been investigated in other dimensions too. In one-space dimension, we refer to [7] and the references therein. In higher dimensions *n* ≥ 3 we are only aware of [8,9]. Similar problems with formally more general equation but with bounded *h* are considered in [10] and [11].

Our point of view is that the nonlinearity may contain local singularities in the space coordinate *x*, and therefore we work in the frame of weighted Lebesgue spaces. These local singularities can be recovered from the scattering amplitude using the method of Born approximation. As a unifying result, we obtain mathematically more general results that have far wider applica‐ bility in physical experiments.

Let us set

$$h\_0(\mathbf{x}) = h(\mathbf{x}, \mathbf{1}).$$

In the subsections that follow we consider the inverse problems of recovering information about *h*0 from the knowledge of full data *D*, backscattering data *DB*, fixed angle data *DA* and fixed energy data *DE*.

#### **2.1. Full scattering data**

The inverse problem with full data *D* was investigated in [5]. Here we summarize the main results without proofs.

**Theorem 1** (Saito's formula). Under the basic assumptions and (7) for the function *h* we have,

$$\lim\_{k \to \infty} k \int\_{\mathbb{S}^1 \times \mathbb{S}^1} e^{-\ell k \left(\theta - \theta', x\right)} A(k, \theta', \theta) d\theta d\theta' = 4\pi \int\_{\mathbb{R}^2} \frac{h\_0(\mathbf{y})}{|\mathbf{x} - \mathbf{y}|} d\mathbf{y}, \qquad \mathbf{x} \in \mathbb{R}^2$$

where the limit is valid in the sense of distributions for 4/3 < *p* ≤ 2 and pointwise (even uniformly) for 2 < *p* ≤ ∞.

**Corollary 1** (Uniqueness). Let *σ* be as in (7). Consider the scattering problems for two sets of potentials *h* and . If the scattering amplitudes coincide for some sequence *kj* → ∞ and for all *θ*′, , then

$$h\_0(x) = \overline{h\_0}(x)$$

holds in the sense of distributions for 4/3 < *p* ≤ ∞.

**Corollary 2** (Representation formula). Let *σ* be as in (7). Then the representation

$$h\_0(\boldsymbol{x}) = \lim\_{k \to \infty} \frac{k^2}{8\pi^2} \int\_{\mathbb{S}^1 \times \mathbb{S}^1} e^{-ik(\boldsymbol{\theta} - \boldsymbol{\theta}', \boldsymbol{x})} A(k, \boldsymbol{\theta}', \boldsymbol{\theta}) |\boldsymbol{\theta} - \boldsymbol{\theta}'| d\boldsymbol{\theta} d\boldsymbol{\theta}'$$

holds in the sense of distributions for 4/3 < *p* ≤ ∞.

**Remark 1.** In addition to providing the above results, Saito's formula might be applied numerically too. It can be written as a convolution equation:

$$4\pi h\_0 \* |x|^{-1} = f,$$

where the function *f* can be computed from the full scattering data in principle. A numerical inversion of this equation would yield a full recovery of *h*<sup>0</sup> but this is an open problem as far as we know. What is more, unlike the representation formula above it holds pointwise in the important case of bounded (*p* = ∞) nonlinearities.

We assume that the function *x* ↦ *h*(*x*, *s*) is real-valued and recall that

$$A(k, \theta', \theta) = \int\_{\mathbb{R}^2} e^{-\mathbb{k}(\theta', y)} h(y, |u|) u(y, k, \theta) dy, \qquad k \ge k\_0.$$

For reasons of purely technical nature we define the scattering solutions *u*(*x*, *k*, *θ*) for negative *k* as

$$u(\mathbf{x},k,\theta) = \overline{u(\mathbf{x},-k,\theta)}, \qquad k < 0.1$$

These are the unique solutions of the integral equation:

$$u(\boldsymbol{x},k,\boldsymbol{\theta}) = e^{ik(\boldsymbol{\chi},\boldsymbol{\theta})} - \int\_{\mathbb{R}^2} \overline{G\_k^+(|\boldsymbol{x}-\boldsymbol{\chi}|)} h(\boldsymbol{\chi},|\boldsymbol{u}|) u(\boldsymbol{\chi},k,\boldsymbol{\theta}) d\boldsymbol{\chi}.$$

provided that *h* is real-valued. This allows us to extend *A* to negative *k* ≤ − *k*0 by

$$A(k, \theta', \theta) = \overline{A(-k, \theta', \theta)}.$$

We also put *A*(*k*, *θ*′, *θ*) = 0 for |*k*| ≤ *k*0. Splitting

$$h(\mathbf{y}, |u|)u = h\_0(\mathbf{x})u\_0 + h(\mathbf{y}, |u|)u - h\_0(\mathbf{x})u\_0$$

we have that

$$A(k, \theta', \theta) = \int\_{\mathbb{R}^2} e^{ik(\theta - \theta', \chi)} h\_0(\chi) d\chi + \int\_{\mathbb{R}^2} e^{-ik(\theta', \chi)} \langle h(\chi, |u|) u(\chi, k, \theta) - h\_0(\chi) u\_0 \rangle d\chi$$

or

Let us set

fixed energy data *DE*.

120 Applied Linear Algebra in Action

**2.1. Full scattering data**

results without proofs.

uniformly) for 2 < *p* ≤ ∞.

*θ*′, , then

In the subsections that follow we consider the inverse problems of recovering information about *h*0 from the knowledge of full data *D*, backscattering data *DB*, fixed angle data *DA* and

The inverse problem with full data *D* was investigated in [5]. Here we summarize the main

**Theorem 1** (Saito's formula). Under the basic assumptions and (7) for the function *h* we have,

where the limit is valid in the sense of distributions for 4/3 < *p* ≤ 2 and pointwise (even

**Corollary 1** (Uniqueness). Let *σ* be as in (7). Consider the scattering problems for two sets of

**Remark 1.** In addition to providing the above results, Saito's formula might be applied

→ ∞ and for all

potentials *h* and . If the scattering amplitudes coincide for some sequence *kj*

**Corollary 2** (Representation formula). Let *σ* be as in (7). Then the representation

holds in the sense of distributions for 4/3 < *p* ≤ ∞.

holds in the sense of distributions for 4/3 < *p* ≤ ∞.

numerically too. It can be written as a convolution equation:

$$A(k, \theta', \theta) = F(h\_0) \{ k(\theta - \theta') \} + A\_{\mathbb{R}}(k, \theta', \theta),$$

where *F* denotes the Fourier transform (8). Using the basic assumptions for the function *h* and (6), we can easily obtain that

$$\begin{split} |A\_{\mathbb{R}}(k,\theta',\theta)| &\leq \int\_{\mathbb{R}^2} |h(\mathsf{y},|u|)u(\mathsf{y},k,\theta) - h\_0(\mathsf{y})u\_0|d\mathsf{y} \leq \mathcal{C} \int\_{\mathbb{R}^2} \{a(\mathsf{y}) + \beta(\mathsf{y})\} |u\_{\mathrm{sc}}(\mathsf{y})|d\mathsf{y} \\ &\leq \mathcal{C} ||u\_{\mathrm{sc}}||\_{\mathrm{op}} \left( ||u||\_1 + ||\beta||\_1 \right) \to 0, \qquad |k| \to \infty. \end{split}$$

We have used here the fact that the basic assumptions for the function *h* guarantee that the functions *α* and *β* both belong to

Hence, for *k* large, we have approximately that

$$A(k, \theta', \theta) \approx F(h\_0) \{ k(\theta - \theta') \}. \tag{10}$$

These considerations and real valuedness of *h* suggest and justify the following definition:

We define the inverse Born approximation *qB* via the equality

$$A(k, \theta', \theta) = F(q\_B)(k(\theta' - \theta))\tag{11}$$

which is understood in the sense of tempered distributions. In order to recover main singu‐ larities of *h*0 from *qB*, we must study their difference and show that it is locally less singular than *h*0. To this end, we have the following main result from [5].

**Theorem 2.** Let *σ* be as in (7). Then

$$q\_B - h\_0 \in H^{\mathfrak{t}}\_{loc}(\mathbb{R}^2),$$

where *t* < 3 − 4/*p* if 4/3 < *p* ≤ 3/2 and *t* < 1 − 1/*p* if 3/2 < *p* ≤ ∞.

**Remark 2.** Theorem 2 means that, for 4/3 < *p* < ∞, the main singularities of *h*0 can be recovered from the inverse scattering Born approximation *qB* with full data *D*. In the case of *p* = ∞, we have no singularities but may have finite jumps. Under such circumstan‐ ces, we record the following special case.

**Corollary 3.** If a piecewise smooth compactly supported function *h*0 contains a jump over a smooth curve, then the curve and the height function of the jump are uniquely determined by the full scattering data. Especially, for the function *h*0 being the characteristic func‐ tion of a smooth bounded domain, this domain is uniquely determined by the full scattering data.

Concluding, in this part of the work, the uniqueness of the direct problem for the nonlinearities *h* satisfying the appropriate properties was proved. These properties allow local singularities and do not require compact support, but rather some sufficient decay at infinity. Under similar properties, we were also able to establish the asymptotic behaviour of scattering solutions, which gives us the scattering data so we can investigate the inverse scattering problems. Note that both results were proved without assuming smallness of the norm of the nonlinearity as is necessary in dimensions three and higher.

What can we regard as the main result of this section is the Saito's formula since it implies a uniqueness result and a representation formula for our unknown function *h*0. In addition, we managed to extract more information about the nonlinearity by applying the method of Born approximation. More precisely, the main singularities (or jumps over smooth curve) of *h*0 can be recovered from the Born approximation *qB* which corresponds to the full scattering data *D*.

#### **2.2. Backscattering and fixed angle data**

In this section, we consider backscattering data *DB* and fixed angle scattering data *DA* following [12]. Using (10) we introduce the inverse backscattering and inverse fixed angle scattering Born approximations *qB b* and *qB θ*0 as follows:

$$A(k, -\theta, \theta) = F(q\_B^b) \\
(2k\theta) \tag{12}$$

and

(10)

(11)

We have used here the fact that the basic assumptions for the function *h* guarantee that the

These considerations and real valuedness of *h* suggest and justify the following definition:

which is understood in the sense of tempered distributions. In order to recover main singu‐ larities of *h*0 from *qB*, we must study their difference and show that it is locally less singular

**Remark 2.** Theorem 2 means that, for 4/3 < *p* < ∞, the main singularities of *h*0 can be recovered from the inverse scattering Born approximation *qB* with full data *D*. In the case of *p* = ∞, we have no singularities but may have finite jumps. Under such circumstan‐

**Corollary 3.** If a piecewise smooth compactly supported function *h*0 contains a jump over a smooth curve, then the curve and the height function of the jump are uniquely determined by the full scattering data. Especially, for the function *h*0 being the characteristic func‐ tion of a smooth bounded domain, this domain is uniquely determined by the full scattering

Concluding, in this part of the work, the uniqueness of the direct problem for the nonlinearities *h* satisfying the appropriate properties was proved. These properties allow local singularities and do not require compact support, but rather some sufficient decay at infinity. Under similar properties, we were also able to establish the asymptotic behaviour of scattering solutions, which gives us the scattering data so we can investigate the inverse scattering problems. Note that both results were proved without assuming smallness of

the norm of the nonlinearity as is necessary in dimensions three and higher.

functions *α* and *β* both belong to

122 Applied Linear Algebra in Action

**Theorem 2.** Let *σ* be as in (7). Then

ces, we record the following special case.

data.

Hence, for *k* large, we have approximately that

We define the inverse Born approximation *qB* via the equality

than *h*0. To this end, we have the following main result from [5].

where *t* < 3 − 4/*p* if 4/3 < *p* ≤ 3/2 and *t* < 1 − 1/*p* if 3/2 < *p* ≤ ∞.

$$A\left(k, \theta', \theta\_0\right) = F\left(q\_{\theta}^{\theta\_0}\right) \left(k\left(\theta' - \theta\_0\right)\right),\tag{13}$$

where *θ*0 is fixed.

Furthermore, we assume in addition that the nonlinearity *h* possesses the Taylor expansion:

$$h(\mathbf{x}, \mathbf{1} + \mathbf{s}) = h(\mathbf{x}, \mathbf{1}) + \partial\_{\mathbf{s}}h(\mathbf{x}, \mathbf{1})\mathbf{s} + \frac{1}{2}\partial\_{\mathbf{s}}^2 h(\mathbf{x}, \mathbf{s}^\*)\mathbf{s}^2, \qquad \mathbf{1} < \mathbf{s}^\* < \mathbf{1} + \mathbf{s}^\*$$

where

$$|\partial\_s h(\mathfrak{x}, \mathbf{1})| \le \eta\_1(\mathfrak{x}), \qquad |\partial\_s^2 h(\mathfrak{x}, \mathbf{s}^\*)| \le \eta\_2(\mathfrak{x})$$

uniformly in *s* ∈ (0, *s*0), *s*0 > 0 and with *η*1, *η*2∈ *L <sup>σ</sup> <sup>p</sup>* (*R* 2), where *σ* as in (7). From this we obtain the expansion:

$$h(x, |u|)u = h\_0(x)u\_0 + g\_1(x)u\_{\rm sc} + g\_2(x)u\_0^2 \overline{u\_{\rm sc}} + \eta(x)O(|u\_{\rm sc}|^2),$$

where *g*<sup>2</sup> (*x*)= <sup>1</sup> <sup>2</sup> ∂*<sup>s</sup> h* (*x*, 1), *g*<sup>1</sup> (*x*)=*h*<sup>0</sup> (*x*) + *g*<sup>2</sup> (*x*) and *η* ∈ *L <sup>σ</sup> <sup>p</sup>* (ℝ<sup>2</sup> ) with the same *σ* as above.

Again, the main result for the recovery of main singularities is formulated as the following theorem.

**Theorem 3.** Let *σ* be as in (7) with 2 < *p* ≤ ∞. Suppose in addition

$$
\widehat{h\_{0\prime}}\_2 \widehat{g\_2} \in \mathcal{W}\_s^1(\mathbb{R}^2), \qquad 1 < s < 2,
$$

and 2 < *s*′ < *p* ≤ ∞, where *s*′ is the Hölder conjugate of *s*. Then

$$q\_B^{\theta\_0} - h\_{0\prime}q\_B^b - h\_0 \in H\_{loc}^t(\mathbb{R}^2)$$

for any *t* < 1 − 1/*p* if 1 < *s* ≤ 4/3 and for any *t* < min{1 − 1/*p*, 4/*s* − 2} if 4/3 < *s* < 2.

Let us sketch the main ideas in the proof of Theorem 3. Using the definition, we may expand the difference in several terms as

$$q\_B^{\theta\_0} - h\_0 = q\_1^{\theta\_0} + q\_\infty^{\theta\_0} + q\_2^{\theta\_0} + q\_R^{\theta\_0}.$$

In straightforward manner one sees that *q*<sup>∞</sup> *<sup>θ</sup>*0∈*<sup>C</sup>* <sup>∞</sup>(ℝ<sup>2</sup> ) and *q*<sup>2</sup> *θ*0 , *qR <sup>θ</sup>*0∈*H <sup>t</sup>* (ℝ<sup>2</sup> ) with *t* < 3 − 4/*p* if 4/3 < *p* ≤ 3/2 and *t* < 1 − 1/*p* if 3/2 < *p* ≤ ∞. The first nonlinear term *q*<sup>1</sup> *<sup>θ</sup>*<sup>0</sup> cannot be analyzed directly from its definition. Instead, we first proved the representation

$$\begin{aligned} q\_1^{\theta\_0}(\mathbf{x}) &= -F\_4^{-1} \Biggl( p\upsilon \frac{\widehat{g\_1}(\xi)\widehat{h\_0}(\eta)(\xi+\eta,\theta\_0)}{(|\eta|^2(\xi+\eta)-|\xi+\eta|^2\eta,\theta\_0)} \Biggr) (\mathbf{x},\mathbf{x}) \\ &- F\_4^{-1} \Bigl( p\upsilon \frac{\widehat{g\_2}(\xi)\widehat{h\_0}(\eta)(\xi+\eta,\theta\_0)}{(|\eta|^2(\xi+\eta)+|\xi+\eta|^2\eta,\theta\_0)} \Bigr) (\mathbf{x},\mathbf{x}) \Bigr) \end{aligned}$$

where *F*<sup>4</sup> -1 denotes the four-dimensional inverse Fourier transform. This formula might have independent interest too, but primarily it allows us to prove the following regularity: the term *q*1 *θ*0 belongs to the space


If we combine all these steps, we obtain Theorem 3 for fixed angle scattering.

The inverse backscattering Born approximation is treated similarly. Namely, we write

$$q\_B^b - h\_0 = q\_1^b + q\_\alpha^b + q\_2^b + q\_R^b.$$

The latter four terms have exactly the same regularity results as their counterparts in fixed angle scattering. For the first nonlinear term, the representation is now

$$q\_1^b(\chi) = 4F\_4^{-1}\left(pv\,\frac{\widehat{g\_1}(\xi)\,\widehat{h\_0}(\eta)}{\langle \xi,\eta \rangle}\right)(\chi,\chi) - 4F\_4^{-1}\left(pv\,\frac{\widehat{g\_2}(\xi)\,\widehat{h\_0}(\eta)}{\langle \xi+2\eta,\eta \rangle}\right)(\chi,\chi).$$

The additional assumption in Theorem 3 implies that *h*0∈ *L* <sup>1</sup> *s* ′ (ℝ<sup>2</sup> ).This explains why we restrict *s*′ < *p*. By Sobolev embedding, we obtain

$$(q\_B^{\theta\_0} - h\_0, q\_B^b - h\_0 \in \mathcal{W}\_p^\epsilon(\mathbb{R}^2))$$

with some positive *∈* < min{1/*p*, 1 − 2/*p*}. Hence, *h*<sup>0</sup> is locally more singular than either of these differences. That's why both Born approximations recover the main singularities of *h*0. On the other hand, we may perform the comparison also in the scale of Sobolev spaces. Indeed, if *h*0∈*Hcomp <sup>r</sup>* (ℝ<sup>2</sup> ) with some 0 < *r* < 1, then

$$q\_B^{\theta\_0} - h\_0, q\_B^b - h\_0 \in H\_{loc}^{\mathfrak{k}}(\mathbb{R}^2)$$

for any *t* < 2*r* if 0 < *r* ≤ 1/3 and for any *t* < (1 + *r*)/2 if 1/3 < *r* < 1. In both cases this *t* can be made bigger than *r*. It means that we can reconstruct all singularities from Sobolev space *Hr* , 0 < *r* < 1 from data *DB* and *DA* by the method of Born approximation.

**Corollary 4.** If a piecewise smooth compactly supported function *h*<sup>0</sup> contains a jump over a smooth curve, then the curve and the height function of the jump are uniquely determined by backscattering and fixed angle scattering data. Especially, for the function *h*<sup>0</sup> being the characteristic function of a smooth bounded domain, this domain is uniquely determined by backscattering and fixed angle scattering data.

Concluding, in this section we proved that all singularities and jumps (in the absence of a uniqueness theorem) of the nonlinearity *h* can be recovered from the inverse scattering Born approximation corresponding to fixed angle scattering and backscattering data *DA* and *DB*, respectively. No assumptions about the smallness of the norm of nonlinearity *h* were used as it were in previous publications even in the linear case.

#### **2.3. Fixed energy data**

and 2 < *s*′ < *p* ≤ ∞, where *s*′ is the Hölder conjugate of *s*. Then

4/3 < *p* ≤ 3/2 and *t* < 1 − 1/*p* if 3/2 < *p* ≤ ∞. The first nonlinear term *q*<sup>1</sup>

from its definition. Instead, we first proved the representation

the difference in several terms as

124 Applied Linear Algebra in Action

where *F*<sup>4</sup>

belongs to the space

**2.** *H* <sup>1</sup> (ℝ<sup>2</sup> ) if *s* = 4/3;

**1.** *C*(ℝ<sup>2</sup> )∩ *L* <sup>∞</sup>(ℝ<sup>2</sup> ) if 1 < *s* < 4/3;

**3.** *H <sup>t</sup>* (ℝ<sup>2</sup> ), *t* <4 /*s* - 2 if 4/3 < *s* < 2.

*q*1 *θ*0

In straightforward manner one sees that *q*<sup>∞</sup>

for any *t* < 1 − 1/*p* if 1 < *s* ≤ 4/3 and for any *t* < min{1 − 1/*p*, 4/*s* − 2} if 4/3 < *s* < 2.

Let us sketch the main ideas in the proof of Theorem 3. Using the definition, we may expand

*<sup>θ</sup>*0∈*<sup>C</sup>* <sup>∞</sup>(ℝ<sup>2</sup> ) and *q*<sup>2</sup>


independent interest too, but primarily it allows us to prove the following regularity: the term

If we combine all these steps, we obtain Theorem 3 for fixed angle scattering.

angle scattering. For the first nonlinear term, the representation is now

The inverse backscattering Born approximation is treated similarly. Namely, we write

The latter four terms have exactly the same regularity results as their counterparts in fixed

*θ*0 , *qR <sup>θ</sup>*0∈*H <sup>t</sup>*

(ℝ<sup>2</sup> ) with *t* < 3 − 4/*p* if

*<sup>θ</sup>*<sup>0</sup> cannot be analyzed directly

The two-dimensional fixed energy problemx1 was a long-standing open problem. In the case of linear Schrödinger operator, the first uniqueness and reconstruction algorithm was proved by Nachman [13] via ∂ ¯ -methods for potentials of conductivity type. Sun and Uhlmann [14] proved uniqueness for potentials satisfying nearness conditions to each other. The question of global uniqueness for the linear Schrödinger equation with fixed energy was settled only in 2008 by Bukhgeim [15] for compactly supported potentials from *Lp* , *p* > 2. This result has recently been improved and extended to related inverse problems (see for example [16] and [17]). Note that Grinevich and Novikov [18] proved that in two dimensions there are nonzero real potentials of the Schwartz class with zero scattering amplitude at fixed positive energy. Thus, the compactness of the supports of the potentials is very natural condition in our considerations.

The results of this section are proved in [19] and they slightly generalize the linear case to a special type of nonlinearity. It turned out that (as we can see in this section) inverse fixed energy scattering problem is much more difficult than the others.

In fixed energy scattering problem, instead of the scattering solutions (4) we need the so-called complex geometrical optics solutions. Complex geometrical optics (CGO) solutions or exponentially growing solutions of the form:

$$u(\mathbf{x}, \mathbf{z}) = e^{i(\mathbf{x}, \mathbf{z})} \{ 1 + R(\mathbf{x}, \mathbf{z}) \}, \quad \mathbf{z} \in \mathbb{C}^2, (\mathbf{z}, \mathbf{z}) = \mathbf{0} \tag{14}$$

with *R* decaying at infinity for |*z*| large for the homogeneous Schrödinger equation

$$-\Delta u(x) + h(x, |u(x)|)u(x) = 0\tag{15}$$

are obtained as follows. Substituting (14) into (15) yields

$$-\Delta R - 2i(\mathbf{z}, \nabla)R + h(\mathbf{x}, \left|e^{i(\mathbf{x}, \mathbf{z})}(\mathbf{1} + R)\right|)(\mathbf{1} + R) = 0.$$

It means that using the Faddeev Green's function

$$g\_{\mathbf{z}}(\mathbf{x}) = \frac{1}{4\pi^2} \int\_{\mathbb{R}^2} \frac{e^{-i(\mathbf{x},\xi)}}{\xi^2 + 2(\mathbf{z},\xi)} d\xi$$

as the fundamental solution of the differential operator

$$-\Delta - 2i(\mathbf{z}, \nabla)$$

we see that the function *R* solves the integral equation

$$R(\mathbf{x}, \mathbf{z}) = -\int\_{\mathbb{R}^2} g\_{\mathbf{z}}(\mathbf{x} - \mathbf{y}) h(\mathbf{y}, |e^{l(\mathbf{y}, \mathbf{z})}(1 + R(\mathbf{y}, \mathbf{z}))|) (1 + R(\mathbf{y}, \mathbf{z})) d\mathbf{y}.\tag{16}$$

It remains to establish unique solvability of this equation. To this end we again use iterations in the sense of next theorem.

We assume that *h* is compactly supported in Ω⊂ℝ2 and

**1.** |*h*(*x*, *s*)| ≤ *α*(*x*) with some *α* ∈ *L*<sup>2</sup> (Ω) and *s* ≥ 0


(14)

(15)

(16)

Thus, the compactness of the supports of the potentials is very natural condition in our

The results of this section are proved in [19] and they slightly generalize the linear case to a special type of nonlinearity. It turned out that (as we can see in this section) inverse fixed energy

In fixed energy scattering problem, instead of the scattering solutions (4) we need the so-called complex geometrical optics solutions. Complex geometrical optics (CGO) solutions or

It remains to establish unique solvability of this equation. To this end we again use iterations

(Ω) and *s* ≥ 0

with *R* decaying at infinity for |*z*| large for the homogeneous Schrödinger equation

scattering problem is much more difficult than the others.

are obtained as follows. Substituting (14) into (15) yields

It means that using the Faddeev Green's function

as the fundamental solution of the differential operator

we see that the function *R* solves the integral equation

We assume that *h* is compactly supported in Ω⊂ℝ2 and

in the sense of next theorem.

**1.** |*h*(*x*, *s*)| ≤ *α*(*x*) with some *α* ∈ *L*<sup>2</sup>

exponentially growing solutions of the form:

considerations.

126 Applied Linear Algebra in Action

**Theorem 4.** Under the above conditions for *h*, there exists a constant *C*0 > 0 such that for all | *z*| ≥ *C*<sup>0</sup> the equation (16) has a unique solution in *L* <sup>∞</sup>(ℝ<sup>2</sup> ) and this solution can be obtained as lim *<sup>j</sup>*→*<sup>∞</sup>Rj* in *L* <sup>∞</sup>(ℝ<sup>2</sup> ), where

$$\mathcal{R}\_{\boldsymbol{\lambda}+1}(\boldsymbol{\chi},\boldsymbol{z}) = -\int\_{\mathbb{R}^2} \mathbf{g}\_z \left( \boldsymbol{\chi} - \boldsymbol{\chi} \right) h \left( \boldsymbol{\chi}, \left| e^{i \left( \boldsymbol{\chi}, \boldsymbol{z} \right)} \left( 1 + \boldsymbol{R}\_{\boldsymbol{\lambda}} \left( \boldsymbol{\chi}, \boldsymbol{z} \right) \right) \right| \right) \left( 1 + \boldsymbol{R}\_{\boldsymbol{\lambda}} \left( \boldsymbol{\chi}, \boldsymbol{z} \right) \right) d\boldsymbol{\upmu}, j = 0, 1, \ldots, n$$

with *R*0 = 0. Moreover, the following estimates hold

$$||R||\_{\infty} \le \frac{c}{|z|^{\mathcal{Y}}}, \left. \left| |R - R\_f| \right| \right|\_{\infty} \le \frac{c\_f}{|z|^{\mathcal{Y}(j+1)}}, \ j = 0, 1, \dots$$

The proof of Theorem 4 is based on the fact that for any *γ* < 1 there is constant *cγ* > 0 such that

$$||g\_z \* f||\_{\infty} \le \frac{c\_\mathcal{V}}{|z|^\mathcal{V}} \left||f||\_{\frac{\mathcal{V}}{2}}, \ |z| > 1.$$

for any *f* ∈ *L*<sup>2</sup> (Ω), see [20].

Turning now to the inverse fixed energy scattering problem, we define the scattering transform by

$$T\_h(\xi) = \int\_{\mathbb{R}^2} e^{i(\chi,\xi)} h(\chi, e\_0 | 1 + R(\chi, z)|) \left(1 + R(\chi, z)\right) d\chi, \qquad |\xi| \ge \sqrt{2} \,\mathcal{C}\_0$$

and *Th*(*ξ*) = 0 for |*ξ*|< <sup>2</sup>*C*0 . Here *<sup>z</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> (*ξ* - *iJξ*),

$$J = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$$

and *e*<sup>0</sup> =|*ei*(*x*,*z*) |=*e* 1 <sup>2</sup> (*x*1*ξ*2-*x*2*ξ*1) . What is more, we have the uniform limit *Th* (*ξ*)=lim *<sup>j</sup>*→*<sup>∞</sup>Th* , *<sup>j</sup>* (*ξ*), where

$$T\_{h,j}(\xi) = \int\_{\mathbb{R}^2} e^{i \langle \chi, \xi \rangle} h(\chi, e\_0 \left| 1 + R\_j(\chi, z) \right|) \left( 1 + R\_j(\chi, z) \right) d\chi.$$

We point out that the scattering transform is somehow an auxiliary object (see *DE* in Introduc‐ tion). But it is connected to the scattering amplitude as follows. It is well known that the scattering amplitude at fixed energy uniquely determines the Dirichlet-to-Neumann map Λ*<sup>h</sup>* -*k*<sup>0</sup> 2 which in turn uniquely determines the scattering transform (see the details, for example, in [19] and [20]). Recall that Λ*<sup>h</sup> f* = ∂*νu*, where *u* satisfies the Dirichlet problem:

$$\begin{cases} -\Delta u + h(x, |u|)u = 0, x \in \Omega\\ u(x) = f(x), x \in \partial\Omega. \end{cases}$$

Here, Ω is a domain with boundary ∂Ω and outward unit normal vector *ν*.

Next, we define the inverse fixed energy Born approximation by

$$q\_B^f(\boldsymbol{x}) = F^{-1}\{T\_h(\boldsymbol{\xi})\}(\boldsymbol{x}).\tag{17}$$

In contrast to the preceding inverse problems, we now set the unknown function to be

$$h\_0(\mathbf{x}) = F^{-1}\left(T\_{h,0}(\xi)\right)(\mathbf{x}).\tag{18}$$

In linear case *h*0 is the actual potential appearing in the Schrödinger equation, but otherwise the connection to physical scatterers is not known to us.

We assume that the nonlinearity *h* admits the Taylor expansion

$$h(\mathbf{x}, e\_0(\mathbf{1} + \mathbf{s})) = h(\mathbf{x}, e\_0) + \partial\_{\mathbf{s}} h(\mathbf{x}, e\_0(\mathbf{1} + \mathbf{s}))|\_{\circ \dots \circ \alpha} \mathbf{s} + \mathcal{O}(\beta\_1(\mathbf{x})\mathbf{s}^2)|\_{\circ}$$

where |∂*sh*(*x*, *e*0(1 + *s*))|*s* = 0| ≤ *β*1(*x*) and *O*(*β*1(*x*)*s*<sup>2</sup> ) with *β*1(*x*) ∈ *L*<sup>2</sup> (Ω) and with small *s* in the neighbourhood of zero and where *O* is uniform in *x* ∈ Ω and such *s*.

Suppose in addition that the nonlinearity *h* satisfies the asymptotic expansions

$$h(\boldsymbol{x}, \boldsymbol{e}\_0) = \sum\_{j=0}^{\infty} \frac{a\_j(\boldsymbol{x})}{|\boldsymbol{\xi}|^j}, \quad h(\boldsymbol{x}, \boldsymbol{e}\_0 | \boldsymbol{1} + \boldsymbol{R}\_1 |) = \sum\_{j=0}^{\infty} \frac{\widetilde{a\_j}(\boldsymbol{x})}{|\boldsymbol{\xi}|^j}.$$

where . Then we have the following main result concerning the recovery of singularities of *h*0(*x*) defined by (18).

**Theorem 5.** Under the foregoing conditions for the potential function *h*, it is true that

$$q\_B^f - h\_0 \in H^t(\mathbb{R}^2)$$

for any *t* < 1 modulo *C* <sup>∞</sup>(ℝ<sup>2</sup> ) - functions.

We point out that the scattering transform is somehow an auxiliary object (see *DE* in Introduc‐ tion). But it is connected to the scattering amplitude as follows. It is well known that the scattering amplitude at fixed energy uniquely determines the Dirichlet-to-Neumann map Λ*<sup>h</sup>* -*k*<sup>0</sup>

which in turn uniquely determines the scattering transform (see the details, for example, in

[19] and [20]). Recall that Λ*<sup>h</sup> f* = ∂*νu*, where *u* satisfies the Dirichlet problem:

Here, Ω is a domain with boundary ∂Ω and outward unit normal vector *ν*.

In contrast to the preceding inverse problems, we now set the unknown function to be

In linear case *h*0 is the actual potential appearing in the Schrödinger equation, but otherwise

) with *β*1(*x*) ∈ *L*<sup>2</sup>

Next, we define the inverse fixed energy Born approximation by

128 Applied Linear Algebra in Action

the connection to physical scatterers is not known to us.

where |∂*sh*(*x*, *e*0(1 + *s*))|*s* = 0| ≤ *β*1(*x*) and *O*(*β*1(*x*)*s*<sup>2</sup>

We assume that the nonlinearity *h* admits the Taylor expansion

neighbourhood of zero and where *O* is uniform in *x* ∈ Ω and such *s*.

Suppose in addition that the nonlinearity *h* satisfies the asymptotic expansions

2

(17)

(18)

(Ω) and with small *s* in the

**Remark 3.** The embedding theorem for Sobolev spaces says that the difference *qB <sup>f</sup>* - *<sup>h</sup>*0 belongs to *Lq* (*R*<sup>2</sup> ) for any *q* < ∞ modulo *C*<sup>∞</sup>(*R*<sup>2</sup> ) functions. It means that all singularities from *L loc <sup>p</sup>* (*R* 2), *p* < ∞ of unknown function *h*0 can be obtained exactly by the Born approximation which corresponds to the inverse scattering problem with fixed positive energy.

We note that under fixed energy data we have some additional assumptions on *h*. This limits the applicability of the main result to, for example, saturation type nonlinearities. In particu‐ lar, cubic nonlinearity is excluded from these considerations. Moreover, the unknown function *h*0 has no direct connection to original scatterers in nonlinear cases.
