**Some Recent Advances in Nonlinear Inverse Scattering in 2D: Theory and Numerics**

Valery Serov, Markus Harju and Georgios Fotopoulos

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/62233

#### **Abstract**

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We survey our recently published results concerning scattering problems for the nonlinear Schrödinger equation

where *h* is a quite general nonlinear analogue of the index of refraction. We will investigate direct scattering problem as well as several inverse scattering problems for this equation. We start by establishing sufficient conditions for the unique solvability of the direct scattering problem. Asymptotic behaviour of the scattering solutions is shown to give rise to scattering amplitude. Inverse problems are formulated as follows: extract information about the nonlinearity *h* from the knowledge of the scattering amplitude. At this point, one must specify more carefully the scattering data sets. We concentrate our attention to full data, backscattering data, fixed angle data and fixed energy data. The latter three data sets are collectively called limited data. For each data set, we define the inverse Born approximation and state theorems, which provide the recovery of main singularities of the function *h*0(*x*) = *h*(*x*, 1). The key idea here is to show that the difference between the Born approximation and *h*0(*x*) is less singular than *h*0(x). For practical applications, one must be able to compute the inverse Born approxima‐ tions numerically. To this end, we proceed as follows. We formulate the computation as a solution of an underdetermined linear system. Due to the ill-posed nature of the system, regularization methods are used. Examples are given to illustrate the effectiveness of the method.

**Keywords:** Inverse scattering, Schrödinger equation, Born approximation, numerical solution, linear inverse problem

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
