# Inverse Conductivity Problem and Nonlinear Integrable Systems

We will discuss two-dimensional inverse conductivity problem and the solution of the Davey-Stewartson II equation. Let $\mathcal O\subset \mathbb R^2$ be a bounded domain with a smooth boundary, and ${\rm div}(\gamma\nabla u) =0,~~ x=(x_1,x_2) \in \mathcal O. $The inverse conductivity problem (of recovering $\gamma$ by the Dirichlet-to Neumann map $\Lambda_\gamma: u|_{\partial\mathcal O}\to \gamma\frac{\partial u}{\partial\nu}|_{\partial\mathcal O}$) will be reduced to the inverse scattering problem for the Dirac equation. The latter problem will be solved using $\overline{\partial}$-method without any assumptions on a symmetry or smallness of the potential. This will allow us to solve the Davey-Stewartson II equation without assumptions on the absence of exceptional points. These results are extensions of our recent results with R.Novikov on $\overline{\partial}$-method for the Schrodinger equation with applications to the Novikov-Veselov equation. See more details in the papers below.

[1] E. Lakshtanov, R. Novikov, B. Vainberg, A global Riemann-Hilbert problem for two-dimensional inverse scattering at fixed energy, to appear in Rendiconti dell'Istituto di Matematica dell'Universita' di Trieste 48, (2016), 1-26, Special issue in honor of G. Alessandrini's 60th birthday.

[2] E. Lakshtanov, B. Vainberg, On reconstruction of complex-valued once differentiable conductivities, J. of Spectral Theory 6, No 4 (2016), Special issue in memory of Yuri Safarov, 881-902.

[3] E. Lakshtanov, B. Vainberg, Solution of the initial value problem for the focusing Davey-Stewartson II system, (2016), arXive:1604.01182.