**Construction of exact solutions in string theory inspired dark energy models**

**Abstract:**

We explore a possibility for the Universe to cross the w=-1 cosmological constant
barrier for the dark energy state parameter. We consider the Universe as a slowly decaying D3-brane. The D3-brane dynamics
is approximately described by a nonlocal string tachyon interaction and a back reaction of gravity is incorporated in the
closed string tachyon dynamics. In a local effective approximation this model contains one phantom component and one usual
field with a polynomial interaction. To understand cosmological properties of this system we study toy models with the same
scalar fields but with modified interactions. These modifications admit polynomial superpotentials. We develop
superpotential method to find exact solutions. Explicit solutions with the dark energy state parameter
crossing/non-crossing the barrier w=-1 at large time are presented.