On elliptic string solutions in AdS3 and dS3

Classical string actions in AdS3 and dS3 can be connected to the sinh-Gordon and cosh-Gordon equations through Pohlmeyer reduction. We show that the problem of constructing a classical string solution with a given static or translationally invariant Pohlmeyer counterpart is equivalent to solving four pairs of effective Schrödinger problems. Each pair consists of a flat potential and an n = 1 Lamé potential whose eigenvalues are connected, and, additionally, the four solutions satisfy a set of constraints. An approach for solving this system is developed by employing an interesting connection between the specific class of classical string solutions and the band structure of the Lamé potential. This method is used for the construction of several families of classical string solutions, one of which turns out to be the spiky strings in AdS3. New solutions include circular rotating strings in AdS3 with singular time evolution of their radius and angular velocity as well as classical string solutions in dS3.