On Elliptic String Solutions in AdS(3) and dS(3)

TL;DR

This paper links classical string solutions in AdS(3) and dS(3) to Schrödinger problems with Lame potentials, developing a method to construct solutions including spiky and rotating strings, revealing new configurations.

Contribution

It introduces a novel approach connecting string solutions to Lame potential band structures, enabling systematic construction of new classical string configurations in AdS(3) and dS(3).

Findings

Constructed families of classical string solutions including spiky strings.

Developed a method linking string solutions to Lame potential band structures.

Discovered new circular rotating string solutions in AdS(3) and dS(3).

Abstract

Classical string actions in AdS(3) and dS(3) 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 Schrodinger problems. Each pair consists of a flat potential and an n = 1 Lame 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 Lame 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 AdS(3). New solutions include circular rotating strings in AdS(3) with singular time evolution of their radius and angular velocity as well as classical string solutions in dS(3).