EXACT INTEGRABILITY OF STRINGS IN D-DIMENSIONAL DESITTER SPACETIME

We show the complete integrability of the string propagation in D-dimensional de Sitter spacetime. We find that the string equations of motion, which correspond to a noncompact O(D, 1)-symmetric sigma model, plus the string constraints, are equivalent to a generalized sinh-Gordon equation. In D = 2 this is the Liouville equation, in D = 3 this is the standard sinh-Gordon equation, and in D = 4 this equation is related to the B2 Toda model. We show that the presence of instability is a general exact feature of strings in de Sitter space, as a direct consequence of the strong instability of the generalized sinh-Gordon Hamiltonian (which is unbounded from below), irrespective of any approximative scheme. We exhibit Backlund transformations for this generalized sinh-Gordon equation, which relate expanding and shrinking string solutions. We find all classical solutions in D = 2 and physically analyze them. In D = 3 and D = 4, we find the asymptotic behaviors of the solutions in the instability regime. The exact solutions exhibit asymptotically all the characteristic features of string instability: namely, the logarithmic dependence of the cosmic time u on the world sheet time tau for u --> +/- infinity, the stretching (or the shrinking) of the proper string size, and the proportionality between tau and the conformal time.