Exact integrability of strings inD-dimensional de Sitter 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 $\mathrm{O}(D, 1)$-symmetric $\sigma$ model, plus the string constraints, are equivalent to a generalized sinh-Gordon equation. In $D=2\mathrm{}$ this is the Liouville equation, in $D=3\mathrm{}$ this is the standard sinh-Gordon equation, and in $D=4\mathrm{}$ this equation is related to the $B_2$ 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 Bäcklund transformations for this generalized sinh-Gordon equation, which relate expanding and shrinking string solutions. We find all classical solutions in $D=2\mathrm{}$ and physically analyze them. In $D=3\mathrm{}$ and $D=4\mathrm{}$, 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 $u\to \pm \infty$, the stretching (or the shrinking) of the proper string size, and the proportionality between $\tau$ and the conformal time.