Multistring solutions by soliton methods in de Sitter spacetime

Abstract

Exact solutions of the string equations of motion and constraints are systematically constructed in de Sitter spacetime using the dressing method of soliton theory. The string dynamics in de Sitter spacetime is integrable due to the associated linear system. We start from an exact string solution ${\mathit{q}}_{\mathrm{}\left(0\right)\mathrm{}}$ and the associated solution of the linear system ${\mathrm{\Psi }}^{\mathrm{}\left(0\right)\mathrm{}}$(λ), and we construct a new solution Ψ(λ) differing from ${\mathrm{\Psi }}^{\mathrm{}\left(0\right)\mathrm{}}$(λ) by a rational matrix in λ with at least four poles ${\lambda }_0$,1/${\lambda }_0$,${\lambda }_0^{\mathrm{*}}$,1/${\lambda }_0^{\mathrm{*}}$. The periodicity condition for closed strings restricts ${\lambda }_0$ to discrete values expressed in terms of Pythagorean numbers. Here we explicitly construct solutions depending on (2+1)-spacetime coordinates, two arbitrary complex numbers (the ‘‘polarization vector’’), and two integers (n,m) which determine the string windings in the space. The solutions are depicted in the hyperboloid coordinates q and in comoving coordinates with the cosmic time T. Despite the fact that we have a single world sheet, our solutions describe multiple (here five) different and independent strings; the world sheet time τ turns out to be a multivalued function of T. (This has no analogue in flat spacetime.) One string is stable (its proper size tends to a constant for T→∞, and its comoving size contracts); the other strings are unstable (their proper sizes blow up for T→∞, while their comoving sizes tend to constants). These solutions (even the stable strings) do not oscillate in time. The interpretation of these solutions and their dynamics in terms of the sinh-Gordon model is particularly enlightening.