Stability of the travelling front of a decaying brane

Debashis Ghoshal,Preeda Patcharamaneepakorn

doi:10.1007/jhep03(2015)159

Abstract

The dynamics (in light-cone time) of the tachyon on an unstable brane in the background of a dilaton linear along a null coordinate is a non-local reaction-diffusion type equation, which admits a travelling front solution. We analyze the (in-)stability of this solution using linearized perturbation theory. We find that the front solution obtained in singular perturbation method is stable. However, these inhomogenous solutions (unlike the homogenous solution) also have Lyapunov exponents corresponding to unstable modes around the (meta-)stable vacuum.

Highlights

Tachyon Fisher equation and the travelling frontWe recall that the dynamics of the open string modes are given by the cubic open string field theory
JHEP03(2015)159 front to a solution of the equations of motion of open string field theory to the first nontrivial order [18]
In terms of the boundary conformal field theory on the worldsheet of the string, which provides the background for the open string field theory, this corresponds to a deformation by a marginal operator which remains marginal when the first stringy corrections are included

Summary

Tachyon Fisher equation and the travelling front

We recall that the dynamics of the open string modes are given by the cubic open string field theory. The leading contribution is the tachyon field φ(xμ) on the unstable brane This is a Klein-Gordon equation with negative mass-square augmented by non-local cubic self-interactions. The singular perturbation analysis that starts with the solution of the homogeneous equation as the seed, and fixes the asymptotic conditions at ξ → ±∞, is not affected by this instability and yields a travelling front solution that converges. This instability could potentially cause the inhomogeneously decaying tachyon to oscillate around φS with increasing amplitude, the behaviour that was seen in the analysis of ref.

Perturbation of the non-local Fisher equation of the tachyon

Euclidean Schrodinger equation

Stability analysis in singular perturbation theory

Conclusions