Stable localized solutions of arbitrary length for the quintic Swift-Hohenberg equation

We show that localized solutions of arbitrary length are stable over a finite parameter interval of subcritical values for the quintic Swift-Hohenberg equation with a destabilizing cubic term. This equation is thought to model a weakly hysteretic transition to stationary patterns. We argue that the stabilization of the localized states of arbitrary length can be traced back to the interaction between long wavelength modulations and spatial variations on the length scale of one unit cell. These results are critically compared with other known mechanisms to stabilize localized states in various situations. We also discuss for which experimental systems the states predicted here could be detected including e.g. the stationary onset of binary fluid convection.