Modulational instabilities in Josephson oscillations of elongated coupled condensates

We study the Josephson oscillations of two coupled elongated condensates. Linearized calculations show that the oscillating mode uniform over the length of the condensates (uniform Josephson mode) is unstable: modes of non zero longitudinal momentum grow exponentially. In the limit of strong atom interactions, we give scaling laws for the instability time constant and unstable wave vectors. Beyond the linearized approach, numerical calculations show a damped recurrence behavior: the energy in the Josephson mode presents damped oscillations. Finally, we derive conditions on the confinement of the condensates to prevent instabilities.