  The instabilities of the nontrivial phase elliptic solutions in a repulsive
Bose-Einstein condensate (BEC) with a periodic potential are investigated.
Based on the defocusing nonlinear Schr\"{o}dinger (NLS) equation with an
elliptic function potential, the well-known modulational instability (MI), the
more recently identified high-frequency instability, and an unprecedented -- to
our knowledge -- variant of the MI, the so-called isola instability are
identified numerically. Upon varying parameters of the solutions, instability
transitions occur through the suitable bifurcations, such as the Hamiltonian
Hope one. Specifically, (i) increasing the elliptic modulus $k$ of the
solutions, we find that MI switches to the isola instability and the dominant
disturbance has twice the elliptic wave's period, corresponding to a Floquet
exponent $\mu=\frac{\pi}{2K(k)}$. The isola instability arises from the
collision of spectral elements at the origin of the spectral plane. (ii) Upon
varying $V_{0}$, the transition between MI and the high-frequency instability
occurs. Differently from the MI and isola instability where the collisions of
eigenvalues happen at the origin, high-frequency instability arises from
pairwise collisions of nonzero, imaginary elements of the stability spectrum;
(iii) In the limit of sinusoidal potential, we show that MI occurs from a
collision of eigenvalues with $\mu=\frac{\pi}{2K(k)}$ at the origin; (iv) we
also examine the dynamic byproducts of the instability in chaotic fields
generated by its manifestation. An interesting observation is that, in addition
to MI, the isola instability could also lead to dark localized events in the
scalar defocusing NLS equation.
