We study the two dimensional non-linear Schrödinger equation with two types of exponential non-linearities. It is well-known by a work of Ruf–Sani (Ruf and Sani, 2013), that such models support solitary wave solutions, which are solutions of some constrained minimization problem. We show that the Ruf–Sani solitons are spectrally unstable and unstable by blow up.
