The nonlinear Schrödinger equation in $\mathbb{R}^{3+1}$ is $$\phi_t + \nabla^2 \phi + f(|\phi|^2)\phi + V(x)\phi= 0, \quad \phi(x,0) = \phi_0(x).$$ (I require $\phi$ to be square-integrable, and also that the solutions vanish at infinity.) For particular $f$, this equation possesses soliton wave solutions of the form $$\phi(t,x) = e^{i\omega t}R(x,\omega), \quad \omega > 0,$$ where $R$ satisfies the elliptic equation $$ -\omega R + \nabla^2 R + f(|R|^2)R + V(x)R = 0. $$ My particular interest is in the case where $$f(|\phi|^2) = \int_{\mathbb{R}^3}d^3y\, \frac{|\phi(y,t)|^2}{|x-y|}.$$ or in terms of the soliton solution, $$f(|R|^2) = \int_{\mathbb{R}^3}d^3y\, \frac{|R(y,\omega)|^2}{|x-y|}.$$ This corresponds to the Pekar-Choquard equation.
In this paper it is proved that when $V(x) = 0$, this equation has orbitally stable soliton solutions. I'm curious to find an instance of this equation where the soliton solutions are not orbitally stable. In particular, I would like the solutions to be exponentially unstable. Are there any simple (and physically realizable, like a harmonic potential, say) choices of the potential that might render this the case? Or perhaps there is another way to achieve that.