Quaternionic non-linear Schrödinger equation

Question

Let $q(t,x)\in{\bf H}$ be a quaternionic number depending of the time $t$ and $x$ a real number. I define the quaternionic non-linear Schrödinger equation:

$$i \frac{\partial q}{\partial t}=-\frac{\partial^2 q}{\partial^2 x}+|q|^2 q$$

Is the quaternionic non-linear Schrödinger equation an integrable system?

Comments

In a similar way, we can define the quaternionic KdV equation:

$$q_t = q_{xxx}+3(q q_x+q_x q)$$

It is surely an integrable system.