**Question:** 

 1. Given a PDE, is there a general method to show that it is *not solvable* using the inverse scattering transform?
 2. Specifically, for the perturbed 1D NLS or the 2D cubic NLS, where was it first shown that these equations can not be solved using *any* form of the inverse scattering transform.

**Background and details:** The cubic 1D nonlinear Schrodinger equation (NLS)
$$ iu_t + u _{xx} + |u | ^2 u = 0$$ 
and the KdV equation $$u_t -6uu_x+u_{xxx} = 0$$ are both known to be integrable, and solvable via the [inverse scattering transform][1]. So, given the initial condition $u(t=0,x)=u_0 (x)$, one can compute these constants and solve an inverse, linear, auxilary problem to find $u$ for all times $t$. For example, for the cubic 1d NLS this is the Zakharov-Shabat equations, and for the KdV it is the linear, time-independant Schrodinger equation. 

The 2D cubic NLS, or almost every perturbation of the 1D case, e.g., $$iu_t +u_{xx} + |u|^2 u -\epsilon |u|^4u = 0 \, ,$$ is known to be *not* solvable using the inverse scattering transform, i.e., not integrable. I didn't find any reference that explains why, however.


  [1]: https://en.wikipedia.org/wiki/Inverse_scattering_transform