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 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. Simply put, https://en.wikipedia.org/wiki/Inverse_scattering_transform. 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.