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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. 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.

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The prolongation structure method developed by Wahlquist and Estabrook is one method to show whether or not a PDE is solvable via the inverse scattering transform. (There are others - refer Y. Kosmann-Schwarzbach, B. Grammaticos, K. M. Tamizhmani (eds.), Integrability of Nonlinear Systems, Lect. Notes Phys. 638 Springer-Verlag 2004.)

For example, in an appendix to their second paper (on the NLS equation - J. Math. Phys. 17 (1976) 1293-1297) Estabrook and Wahlquist analyse the KdV-like equation $u_t + u_{xxx} + f(u)\,u_x = 0$, concluding that it admits a non-trivial prolongation structure (i.e. is integrable via IST) only if $f(u) = 2\alpha + 6\beta u+12\gamma u^2$, for some constants $\alpha$, $\beta$, $\gamma$.

Dodd and Fordy "The prolongation structures of quasi-polynomial flows" Proc Roy. Soc. Lond. A385 (1983) 389-429 discuss methods for dealing with a general class of equations that includes your variants of the NLS. Applying their methods to your quoted example would show why it does not produce a non-trivial prolongation structure.

As far as I can see from the literature, perturbations of the type you describe are discussed as perturbations of the IST solution of the unperturbed equation, e.g. V. I. Karpman "Soliton Evolution in the Presence of Perturbation" Physica Scripta 20 (1979) 462

Edit: most, if not all of the equations in two spatial dimensions - such as the Kadomtsev–Petviashvili equation - place heavy restrictions on soliton behaviour. They've sometimes been described as "one and a half dimensional" problems.

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  • $\begingroup$ Thank you, that's a lot to work with. I haven't read them all, but while Karpman's paper should be relevant for the perturbed 1D NLS, I don't think it can work for the 2D NLS, as it is not a perturbation of any integrable system. $\endgroup$ – Amir Sagiv Jun 25 '17 at 8:56
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    $\begingroup$ @AmirSagiv B.K. Harrison "On Methods of Finding Bäcklund Transformations in Systems with More than Two Independent Variables" Nonlinear Mathematical Physics 1995, V.2, N 3–4, 201–215 discusses extensions of the Wahlquist-Estabrook method to equations with more than one spatial dimension. $\endgroup$ – Phil Harmsworth Jun 26 '17 at 8:11
  • $\begingroup$ Whalquist-Estanbrook method is the prolongation method? $\endgroup$ – Amir Sagiv Jun 26 '17 at 15:17
  • $\begingroup$ @AmirSagiv Yes, that's right. Not to be confused with Cartan-Kuranishi prolongation. $\endgroup$ – Phil Harmsworth Jun 27 '17 at 9:20
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Another method for testing integrability is the Painleve test, see e.g. these lecture notes and references therein. It has some caveats: for example, certain changes of variables do not preserve the Painleve property. Yet another possibility for integrability testing is using higher symmetries, see e.g. here.

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