This paper corrects several errors in the author's previous papers (Journal of Spectral Theory 2016, Analysis and PDE 2014) on the Davey-Stewartson II (DS II) and modified Novikov-Veselov (mNV) equations. In each of these papers a proof was given that the solution by inverse scattering yields a classical solution to the PDE. The mNV equation lies in the integrable hierarchy of the DS II equation, so the same scattering transform may be used in both cases. In the 2014 paper, an incorrect formula is given for the nonlinearity in the mNV equation. Here we correct errors in the proof and obtain a correct statement of the mNV equation as solved by inverse scattering.

We prove large time asymptotics for solutions of the KP I equation with small initial data. Our assumptions on the initial data rule out lump solutions but give a precise description of the radiation field at large times. Our analysis uses the inverse scattering method and involves large-time asymptotics for solutions to a non-local Riemann-Hilbert problem.

The fifth-order KP II equation ∂_t u + α∂_x^3 u + β∂_x^5 u + u ∂_x u + ∂_x^-1∂_y^2u=0 (β<0, α>0) is a nonlinear dispersive equation that models long dispersive waves in two space dimensions. We prove that solutions of the fifth-order KP II equation scatter to solutions of the corresponding linear equation ∂_t v + α∂_x^3 v + β∂_x^5 v + ∂_x^-1∂_y^2 v = 0 for small data. Our proof uses builds on Hadac, Herr, and Koch's work (see ArXiv:0708.2011) on the third-order KP II equation.

We show that Nachman's integral equations for the Calderon problem, derived for conductivities in W-2,W-p(Omega), still hold for L-infinity conductivities which are 1 in a neighborhood of the boundary. We also prove convergence of scattering transforms for smooth approximations to the scattering transform of L-infinity conductivities. We rely on Astala-Paivarinta's formulation of the Calderon problem for a framework in which these convergence results make sense.

We show that the derivative nonlinear Schrodinger (DNLS) equation is globally well-posed in the weighted Sobolev space H-2,H-2 (R) Our result exploits the complete integrability of the DNLS equation and removes certain spectral conditions on the initial data required by our previous work, thanks to Zhou's analysis (Comm. Pure Appl. Math. 42:7 (1989), 895-938) on spectral singularities in the context of inverse scattering.

We review recent results on global wellposedness and long-time behavior of smooth solutions to the derivative nonlinear Schrodinger (DNLS) equation. Using the integrable character of DNLS, we show how the inverse scattering tools and the method of Zhou for treating spectral singularities lead to global wellposedness for general initial conditions in the weighted Sobolev space $H^{2,2}(\mathbb{R})$. For generic initial data that can support bright solitons but exclude spectral singularities, we prove the soliton resolution conjecture: the solution is asymptotic, at large times, to a sum of localized solitons and a dispersive component, Our results also show that soliton solutions of DNLS are asymptotically stable.

These notes are a considerably revised and expanded version of expository lectures given at the Fields Institute Workshop on "Nonlinear Dispersive Partial Differential Equations and Inverse Scattering" in August 2017. We give a complete and self-contained treatment of inverse scattering for the defocussing cubic NLS in one-dimension, following the 2003 paper of Deift and Zhou, and the defocussing Davey-Stewartson equation in two space dimensions, following the work of Perry and more recent work of Nachman, Regev, and Tataru.

We give a rigorous mathematical analysis of the one-soliton solution of the focusing Davey-Stewartson II equation and a proof of its instability under perturbation. Building on the fundamental perturbation analysis of Gadyl'shin and Kiselev, we verify their Assumption 1 and use Fredholm determinants to globalize their perturbation analysis.

Using the inverse scattering method, we construct global solutions to the Novikov-Veselov equation for real-valued decaying initial data q with the property that the associated Schrodinger operator with potential q is nonnegative. Such initial data are either critical (an arbitrarily small perturbation of the potential makes the operator nonpositive) or subcritical (sufficiently small perturbations of the potential preserve non-negativity of the operator). Previously, Lassas, Mueller, Siltanen and Stahel proved global existence for critical potentials, also called potentials of conductivity type. We extend their results to include the much larger class of subcritical potentials. We show that the subcritical potentials form an open set and that the critical potentials form the nowhere dense boundary of this open set. Our analysis draws on previous work of the first author and on ideas of P. G. Grinevich and S. V. Manakov.

We study the derivative nonlinear Schrodinger (DNLS) equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but exclude spectral singularities). We show that the set of such initial data is open and dense in a weighted Sobolev space, and includes data of arbitrarily large L-2-norm. We prove global well-posedness on this open and dense set. In a subsequent paper, we will use these results and a steepest descent analysis to prove the soliton resolution conjecture for the DNLS equation with the initial data considered here and asymptotic stability of N-soliton solutions.

We study the derivative nonlinear Schrödinger equation for generic initial data in a weighted Sobolev space that can support bright solitons (but exclude spectral singularities). Drawing on previous well-posedness results, we give a full description of the long-time behavior of the solutions in the form of a finite sum of localized solitons and a dispersive component. At leading order and in space-time cones, the solution has the form of a multi-soliton whose parameters are slightly modified from their initial values by soliton–soliton and soliton–radiation interactions. Our analysis provides an explicit expression for the correction dispersive term. We use the nonlinear steepest descent method of Deift and Zhou (Commun Pure Appl Math 56:1029–1077, 2003) revisited by the \({\overline{\partial}}\)-analysis of McLaughlin and Miller (IMRP Int Math Res Pap 48673:1–77, 2006) and Dieng and McLaughlin (Long-time asymptotics for the NLS equation via dbar methods. Preprint, arXiv:0805.2807, 2008), and complemented by the recent work of Borghese et al. (Ann Inst Henri Poincaré Anal Non Linéaire,  https://doi.org/10.1016/j.anihpc.2017.08.006, 2017) on soliton resolution for the focusing nonlinear Schrödinger equation. Our results imply that N-soliton solutions of the derivative nonlinear Schrödinger equation are asymptotically stable.

The large-time behavior of solutions to the derivative nonlinear Schrödinger equation is established for initial conditions in some weighted Sobolev spaces under the assumption that the initial conditions do not support solitons. Our approach uses the inverse scattering setting and the nonlinear steepest descent method of Deift and Zhou as recast by Dieng and McLaughlin.

This lecture reports on joint work with Robert Jenkins, Jiaqi Liu, and Catherine Sulem. We illustrate the strengths of the inverse scattering method for addressing large-time behavior of completely integrable dispersive PDE’s by proving global well-posedness and determining large-time asymptotic behavior for the Derivative Nonlinear Schrödinger equation (DNLS) for soliton-free initial data. Our work uses techniques from the work of Deift and Zhou on the defocussing NLS together with further developments due to Dieng and McLaughlin.

We show that the inverse scattering map for the linear system associated with the defocussing Davey-Stewartson II equation is locally Lipschitz continuous with locally Lipschitz continuous inverse on $H^{1,1}(R^2)$. From the inverse scattering method we then obtain global well-posedness for the defocussing Davey-Stewartson II equation. We show that these global solutions are dispersive by computing their leading asymptotic behavior as $t \rightarrow \infty$ in terms of an associated linear problem.

We develop inverse scattering for the derivative nonlinear Schrodinger equation (DNLS) on the line using its gauge equivalence with a related nonlinear dispersive equation. We prove Lipschitz continuity of the direct and inverse scattering maps from the weighted Sobolev spaces H-2,H-2() to itself. These results immediately imply global existence of solutions to the DNLS for initial data in a spectrally determined (open) subset of H-2,H-2() containing a neighborhood of 0. Our work draws ideas from the pioneering work of Lee and from more recent work of Deift and Zhou on the nonlinear Schrodinger equation.

We use the (partial derivative) over bar -inverse scattering method to obtain global well-posedness and large-time asymptotics for the defocussing Davey-Stewartson II equation. We show that these global solutions are dispersive by computing their leading asymptotic behavior as t -> infinity in terms of an associated linear problem. These results appear to be sharp.

We consider a scattering map that arises in the ∂¯ approach to the scattering theory for the Davey–Stewartson II equation and show that the map is an invertible map between certain weighted L2 Sobolev spaces.

Recent progress in the theory and computation for the Novikov-Veselov (NV) equation is reviewed with initial potentials decaying at infinity, focusing mainly on the zero-energy case. The inverse scattering method for the zero-energy NV equation is presented in the context of Manakov triples, treating initial data of conductivity type rigorously. Special closed-form solutions are presented, including multisolitons, ring solitons, and breathers. The computational inverse scattering method is used to study zero-energy exceptional points and the relationship between supercritical, critical, and subcritical potentials.