Journées Équations aux dérivées partielles, Jan 31, 2017
This lecture reports on joint work with Robert Jenkins, Jiaqi Liu, and Catherine Sulem. We illust... more 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.
Communications in Mathematical Physics, May 3, 2018
We study the derivative nonlinear Schrödinger equation for generic initial data in a weighted Sob... more 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 multisoliton 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 [8] revisited by the B-analysis of McLaughlin-Miller [24] and Dieng-McLaughlin [9], and complemented by the recent work of Borghese-Jenkins-McLaughlin [1] 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. C.
We study the Derivative Nonlinear Schrödinger equation for general initial conditions in weighted... more We study the Derivative Nonlinear Schrödinger equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but excluding spectral singularities). We prove global well-posedness and 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 solitons-solitons and solitons-radiation interactions. Our analysis provides an explicit expression for the correction dispersive term. We use the nonlinear steepest descent method of Deift and Zhou [9] revisited by the B-analysis of Dieng-McLaughlin [10] and complemented by the recent work of Borghese-Jenkins-McLaughlin [2] on soliton resolution for the focusing nonlinear Schrödinger equation.
This is the second in a series of papers on scattering theory for one-dimensional Schrödinger ope... more This is the second in a series of papers on scattering theory for one-dimensional Schrödinger operators with Miura potentials admitting a Riccati representation of the form q = u ′ + u 2 for some u ∈ L 2 (R). We consider potentials for which there exist 'left' and 'right' Riccati representatives with prescribed integrability on half-lines. This class includes all Faddeev-Marchenko potentials in L 1 `R, (1 + |x|)dx ´generating positive Schrödinger operators as well as many distributional potentials with Dirac delta-functions and Coulomb-like singularities. We completely describe the corresponding set of reflection coefficients r and justify the algorithm reconstructing q from r.
We construct surfaces TP0 with volume, first eigenvalue, and L p norm of the curvature bounded in... more We construct surfaces TP0 with volume, first eigenvalue, and L p norm of the curvature bounded independent of x0, but whose Cheeger constant tends to 0 as x0 tends to infinity. In [BPP], the authors showed, for p > n/2, the existence of a constant K(n, p) such that if M is an n-manifold satisfying IlRiccllp > K(n,p)(vol(M))l/p' then the Cheeger constant h(M) is bounded from below in terms of AI(M), vol(M), and I IRiccl Ip, where Rice denotes the Ricci tensor. In particular, in the case n = 2 or 3, IlRiccll2 is spectrally bounded, so we find purely spectral conditions under which we may bound the Cheeger constant spectrally. In this paper, we show by example that K(n, p) cannot be taken arbitrarily small, for any finite p. This is in striking contrast to the case (p = e~) when one has L ~ curvature bounds, where Buser's inequality [Bu] tells us that one can take K(n, e~) = O.
We study the number of permutations in the symmetric group on n elements that avoid consecutive p... more We study the number of permutations in the symmetric group on n elements that avoid consecutive patterns S. We show that the spectrum of an associated integral operator on the space L 2 [0, 1] m determines the asymptotic behavior of such permutations. Moreover, using an operator version of the classical Frobenius-Perron theorem due to Kreȋn and Rutman, we prove asymptotic results for large classes of patterns S. This extends previously known results of Elizalde. Résumé. Nousétudions le nombre de permutations dans le groupe symétrique sur néléments quiévitent des motifs S consécutifs. Nous montrons que le spectre d'un opérateur intégral associé sur L 2 [0, 1] m détermine le comportement asymptotique de telles permutations. Utilisant de plus une version d'opérateur du théorme classique de Frobenius-Perron en raison de Kreȋn et Rutman, nous donnons des résultats asymptotiques pour les grandes classes de motifs S. Ceciétend résultats précédemment des connus de Elizalde.
Communications in Partial Differential Equations, Aug 3, 2018
We study the derivative nonlinear Schr€ odinger (DNLS) equation for general initial conditions in... more We study the derivative nonlinear Schr€ odinger (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.
Communications in Partial Differential Equations, Aug 30, 2016
We develop inverse scattering for the derivative nonlinear Schrödinger equation (DNLS) on the lin... more We develop inverse scattering for the derivative nonlinear Schrödinger 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 2,2 pRq to itself. These results immediately imply global existence of solutions to the DNLS for initial data in a spectrally determined (open) subset of H 2,2 pRq 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 Schrödinger equation.
We construct continuous families of non-isometric metrics on simply connected manifolds of dimens... more We construct continuous families of non-isometric metrics on simply connected manifolds of dimension n ≥ 9 which have the same scattering phase, the same resolvent resonances, and strictly negative sectional curvatures. This situation contrasts sharply with the case of compact manifolds of negative curvature, where Guillemin/Kazhdan, Min-Oo, and Croke/Sharafutdinov showed that there are no nontrivial isospectral deformations of such metrics.
Asymptotic Completeness for Few Body Schrödinger Operators
North-holland Mathematics Studies, 1984
We discuss a method for proving asymptotic completeness for many body quantum mechanics. The main... more We discuss a method for proving asymptotic completeness for many body quantum mechanics. The main idea depends on, an analysis of the time rates of decay of the wave function in certain regions of configuration space. These rates of decay depend on the scattering channel to which the wave function belongs. The method has been applied to prove asymptotic completeness for generic three and four body systems with short range two body potentials.
We consider a scattering map that arises in the∂ approach to the scattering theory for the Davey-... more 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 L 2 Sobolev spaces.
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Papers by Peter Perry