Quasi-invariant Gaussian measures for the cubic nonlinear Schrödinger equation with third-order dispersion

Abstract

In this paper, we consider the cubic nonlinear Schrödinger equation with third order dispersion on the circle. In the non-resonant case, we prove that the meanzero Gaussian measures on Sobolev spaces H s (T), s > 3 4 , are quasi-invariant under the flow. In establishing the result, we apply gauge transformations to remove the resonant part of the dynamics and use invariance of the Gaussian measures under these gauge transformations.Résumé. Dans cet article, nous considérons l'équation de Schrödinger non linéaire cubique avec dispersion d'ordre trois sur le cercle. Dans le cas non résonant, nous prouvons que les mesures gaussiennes de moyenne nulle sur les espaces de Sobolev H s (T), s > 3 4 , sont quasiinvariantes par le flot. Enétablissant le résultat, nous appliquons des transformations de gauge pouréliminer la partie résonante de la dynamique et nous utilisons l'invariance des mesures gaussiennes par rapportà ces transformations de gauge.2010 Mathematics Subject Classification. 35Q55.

On the transport of Gaussian measures under the one-dimensional fractional nonlinear Schrödinger equations

“…In particular, when α = 2, this improves upon the result in [33] of s > 3 4 . However, as remarked in [32], this same result of s > 2 3 for 4NLS could be obtained by using Method 1 and an additional novel gauge transformation introduced in that same paper. For FNLS (1.1) with α > 1 (and large enough), we expect the optimal result s > 1 2 could be obtained by using a finer modified energy arising from an infinite sequence of normal form reductions.…”

“…The quasi-invariance of Gaussian measures supported on periodic functions under the flow of 4NLS was recently studied by Oh and Tzvetkov [33] and Oh, Sosoe and Tzvetkov [31]. See also Remark 1.5 and the recent work [32] on Schrödinger-type equations. Our main goal is to extend these quasi-invariance results to more general values of dispersion α.…”

“…Methodology and discussion. Theorem 1.2 is an addition to a recent program initiated by Tzvetkov [43] based on understanding the role dispersion has on the transport properties of Gaussian measures under the flow of nonlinear Hamiltonian PDEs; see also [33,35,31,32,38,19]. Within the context of abstract Wiener space, the classical work of Ramer [39] (see also [7]) studied the quasi-invariance of Gaussian measures under general nonlinear transformations.…”

Transport of Gaussian measures by the flow of the nonlinear Schrödinger equation

“…This will allow to prove quasiinvariance for a family of natural gaussian measures under the flow of the 1d quintic defocusing NLS on the one dimensional torus. This is a significant generalization of the recent works [16,17,18,19,22] since the NLS case was out of reach of the techniques used there. Moreover, in the focusing case we get local in time quasi-invariance, thus answering a question first raised by Bourgain in [3, page 28].…”

“…In [22] the author introduced a new method (inspired by [6,23,24]) to study quasi-invariance of gaussian measures along the flow associated with dispersive equations. This approach was further generalized to much more involved situations in [16,17,18,19]. In particular, in [18] a multi-linear stochastic argument was introduced.…”