Analysis Seminar

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Scattering of the Gross-Pitaevskii Equation in the 3D Radial Energy Space

Speaker: Zaher Hani, Georgia Tech

Location: Warren Weaver Hall 1302

Date: Thursday, March 30, 2017, noon

Synopsis:

We consider the 3D Gross-Pitaevskii (GP) equation, or the nonlinear Schrödinger (NLS) equation with non-vanishing constant amplitude at spatial infinity. We are interested in the asymptotic stability of the plane wave solutions. This was first answered positively by Gustafson, Nakanishi, and Tsai assuming that the perturbation belongs to a weighted Sobolev space. Such a control is not provided by the conserved energy of the system, so the natural question is whether this stability holds for perturbations in the energy space. Such a result is crucial to addressing the large-data theory. We give a positive answer under the assumption of radial symmetry (or angular regularity), and prove small-data scattering to appropriate free states. The interaction with the stationary plane wave is long-range, which makes scattering for GP much harder than that for NLS (i.e. for data vanishing at spatial infinity). The proof relies on: A) a normal forms transformation, which is a modification of that used by Gustafson, Nakanishi, and Tsai, that introduces additional null structures to the problem, and b) improved Strichartz estimates for the linear evolution assuming angular regularity. As mentioned above, this result gives access to the large-data scattering problem in the radial setting, which features a rather intriguing phenomenon: We know that such large-data scattering cannot happen in all energy space due to the presence of traveling wave solutions; however, such solutions are far from being radial. So, one might expect that scattering should happen in the radial setting without any obstruction.

This is joint work with Zihua Guo and Kenji Nakanishi.