On 2D Nonlinear Schrödinger Equations with Data on R×T☆

By Hideo Takaoka, N. Tzvetkov. 2001

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MODIFIED SCATTERING FOR THE CUBIC SCHRÖDINGER EQUATION ON PRODUCT SPACES AND APPLICATIONS (2013, background): “In light of the above sharp contrast in behavior between Rd and its compact quotient Td, considerable interest has emerged in the past few years to study questions of long-time behavior on “in between” manifolds, like the ones presented by the non-compact quotients of Euclidean space [52, 59, 60, 84”
Strichartz estimates for partially periodic solutions to Schr (2010, background): “Following [9], let us define h(x) = |{(ξ, n) ∈ R× Z : ξ(2) + (n− e)(2) ≤ x}|, x ≥ 1.”