Birkhoff Coordinates for the Focusing NLS Equation

Abstract

Abstract:In this paper we construct Birkhoff coordinates for the focusing nonlinear Schrödinger equation near the zero solution.

On the spectral problem associated with the time-periodic nonlinear Schrödinger equation

“…These results are needed for the establishment of local Birkhoff coordinates and shall serve as a solid foundation for future investigations in this direction. Theorem 4.5 is inspired by [20,Proposition 2.6], which establishes similar properties for the x-periodic potentials of imaginary type for the focusing 4 NLS. Our proof makes use of the ideas and techniques of [20].…”

“…For z ∈ C, we define the linear map e zσ 3 on the space of complex 2 For λ ∈ C and ψ ∈ X 1 , let M be the fundamental solution of (2.5) on the unit interval. If |λ| is so large that Z −1 p exists for all t ∈ [0, 1], then M + satisfies 20) and M − satisfies…”

“…We use the notation p,s n to denote the n-th coordinate of a sequence ( p,s n ) n∈Z in p,s C , see [13,18,20].…”

“…These asymptotics are a key ingredient for the subsequent two sections. The discussion of the asymptotic localization of the Dirichlet eigenvalues, Neumann eigenvalues and periodic eigenvalues in Section 3, as well as the study of the zero set of the imaginary part of the discriminant for potentials of real and imaginary type (corresponding to the defocusing and focusing NLS, respectively) in Section 4 then follow closely [13] and, respectively, [20]. In Section 5, we consider the special (but important) case of single-exponential potentials for which the fundamental matrix solution permits an exact formula.…”

On the characterization of the smoothness of skew-adjoint potentials in periodic Dirac operators

“…In subsequent work we will use the theorems stated above to construct Birkhoff coordinates for the fNLS-equation -see [10].…”