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The aim of this paper is to solve an important inverse source problem which arises from the well-known inverse scattering problem. We propose to truncate the Fourier series of the solution to the governing equation with respect to a special basis of L2. By this, we obtain a system of linear elliptic equations. Solutions to this system are the Fourier coefficients of the solution to the governing equation. After computing these Fourier coefficients, we can directly find the desired source function. Numerical examples are presented.
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Onset of the wave turbulence description of the longtime behavior of the nonlinear Schrödinger equation
Abstract Consider the cubic nonlinear Schrödinger equation set on a d -dimensional torus, with data whose Fourier coefficients have phases which are uniformly distributed and independent. We show that, on average, the evolution of the moduli of the Fourier coefficients is governed by the so-called wave kinetic equation , predicted in wave turbulence theory, on a nontrivial timescale.
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- PAR ID:
- 10291000
- Date Published:
- Journal Name:
- Inventiones mathematicae
- Volume:
- 225
- Issue:
- 3
- ISSN:
- 0020-9910
- Page Range / eLocation ID:
- 787 to 855
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00222-021-01039-z
