Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations

Existence of 'breathers', that is, time-periodic, spatially localized solutions, is proved for a broad range of time-reversible or Hamiltonian networks of weakly coupled oscillators. Some of their properties are discussed, some generalizations suggested, and several open questions raised.

https://iopscience.iop.org/article/10.1088/0951-7715/7/6/006/pdf

Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators

This book presents the main results and methods on inverse spectral problems for Sturm-Liouville differential operators and their applications. Inverse problems of spectral analysis consist in recovering operators from their spectral characteristics. Such problems often appear in mathematics, mechanics, physics, electronics, geophysics, meteorology and other branches of natural sciences. Inverse problems also play an important role in solving non-linear evolution equations in mathematical physics. Interest in this subject has been increasing permanently because of the appearance of new important applications, resulting in intensive study of inverse problem theory all over the world.

http://www.gbv.de/dms/goettingen/326677410.pdf

Inverse Sturm-Liouville problems and their applications

Frequency analysis for multi-dimensional systems: global dynamics and diffusion

We prove the existence of periodic solutions of the nonlinear wave equation 






satisfying either Dirichlet or periodic boundary conditions on the interval [O, π]. The coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation. Using a version of Newton's method, we show that this equation has solutions provided the nonlinearity g(x, u) satisfies certain generic conditions of nonresonance and genuine nonlinearity. © 1993 John Wiley & Sons, Inc.

Newton's method and periodic solutions of nonlinear wave equations

Inverse spectral theory

*This paper was written while both authors were guests of the Forschungsinstitut fur Mathematik at the ETH Zurich, and we thank the institute for its hospitality, pleasant working atmosphere and helpful staff. In particular, we thank Jiirgen Moser for numerous stimulating discussions on the subject. We also benefitted from a remark by Sigurd Angenent concerning the analyticity of the solutions. The second author also thanks the Deutsche Forschungsgemeinschaft for their financial support through a Heisenberg grant.

https://www.jstor.org/stable/pdfplus/2118656.pdf

Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrodinger equation

Integrability of hamiltonian systems on cantor sets

On Elliptic Lower Dimensional Tori in Hamiltonian Systems.

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Jürgen Pöschel

Papers

Inverse spectral theory

Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrodinger equation

Integrability of hamiltonian systems on cantor sets

On Elliptic Lower Dimensional Tori in Hamiltonian Systems.

A KAM-theorem for some nonlinear partial differential equations