Arnold--Liouville Theorem for Integrable PDEs: A Case Study of the Focusing NLS Equation

DOI10.1137/21M1443169zbMath1496.35362arXiv2002.11638WikidataQ113779021 ScholiaQ113779021MaRDI QIDQ5092893

Peter J. Topalov, Thomas Kappeler

Publication date: 26 July 2022

Published in: SIAM Journal on Mathematical Analysis (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2002.11638

zbMATH Keywords

normal formsBirkhoff coordinatesArnold-Liouville theoremfocusing NLS equationfocusing mKdV equation

Mathematics Subject Classification ID

Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) KdV equations (Korteweg-de Vries equations) (35Q53) NLS equations (nonlinear Schrödinger equations) (35Q55) Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions (37K20)

Related Items

Spatially quasi-periodic solutions of the Euler equation, On the Nonexistence of Local, Gauge-Invariant Birkhoff Coordinates for the Focusing NLS Equation

Cites Work

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• On normalized differentials on hyperelliptic curves of infinite genus
• Birkhoff coordinates for the focusing NLS equation
• Global wellposedness of KdV in \(H^{-1}(\mathbb T,\mathbb R)\)
• Normal forms for Hamiltonian systems with Poisson commuting integrals - elliptic case
• On the characterization of the smoothness of skew-adjoint potentials in periodic Dirac operators
• A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I
• Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I: Schrödinger equations
• Morse and Melnikov functions for NLS PDE's
• A note on focus-focus singularities
• Scattering-like phenomena of the periodic defocusing NLS equation
• Asymptotics of spectral quantities of Zakharov-Shabat operators
• Perturbations of the defocusing nonlinear Schrödinger equation
• On nomalized differentials on spectral curves associated with the sinh-Gordon equation
• Generic non-selfadjoint Zakharov-Shabat operators
• Convergence versus integrability in Birkhoff normal form
• The defocusing NLS equation and its normal form
• Canonically Conjugate Variables for the Korteweg-de Vries Equation and the Toda Lattice with Periodic Boundary Conditions
• On normalized differentials on families of curves of infinite genus
• Algebraic-geometric principles of superposition of finite-zone solutions of integrable non-linear equations
• On global action-angle coordinates
• Periodic solutions of the KdV equation
• Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points
• Sur Certains Systemes Dynamiques Separables
• [https://portal.mardi4nfdi.de/wiki/Publication:4342735 Action-angle variables for the cubic Schr�dinger equation]
• Riemann Surfaces of Infinite Genus
• Semi-linearity of the Non-linear Fourier Transform of the Defocusing NLS Equation
• On the Nonexistence of Local, Gauge-Invariant Birkhoff Coordinates for the Focusing NLS Equation
• On an Arnold-Liouville Type Theorem for the Focusing NLS and the Focusing mKdV Equations
• On Spectral Properties of the L Operator in the Lax Pair of the sine-Gordon Equation
• On the Well-Posedness of the Defocusing mKdV Equation Below $L^{2}$