Analysis of Hamiltonian PDEs

By Sergei B. Kuksin. 2000

294 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/e8b6c228cb227a2364efa0971a9a7feb456cabed

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

Sobolev Periodic Solutions of Nonlinear Wave Equations in Higher Spatial Dimensions (2010, background): “In this case one can impose strong non-resonance conditions between the “tangential” and the “normal” frequencies of the expected KAM torus (the so-called second-order Melnikov non-resonance conditions) to solve the linear homological equations which arise at each step of the KAM iteration, see also”
Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type (2012, methodology): “In fact, the wide existing literature on KAM and Nash-Moser theory mainly deals with problems where the perturbation is bounded (see Kuksin [27], Craig [15], Berti [6] for a survey.”
Quasi-periodic solutions with Sobolev regularity of NLS on T^d with a multiplicative potential (2010, background): “[29], [30], [16]), but much work remains to be done for PDEs in higher space dimensions, due to the more complex properties of the eigenfunctions and eigenvalues of”
Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces (2011, methodology): “Their method was based on KAM theory (see also the works by Kuksin and Pöschel [32]-[28]-[29]) and strongly exploited that the eigenvalues of the Laplacian are simple.”
Birkhoff Normal Form and Hamiltonian PDEs (2006, methodology): “For a general overview on Hamiltonian PDEs, the reader may consult the recent monographies by Craig [Cra00], by Kuksin [Kuk00], by Bourgain [Bou05a] and by Kappleler and Pöschel [KP03].”
KAM for autonomous quasi-linear perturbations of mKdV (2014, methodology): “The choice γ > ε2b reduces to study perturbations of an isochronous system (as in [23,25,28]), and it is convenient to introduce ξ(ω) as a variable.”
Nonlinear dispersive equations : local and global analysis (2006, methodology): “See [Kuk3] for a more rigorous treatment of this infinite-dimensional Hamiltonian perspective, and [SSul] for a Lagrangian perspective of the material in this section.”
Branching of Cantor Manifolds of Elliptic Tori and Applications to PDEs (2011, methodology): “The iterative scheme in [24,26] would not converge assuming only (H1) or (H2).”
KAM for Reversible Derivative Wave Equations (2012, background): “For Hamiltonian perturbations of KdV, the existence and stability of quasi-periodic solutions was first proved by Kuksin [18,19] in the late 1990s, see also Kappeler–Pöschel [16].”
An abstract Nash-Moser theorem with parameters and applications to PDEs (2010, methodology): “This problem has been handled for PDEs with analytic nonlinearities via KAM methods, see e.g. Kuksin [22,23], Wayne [29], Pöschel [27], Eliasson and Kuksin [15], or via Newton-type iterative schemes as developed in Craig and Wayne [14] and Bourgain [7–10].”
KAM for the nonlinear beam equation (2016, methodology): “of the KdV or Sine-Gordon equations, see [26].”
Why are solitons stable (2008, unknown): “adopt this (important) perspective here; see [38] for further discussion.”
A KAM Theorem for Hamiltonian Partial Differential Equations with Unbounded Perturbations (2011, methodology): “To avoid this plight, Kuksin[7] smartly put the whole R j j(x) rather than R̂ j j(0) into Ω j as a modification of the normal form Nν so that it is not necessary to solve the equation for Fj j(x).”
Invariance of the Gibbs measure for the periodic quartic gKdV (2012, background): “1This is at least formally correct, for the rigorous definition of gKdV as a Hamiltonian system with Poisson structure J = ∂ ∂x on the Sobolev space H − 1 2 0 (T) (mean-zero functions) equipped with a compatible symplectic form, see [16].”