Birkhoff Normal Form for Some Nonlinear PDEs

By Dario Bambusi. 2003

221 citations recorded by the source corpus.

Subfields: Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/2a1a9e00989d7b1348cccc266ff79b1b0925b8fd

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In Numerical Analysis

In Statistical and Nonlinear Physics

Citations in the corpus

Birkhoff normal form for partial differential equations with tame modulus (2006, methodology): “Such a result is proved by exploiting the idea that most monomials in the nonlinearity are already small (as in the present paper and in [ Bam03 ]).”
Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation (2010, unknown): “[4, 6, 61, 73] and the surveys [5, 47]).”
Branching of Cantor Manifolds of Elliptic Tori and Applications to PDEs (2011, methodology): “12), that are similar to those used in [1]-[3] close to an elliptic equilibrium.”
A KAM Theorem for Hamiltonian Partial Differential Equations in Higher Dimensional Spaces (2006, methodology): “This fact has been used by many authors when transforming the leading nonlinearity in the perturbation into the partial Birkhoff normal form under Cartesian coordinate systems (see Kuksin–Pöschel [16], Pöschel [17], Craig–Worfolk [10], Bourgain [7, 8], Bambusi [1], Bambusi–Berti [2], and Geng–”
KAM for the nonlinear beam equation (2016, background): “There exists a zero-measure Borel set C ⊂ [1, 2] such that for any m / ∈ C, any admissible set A, any c∗ ∈ (0, 1/2] and any analytic nonlinearity of the form (1.”
A Birkhoff normal form theorem for some semilinear PDEs (2008, methodology): “The theory presented here was developed in [ Bam03 , BG06, DS06, BDGS07, Gre06].”
Long time existence for fully nonlinear NLS with small Cauchy data on the circle (2018, result): “We quote for instance the papers by Bambusi [4], Bambusi-Grebert [6] and by Delort-Szeftel [12, 13].”
A KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions (2005, unknown): “In the following Lemma, we prove the coefficients of nonlinearity G have decay properties (similar proofs see Bambusi [1,2]).”
Almost global existence for Hamiltonian semilinear Klein‐Gordon equations with small Cauchy data on Zoll manifolds (2005, result): “In the case of one dimensional compact manifolds, similar statements have been obtained by Bourgain [5, 7] (with a loss on the number of derivatives of the solution with respect to those of the data), by Bambusi [4] and by Bambusi-Grébert [2].”
Long-time existence for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds (2006, background): “In the framework of compact manifolds, the first results have been obtained by Bourgain [2] and Bambusi [ 1 ].,• In the Hamiltonian case, f ≡ f (u), Bourgain [2] and Bambusi [ 1 ] proved for the one dimensional problem on an interval with Dirichlet boundary conditions a result of almost global exist”