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Ioan Marcut - One of the best experts on this subject based on the ideXlab platform.
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Normal Forms for poisson maps and symplectic groupoids around poisson transversals
Letters in Mathematical Physics, 2018Co-Authors: Pedro Frejlich, Ioan MarcutAbstract:Poisson transversals are submanifolds in a Poisson manifold which intersect all symplectic leaves transversally and symplectically. In this communication, we prove a Normal Form Theorem for Poisson maps around Poisson transversals. A Poisson map pulls a Poisson transversal back to a Poisson transversal, and our first main result states that simultaneous Normal Forms exist around such transversals, for which the Poisson map becomes transversally linear, and intertwines the Normal Form data of the transversals. Our second result concerns symplectic integrations. We prove that a neighborhood of a Poisson transversal is integrable exactly when the Poisson transversal itself is integrable, and in that case we prove a Normal Form Theorem for the symplectic groupoid around its restriction to the Poisson transversal, which puts all structure maps in Normal Form. We conclude by illustrating our results with examples arising from Lie algebras.
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Normal Forms for poisson maps and symplectic groupoids around poisson transversals
arXiv: Symplectic Geometry, 2015Co-Authors: Pedro Frejlich, Ioan MarcutAbstract:Poisson transversals are those submanifolds in a Poisson manifold which intersect all symplectic leaves transversally and symplectically. In a previous note we proved a Normal Form Theorem around such submanifolds. In this communication, we promote that result to a Normal Form Theorem for Poisson maps around Poisson transversals. A Poisson map pulls a Poisson transversal back to a Poisson transversal, and our first main result states that simultaneous Normal Forms exist around such transversals, for which the Poisson map becomes transversally linear, and intertwines the Normal Form data of the transversals. Our second main result concerns symplectic integrations. We prove that a neighborhood of a Poisson transversal is integrable exactly when the Poisson transversal itself is integrable, and in that case we prove a Normal Form Theorem for the symplectic groupoid around its restriction to the Poisson transversal, which puts all its structure maps in Normal Form. We conclude the paper by illustrating our results with examples arising from Lie algebras.
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the Normal Form Theorem around poisson transversals
arXiv: Symplectic Geometry, 2013Co-Authors: Pedro Frejlich, Ioan MarcutAbstract:We prove a Normal Form Theorem for Poisson structures around Poisson transversals (also called cosymplectic submanifolds), which simultaneously generalizes Weinstein's symplectic neighborhood Theorem from symplectic geometry and Weinstein's splitting Theorem. Our approach turns out to be essentially canonical, and as a byproduct, we obtain an equivariant version of the latter Theorem.
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poisson transversals i the Normal Form Theorem
arXiv: Symplectic Geometry, 2013Co-Authors: Pedro Frejlich, Ioan MarcutAbstract:We prove a local Normal Form Theorem for Poisson structures around Poisson transversals (also called cosymplectic submanifolds), which simultaneously generalizes Weinstein's splitting Theorem and Weinstein's symplectic neighborhood Theorem from symplectic geometry. The construction turns out to be essentially canonical, and, in so being, it can be easily adapted to the equivariant setting.
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A Normal Form Theorem around symplectic leaves
Journal of Differential Geometry, 2012Co-Authors: Marius Crainic, Ioan MarcutAbstract:We prove the Poisson geometric version of the Local Reeb Stability (from foliation theory) and of the Slice Theorem (from equivariant geometry), which is also a generalization of Conn’s linearization Theorem.
Pedro Frejlich - One of the best experts on this subject based on the ideXlab platform.
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Normal Forms for poisson maps and symplectic groupoids around poisson transversals
Letters in Mathematical Physics, 2018Co-Authors: Pedro Frejlich, Ioan MarcutAbstract:Poisson transversals are submanifolds in a Poisson manifold which intersect all symplectic leaves transversally and symplectically. In this communication, we prove a Normal Form Theorem for Poisson maps around Poisson transversals. A Poisson map pulls a Poisson transversal back to a Poisson transversal, and our first main result states that simultaneous Normal Forms exist around such transversals, for which the Poisson map becomes transversally linear, and intertwines the Normal Form data of the transversals. Our second result concerns symplectic integrations. We prove that a neighborhood of a Poisson transversal is integrable exactly when the Poisson transversal itself is integrable, and in that case we prove a Normal Form Theorem for the symplectic groupoid around its restriction to the Poisson transversal, which puts all structure maps in Normal Form. We conclude by illustrating our results with examples arising from Lie algebras.
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Normal Forms for poisson maps and symplectic groupoids around poisson transversals
arXiv: Symplectic Geometry, 2015Co-Authors: Pedro Frejlich, Ioan MarcutAbstract:Poisson transversals are those submanifolds in a Poisson manifold which intersect all symplectic leaves transversally and symplectically. In a previous note we proved a Normal Form Theorem around such submanifolds. In this communication, we promote that result to a Normal Form Theorem for Poisson maps around Poisson transversals. A Poisson map pulls a Poisson transversal back to a Poisson transversal, and our first main result states that simultaneous Normal Forms exist around such transversals, for which the Poisson map becomes transversally linear, and intertwines the Normal Form data of the transversals. Our second main result concerns symplectic integrations. We prove that a neighborhood of a Poisson transversal is integrable exactly when the Poisson transversal itself is integrable, and in that case we prove a Normal Form Theorem for the symplectic groupoid around its restriction to the Poisson transversal, which puts all its structure maps in Normal Form. We conclude the paper by illustrating our results with examples arising from Lie algebras.
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the Normal Form Theorem around poisson transversals
arXiv: Symplectic Geometry, 2013Co-Authors: Pedro Frejlich, Ioan MarcutAbstract:We prove a Normal Form Theorem for Poisson structures around Poisson transversals (also called cosymplectic submanifolds), which simultaneously generalizes Weinstein's symplectic neighborhood Theorem from symplectic geometry and Weinstein's splitting Theorem. Our approach turns out to be essentially canonical, and as a byproduct, we obtain an equivariant version of the latter Theorem.
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poisson transversals i the Normal Form Theorem
arXiv: Symplectic Geometry, 2013Co-Authors: Pedro Frejlich, Ioan MarcutAbstract:We prove a local Normal Form Theorem for Poisson structures around Poisson transversals (also called cosymplectic submanifolds), which simultaneously generalizes Weinstein's splitting Theorem and Weinstein's symplectic neighborhood Theorem from symplectic geometry. The construction turns out to be essentially canonical, and, in so being, it can be easily adapted to the equivariant setting.
Benoit Grebert - One of the best experts on this subject based on the ideXlab platform.
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kam for the non linear beam equation 2 a Normal Form Theorem
arXiv: Analysis of PDEs, 2015Co-Authors: Hakan L Eliasson, Benoit Grebert, Sergei KuksinAbstract:We prove an abstract KAM Theorem adapted to space-multidimensional hamiltonian PDEs with regularizing nonlinearities. It applies in particular to the singular perturbation problem studied in the first part of this work.
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Birkhoff Normal Form and Hamiltonian PDEs
2007Co-Authors: Benoit GrebertAbstract:These notes are based on lectures held at the Lanzhou university (China) during a CIMPA summer school in july 2004 but benefit from recent devellopements. Our aim is to explain some perturbations technics that allow to study the long time behaviour of the solutions of Hamiltonian perturbations of integrable systems. We are in particular interested with stability results. Our approach is centered on the Birkhoff Normal Form Theorem that we first proved in finite dimension. Then, after giving some exemples of Hamiltonian PDEs, we present an abstract Birkhoff Normal Form Theorem in infinite dimension and discuss the dynamical consequences for Hamiltonian PDEs.
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birkhoff Normal Form for partial differential equations with tame modulus
Duke Mathematical Journal, 2006Co-Authors: Dario Bambusi, Benoit GrebertAbstract:We prove an abstract Birkhoff Normal Form Theorem for Hamiltonian Partial Differential Equations. The Theorem applies to semilinear equations with nonlinearity satisfying a property that we call of Tame Modulus. Such a property is related to the classical tame inequality by Moser. In the nonresonant case we deduce that any small amplitude solution remains very close to a torus for very long times. We also develop a general scheme to apply the abstract theory to PDEs in one space dimensions and we use it to study some concrete equations (NLW,NLS) with different boundary conditions. An application to a nonlinear Schrodinger equation on the $d$-dimensional torus is also given. In all cases we deduce bounds on the growth of high Sobolev norms. In particular we get lower bounds on the existence time of solutions.
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birkhoff Normal Form for partial differential equations with tame modulus
Duke Mathematical Journal, 2006Co-Authors: Dario Bambusi, Benoit GrebertAbstract:We prove an abstract Birkhoff Normal Form Theorem for Hamiltonian partial differential equations (PDEs). The Theorem applies to semilinear equations with nonlinearity satisfying a property that we call tame modulus. Such a property is related to the classical tame inequality by Moser. In the nonresonant case we deduce that any small amplitude solution remains very close to a torus for very long times. We also develop a general scheme to apply the abstract theory to PDEs in one space dimensions, and we use it to study some concrete equations (nonlinear wave (NLW) equation, nonlinear Schrodinger (NLS) equation) with different boundary conditions. An application to an NLS equation on the d-dimensional torus is also given. In all cases we deduce bounds on the growth of high Sobolev norms. In particular, we get lower bounds on the existence time of solutions
Dario Bambusi - One of the best experts on this subject based on the ideXlab platform.
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a birkhoff Normal Form Theorem for some semilinear pdes
2008Co-Authors: Dario BambusiAbstract:In these lectures we present an extension of Birkhoff Normal Form Theorem to some Hamiltonian PDEs. The Theorem applies to semilinear equations with non- linearity of a suitable class. We present an application to the nonlinear wave equation on a segment or on a sphere. We also give a complete proof of all the results.
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birkhoff Normal Form for partial differential equations with tame modulus
Duke Mathematical Journal, 2006Co-Authors: Dario Bambusi, Benoit GrebertAbstract:We prove an abstract Birkhoff Normal Form Theorem for Hamiltonian Partial Differential Equations. The Theorem applies to semilinear equations with nonlinearity satisfying a property that we call of Tame Modulus. Such a property is related to the classical tame inequality by Moser. In the nonresonant case we deduce that any small amplitude solution remains very close to a torus for very long times. We also develop a general scheme to apply the abstract theory to PDEs in one space dimensions and we use it to study some concrete equations (NLW,NLS) with different boundary conditions. An application to a nonlinear Schrodinger equation on the $d$-dimensional torus is also given. In all cases we deduce bounds on the growth of high Sobolev norms. In particular we get lower bounds on the existence time of solutions.
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birkhoff Normal Form for partial differential equations with tame modulus
Duke Mathematical Journal, 2006Co-Authors: Dario Bambusi, Benoit GrebertAbstract:We prove an abstract Birkhoff Normal Form Theorem for Hamiltonian partial differential equations (PDEs). The Theorem applies to semilinear equations with nonlinearity satisfying a property that we call tame modulus. Such a property is related to the classical tame inequality by Moser. In the nonresonant case we deduce that any small amplitude solution remains very close to a torus for very long times. We also develop a general scheme to apply the abstract theory to PDEs in one space dimensions, and we use it to study some concrete equations (nonlinear wave (NLW) equation, nonlinear Schrodinger (NLS) equation) with different boundary conditions. An application to an NLS equation on the d-dimensional torus is also given. In all cases we deduce bounds on the growth of high Sobolev norms. In particular, we get lower bounds on the existence time of solutions
Michela Procesi - One of the best experts on this subject based on the ideXlab platform.
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an abstract birkhoff Normal Form Theorem and exponential type stability of the 1d nls
Communications in Mathematical Physics, 2020Co-Authors: Luca Biasco, Jessica Elisa Massetti, Michela ProcesiAbstract:We study stability times for a family of parameter dependent nonlinear Schrodinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we prove a rather flexible Birkhoff Normal Form Theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively.
