KAM for the nonlinear beam equation
In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus u t t + Δ 2 u + m u + ∂ u G ( x , u ) = 0 , t ∈ R , x ∈ T d , ( ∗ ) where G ( x , u ) = u 4 + O ( u 5 ) . Namely, we show that, for generic m , many of the small amplitude in...
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| Published in | Geometric and functional analysis Vol. 26; no. 6; pp. 1588 - 1715 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
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Springer International Publishing
01.12.2016
Springer Nature B.V Springer Verlag |
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| Abstract | In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus
u
t
t
+
Δ
2
u
+
m
u
+
∂
u
G
(
x
,
u
)
=
0
,
t
∈
R
,
x
∈
T
d
,
(
∗
)
where
G
(
x
,
u
)
=
u
4
+
O
(
u
5
)
. Namely, we show that, for generic
m
, many of the small amplitude invariant finite dimensional tori of the linear equation
(
∗
)
G
=
0
, written as the system
u
t
=
-
v
,
v
t
=
Δ
2
u
+
m
u
,
persist as invariant tori of the nonlinear equation
(
∗
)
, re-written similarly. The persisted tori are filled in with time-quasiperiodic solutions of
(
∗
)
. If
d
≥
2
, then not all the persisted tori are linearly stable, and we construct explicit examples of partially hyperbolic invariant tori. The unstable invariant tori, situated in the vicinity of the origin, create around them some local instabilities, in agreement with the popular belief in the nonlinear physics that small-amplitude solutions of space-multidimensional Hamiltonian PDEs behave in a chaotic way. |
|---|---|
| AbstractList | In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus u t t + Δ 2 u + m u + ∂ u G ( x , u ) = 0 , t ∈ R , x ∈ T d , ( ∗ ) where G ( x , u ) = u 4 + O ( u 5 ) . Namely, we show that, for generic m, many of the small amplitude invariant finite dimensional tori of the linear equation ( ∗ ) G = 0 , written as the system u t = - v , v t = Δ 2 u + m u , persist as invariant tori of the nonlinear equation ( ∗ ) , re-written similarly. The persisted tori are filled in with time-quasiperiodic solutions of ( ∗ ) . If d ≥ 2 , then not all the persisted tori are linearly stable, and we construct explicit examples of partially hyperbolic invariant tori. The unstable invariant tori, situated in the vicinity of the origin, create around them some local instabilities, in agreement with the popular belief in the nonlinear physics that small-amplitude solutions of space-multidimensional Hamiltonian PDEs behave in a chaotic way. In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus u t t + Δ 2 u + m u + ∂ u G ( x , u ) = 0 , t ∈ R , x ∈ T d , ( ∗ ) where G ( x , u ) = u 4 + O ( u 5 ) . Namely, we show that, for generic m , many of the small amplitude invariant finite dimensional tori of the linear equation ( ∗ ) G = 0 , written as the system u t = - v , v t = Δ 2 u + m u , persist as invariant tori of the nonlinear equation ( ∗ ) , re-written similarly. The persisted tori are filled in with time-quasiperiodic solutions of ( ∗ ) . If d ≥ 2 , then not all the persisted tori are linearly stable, and we construct explicit examples of partially hyperbolic invariant tori. The unstable invariant tori, situated in the vicinity of the origin, create around them some local instabilities, in agreement with the popular belief in the nonlinear physics that small-amplitude solutions of space-multidimensional Hamiltonian PDEs behave in a chaotic way. In this paper we prove a KAM result for the non linear beam equation on the d-dimensional torusutt+∆2u+mu+g(x,u)=0, t∈R,x∈Td, (∗)where g(x,u) = 4u3 +O(u4). Namely, we show that, for generic m, many of the small amplitude invariant finite dimensional tori of the linear equation (∗)g=0, written as the systemut =−v, vt =∆2u+mu,persist as invariant tori of the nonlinear equation (∗), re-written similarly. If d ≥ 2, then not all the persisted tori are linearly stable, and we construct explicit examples of partially hyperbolic invariant tori. The unstable invariant tori, situated in the vicinity of the origin, create around them some local instabilities, in agreement with the popular belief in the nonlinear physics that small-amplitude solutions of space-multidimensonal hamiltonian PDEs behave in a chaotic way. |
| Author | Kuksin, Sergei B. Grébert, Benoît Eliasson, L. Hakan |
| Author_xml | – sequence: 1 givenname: L. Hakan surname: Eliasson fullname: Eliasson, L. Hakan organization: Université Paris Diderot, Sorbonne Paris Cité, Institut de, Mathémathiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS, Sorbonne Universités, UPMC Univ. Paris 06 – sequence: 2 givenname: Benoît surname: Grébert fullname: Grébert, Benoît organization: Laboratoire de Mathématiques Jean Leray, Université de Nantes, UMR CNRS 6629 – sequence: 3 givenname: Sergei B. surname: Kuksin fullname: Kuksin, Sergei B. email: kuksin@gmail.com, sergei.kuksin@imj-prg.fr organization: CNRS, Institut de Mathémathiques de Jussieu-Paris Rive Gauche, UMR 7586, Université Paris Diderot, Sorbonne Paris Cité, Sorbonne Universités, UPMC Univ. Paris 06 |
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| Keywords | Hamiltonian systems Beam equation KAM theory hamiltonian systems |
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| Snippet | In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus
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,... In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus u t t + Δ 2 u + m u + ∂ u G ( x ,... In this paper we prove a KAM result for the non linear beam equation on the d-dimensional torusutt+∆2u+mu+g(x,u)=0, t∈R,x∈Td, (∗)where g(x,u) = 4u3 +O(u4).... |
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| SubjectTerms | Analysis Analysis of PDEs Invariants Mathematics Mathematics and Statistics Nonlinear equations Toruses |
| Title | KAM for the nonlinear beam equation |
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