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23
Cantor families of periodic solutions for completely resonant nonlinear wave equations.
 Duke Math. J.
, 2006
"... Abstract We prove the existence of small amplitude, (2π/ω) ..."
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Cited by 31 (9 self)
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Abstract We prove the existence of small amplitude, (2π/ω)
Periodic Solutions of Nonlinear Wave Equations with General Nonlinearities
"... We prove the existence of small amplitude periodic solutions, with strongly irrational frequency # close to one, for completely resonant nonlinear wave equations. We provide multiplicity results for both monotone and nonmonotone nonlinearities. For # close to one we prove the existence of a large nu ..."
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Cited by 28 (6 self)
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We prove the existence of small amplitude periodic solutions, with strongly irrational frequency # close to one, for completely resonant nonlinear wave equations. We provide multiplicity results for both monotone and nonmonotone nonlinearities. For # close to one we prove the existence of a large number N# of 2#/#periodic in time solutions u 1 , . . . , un , . . . , uN : N# 1. The minimal period of the nth solution un is proved to be 2#/n#. The proofs are based on a LyapunovSchmidt reduction and variational arguments.
Quasiperiodic solutions of completely resonant forced wave equations, preprint Sissa
, 2005
"... Abstract. We prove existence of quasiperiodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases the forcing frequency is: (Case A) a rational number and (Case B) an irrational num ..."
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Cited by 7 (2 self)
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Abstract. We prove existence of quasiperiodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases the forcing frequency is: (Case A) a rational number and (Case B) an irrational number.
DIOPHANTINE CONDITIONS IN WELLPOSEDNESS THEORY OF COUPLED KDVTYPE SYSTEMS: LOCAL THEORY
, 2009
"... We consider the local wellposedness problem of a oneparameter family of coupled KdVtype systems both in the periodic and nonperiodic setting. In particular, we show that certain resonances occur, closely depending on the value of a coupling parameter α when α = 1. In the periodic setting, we u ..."
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Cited by 5 (2 self)
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We consider the local wellposedness problem of a oneparameter family of coupled KdVtype systems both in the periodic and nonperiodic setting. In particular, we show that certain resonances occur, closely depending on the value of a coupling parameter α when α = 1. In the periodic setting, we use the Diophantine conditions to characterize the resonances, and establish sharp local wellposedness of the system in H s (Tλ), s ≥ s ∗, where s ∗ = s ∗ (α) ∈ ( 1,1] is determined by the Diophantine characterization of certain 2 constants derived from the coupling parameter α. We also present a sharp local (and global) result in L 2 (R). In the appendix, we briefly discuss the local wellposedness result in H −1 2 (Tλ) for α = 1 without the mean 0 assumption, by introducing the vectorvalued X s,b spaces.
Conservation of resonant periodic solutions for the onedimensional nonlinear Schrödinger equation
 Comm. Math. Phys
, 2006
"... Abstract. We consider the onedimensional nonlinear Schrödinger equation with Dirichlet boundary conditions in the fully resonant case (absence of the mass term). We investigate conservation of small amplitude periodic solutions for a large measure set of frequencies. In particular we show that ..."
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Abstract. We consider the onedimensional nonlinear Schrödinger equation with Dirichlet boundary conditions in the fully resonant case (absence of the mass term). We investigate conservation of small amplitude periodic solutions for a large measure set of frequencies. In particular we show that there are infinitely many periodic solutions which continue the linear ones involving an arbitrary number of resonant modes, provided the corresponding frequencies are large enough, say greater than a certain threshold value depending on the number of resonant modes. If the frequencies of the latter are close enough to such a threshold, then they can not be too distant from each other. Hence we can interpret such solutions as perturbations of wave packets with large wave number. 1. Introduction and
EXISTENCE AND CONTINUOUS APPROXIMATION OF SMALL AMPLITUDE BREATHERS IN 1D AND 2D KLEIN–GORDON LATTICES
, 909
"... Abstract. We construct small amplitude breathers in 1D and 2D Klein–Gordon infinite lattices. We also show that the breathers are well approximated by the ground state of the nonlinear Schrödinger equation. The result is obtained by exploiting the relation between the Klein Gordon lattice and the di ..."
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Cited by 2 (0 self)
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Abstract. We construct small amplitude breathers in 1D and 2D Klein–Gordon infinite lattices. We also show that the breathers are well approximated by the ground state of the nonlinear Schrödinger equation. The result is obtained by exploiting the relation between the Klein Gordon lattice and the discrete Non Linear Schrödinger lattice. The proof is based on a LyapunovSchmidt decomposition and continuum approximation techniques introduced in [9], actually using its main result as an important lemma. 1.
Periodic solutions of nonlinear Schrödinger equations: a paradifferential approach
, 2011
"... This paper is devoted to the construction of periodic solutions of nonlinear Schrödinger equations on the torus, for a large set of frequencies. Usual proofs of such results rely on the use of Nash–Moser methods. Our approach avoids this, exploiting the possibility of reducing, through paradifferent ..."
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This paper is devoted to the construction of periodic solutions of nonlinear Schrödinger equations on the torus, for a large set of frequencies. Usual proofs of such results rely on the use of Nash–Moser methods. Our approach avoids this, exploiting the possibility of reducing, through paradifferential conjugation, the equation under study to an equivalent form for which periodic solutions may be constructed by a classical iteration scheme.
Quasiperiodic solutions for 1D nonlinear wave equation with a general . . .
, 2004
"... In this paper, one–dimensional (1D) wave equation with a general nonlinearity utt −uxx +mu+f(u) = 0, m> 0 under Dirichlet boundary conditions is considered; the nonlinearity f is a real analytic, odd function and f(u) = au2¯r+1 + f2k+1u2k+1, a = 0 and ¯r ∈ N. It is proved that k≥¯r+1 for almos ..."
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Cited by 1 (1 self)
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In this paper, one–dimensional (1D) wave equation with a general nonlinearity utt −uxx +mu+f(u) = 0, m> 0 under Dirichlet boundary conditions is considered; the nonlinearity f is a real analytic, odd function and f(u) = au2¯r+1 + f2k+1u2k+1, a = 0 and ¯r ∈ N. It is proved that k≥¯r+1 for almost all m> 0 in Lebesgue measure sense, the above equation admits smallamplitude quasiperiodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system. The proof is based on infinite dimensional KAM theorem, partial normal form and scaling skills.