A variational approach to homoclinic orbits in Hamiltonian systems

Zelati, Vittorio Coti; Ekeland, Ivar; Séré, Éric

doi:10.1007/bf01444526

Cited by 245 publications

(178 citation statements)

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“…The first paper where modern global variational methods have been employed in order to find homoclinic solutions for a Hamiltonian system seems to be [12]. The Hamiltonian considered there was of the form H(z, t) = 1 2 Az · z + G(z, t), with G periodic in t, strictly convex in z and G z superlinear.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Multibump solutions and critical groups

Arioli

Szulkin

Zou

2009

Trans. Amer. Math. Soc.

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Abstract. We consider the Newtonian system −q + B(t)q = W q (q, t) with B, W periodic in t, B positive definite, and show that for each isolated homoclinic solution q 0 having a nontrivial critical group (in the sense of Morse theory), multibump solutions (with 2 ≤ k ≤ ∞ bumps) can be constructed by gluing translates of q 0 . Further we show that the collection of multibumps is semiconjugate to the Bernoulli shift. Next we consider the Schrödinger equa-

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Multibump solutions and critical groups

Arioli

Szulkin

Zou

2009

Trans. Amer. Math. Soc.

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“…By (L), 2 1 00 2 Ilqll = _oo(lql +L(t)q·q)dt can and will be taken as an equivalent norm on E . Hence I can be written as (0.2) I(q) = ~lIq112 -i: V(t, q) dt.…”

Section: I(q) = I: [~(Lqi2 + L(t)q· Q) -V(t Q)] Dtmentioning

confidence: 99%

Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials

Zelati¹,

Rabinowitz²

1991

J. Amer. Math. Soc.

319

175

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The goal of this paper is to establish the existence of infinitely many homoclinic orbits for a class of second order Hamiltonian systems of the form:Here q E R n , and we assume the n x n matrix L(t) satisfies (L) is T-periodic in t, and is symmetric and positive definite uniformly for Integrating (V 3 ) shows V(t, q) = o( lqI2) as Iql ~ 0 and V(t, q)lql-2 ~ 00 as Iql ~ 00 , i.e., V is a "superquadratic" potential.Our approach to (HS) involves the use of variational methods of a mini-max nature. To describe them more fully, let E = Wi ,2(R, Rn) under the usual normThus E is a Hilbert space and it is not difficult to show that E C CO (R, Rn) , the space of continuous functions q on R such that q(t) ~ 0 as It I ~ 00 (see,

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“…Existence of solitary waves has been proved in many other physical systems exhibiting a similar cubic expansion of the dispersion relation at the origin. Methods include calculus of variations [8], abstract fixed point arguments and implicit function theorem [3,11], and spatial dynamics [19,20]. However, proofs of the stability of solitary waves rely almost exclusively on variational methods, exploiting definiteness of the energy restricted to a fixed value of the impulse functional [5,6].…”

Section: Introductionmentioning

confidence: 99%