Periodic solutions for completely resona
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Periodic solutions for completely resona
arXiv:math/0402262vl [math.DS] 16 Feb 2004Periodic solutions for completely resonant nonlinear wave equationsGuido Gentile1, Vieri Mastropietro*, Mich Periodic solutions for completely resona hela Procesi*1 Dipartimento di Matematica, Vniversità di Roina Tre. Roma, 1-00146* Dipartiincnto di Matcmatica, Università di Roma "Tor Vergata”, Roma, 1-00133* SISSA, TYieste, 1-34014Abstract. We consider the nonlinear string equation with Dirichlet boundary conditions uxx — Utt = wilhPeriodic solutions for completely resona
o 1. This extends prernous results where only a sero-measure set of frequencies could be treated (the ones for which no small divisors appear). The prarXiv:math/0402262vl [math.DS] 16 Feb 2004Periodic solutions for completely resonant nonlinear wave equationsGuido Gentile1, Vieri Mastropietro*, Mich Periodic solutions for completely resona Fourier components, respectively the Q and the p equations, with resummation techniques of divergent powers series, allowing US to control the small divisors problem. The main difficulty with respect the nonlinear wave equations Uxx - Uti + Mu = I^(a), M Ỷ 0, « that not. only the p equation bat also Periodic solutions for completely resona the Q equation is infinite-dimensional.1. IntroductionWe consider the nonlinear wave equation in d = 1 given by/ t*M - Ur, = y(u),H O(u(0,i) = u(jr,tPeriodic solutions for completely resona
) = 0,where Dirichlet boundary conditions allow US to use as a basis in £2([0, ir]) the set of functions {sin mx,m 6 N), and ự>(u) is any odd analyticarXiv:math/0402262vl [math.DS] 16 Feb 2004Periodic solutions for completely resonant nonlinear wave equationsGuido Gentile1, Vieri Mastropietro*, Mich Periodic solutions for completely resona resonant case for the nonliueiu wave equation as in the absence of non linealities all the frequencies are resonant .In the finite dimensional case the problem lias its analogous in the study of periodic orbits close to elliptic equilibrium points: results of existence have been obtained in such a c Periodic solutions for completely resona ase by Lyapunov [27] in the nonresonant case, by Birkhoff anti Lewis [6] in case of resonances of order greater t han four, and by Weinstein [33] in cPeriodic solutions for completely resona
ase of any kind of resonances. Systems with infinitely many degrees of freedom (as the nonlinear wave equation, the nonlinear Schrodinger equation andarXiv:math/0402262vl [math.DS] 16 Feb 2004Periodic solutions for completely resonant nonlinear wave equationsGuido Gentile1, Vieri Mastropietro*, Mich Periodic solutions for completely resona h is absent in the finite dimensional case. For the nonlinear wave equations Uli — urt + Mu = ¥>(«), with mass M strictly positive, existence of periodic solutions has been proved by Craig and Wayne [13], by Poschel [29] (by adapting the analogous result found by Kuksin anti Poschcl [25] for the non Periodic solutions for completely resona linear Schrodinger equation) and1https://khothuvien.cori!by lỉourgain [8] (see also the review [12]). In order to solve the small divisors problem onePeriodic solutions for completely resona
has to require that the amplitude and frequency of the solution must belong to a Cantor set. and the main difficulty is to prove that such a set can arXiv:math/0402262vl [math.DS] 16 Feb 2004Periodic solutions for completely resonant nonlinear wave equationsGuido Gentile1, Vieri Mastropietro*, Mich Periodic solutions for completely resona in many other papers the case in which the coefficient M of the linear term is replaced by a function depending on parameters is considered; see for instance [32], [7] ami the reviews [23], Í24])-In all the quoted jMipers only non-resonant cases arc considered. Some cases with some low-order resonan Periodic solutions for completely resona ces between the frequencies have been studied by Craig ami Wayne [11], The completely resonant case (1.1) has 1>CCI1 studied with variational methodsPeriodic solutions for completely resona
starting from Rabinowitz [30], [31], [11], [10], [15], where periodic solutions with period which is a rational multiple of ĨĨ have been obtained; sucarXiv:math/0402262vl [math.DS] 16 Feb 2004Periodic solutions for completely resonant nonlinear wave equationsGuido Gentile1, Vieri Mastropietro*, Mich Periodic solutions for completely resona sure of values, has been studied so far only under strong Diophantine conditions (as the ones introduced in [2]) which essentially remove the small divisors problem leaving in fact again a zero-measure set of values [26]. [3], [4]. It is however conjectured that also for A/ = 0 periodic solutions sh Periodic solutions for completely resona ould exist for a large measure set of values of the amplitudes, see for instance [24], and indeed we prove in this paper that this is actually the casPeriodic solutions for completely resona
e: the unperturbed periodic solutions with periods = 2x/j can be continued into periodic solutions with period Wf j close to Wj. We rely on the RenormarXiv:math/0402262vl [math.DS] 16 Feb 2004Periodic solutions for completely resonant nonlinear wave equationsGuido Gentile1, Vieri Mastropietro*, Mich Periodic solutions for completely resona s us to control the small divisors problem.Note that a naive implementation of the Craig and Wayne approach to the resonant case encounters some obvious difficulties. Also in such an approach the first step consists in a Lyapunov-Schmidt decomposition, which leads to two separate sets of equations d Periodic solutions for completely resona ealing with the resonant and nonresonant Fourier components, respectively the Q and the p equations. In the nonresonant case, by calling V the solutioPeriodic solutions for completely resona
n of the Q equation and w the solution of the p equation, if one suppose to fix V in the p equation then one gets a solution u> = w(v), which insertedarXiv:math/0402262vl [math.DS] 16 Feb 2004Periodic solutions for completely resonant nonlinear wave equationsGuido Gentile1, Vieri Mastropietro*, Mich Periodic solutions for completely resona the solution w(v) of the p equation by assuming that the Fourier coefficients of I' are exponentially decreasing. However one “consumes” part of the exponential decay to Ixmnd the small divisors, hence the function u.’(v) has Fourier coefficients decaying with a smaller rate, and when inserted into Periodic solutions for completely resona the Q equation the V solution has not the properties assumed at the beginning. Such a problem can l>e avoided if there are no small divisors, as in suPeriodic solutions for completely resona
ch a case even a power law decay is sufficient to find u?(u) and there is no loss of smoothness; this is what is done in the quoted papers [26]. [3] aarXiv:math/0402262vl [math.DS] 16 Feb 2004Periodic solutions for completely resonant nonlinear wave equationsGuido Gentile1, Vieri Mastropietro*, Mich Periodic solutions for completely resona olving them toget her. As in [3] ami [5] we also consider t he problem of finding how many solutions can be obtained with given period, and we study their minimal period.If = 0 every real solution of (1.1) can be written asu(x, t) =ư„ sin iixcoa(iV„t + ớn),-1.2tt—1wherePeriodic solutions for completely resona
3where Ofc-2) denotes un analytic function of u and 4’ OÍ order at least 2 in 4', and we deline O’* v'l ÀỈ.-, with A < R. so that Wk 1 for 4’ 0.As thearXiv:math/0402262vl [math.DS] 16 Feb 2004Periodic solutions for completely resonant nonlinear wave equationsGuido Gentile1, Vieri Mastropietro*, Mich Periodic solutions for completely resona sider small and we shall show that there exists a solution of (1.3). which is 'ItỉỊúìị periodic in t and close to the functioniio(x,Lứet) = Oo(a}:t + «) —- «).-1.4provided that r. is in an appropriate Cantor set and rto(i) is the odd 2r periodic solution of the Integra differential equationAno = -3( Periodic solutions for completely resona 05)no - «0.(I-r»)where the dot denotes derivative with respect to f, and, given any periodic function /■’(£) with period T, we denote by(,fi)its averaPeriodic solutions for completely resona
ge. Then a 2it/wf-pcnodic solution of (1.1) is simply obtained by scaling hack the solution of (1.3).arXiv:math/0402262vl [math.DS] 16 Feb 2004Periodic solutions for completely resonant nonlinear wave equationsGuido Gentile1, Vieri Mastropietro*, MicharXiv:math/0402262vl [math.DS] 16 Feb 2004Periodic solutions for completely resonant nonlinear wave equationsGuido Gentile1, Vieri Mastropietro*, MichGọi ngay
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