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Resonances, Radiation Damping and Instability in Hamiltonian Nonlinear

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Resonances, Radiation Damping and Instability in Hamiltonian Nonlinear Resonances, Radiation Damping and Instability in Hamiltonian Nonlinear Wave Equations y A. So er and M.I. Weinstein Abstract We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are p erturbations of linear disp ersive equations. The unp erturb ed dynamical system has a b ound state, a spatially lo calized and time p erio dic solution. We show that, for generic nonlinear Hamiltonian p erturbations, all small amplitude solutions decay to zero as time tends to in nityat an anomalously slow rate. In particular, spatially lo calized and time- p erio dic solutions of the linear problem are destroyed by generic nonlinear Hamiltonian p erturbations via slow radiation of energy to in nity. These solutions can therefore be thought of as metastable states. The main mechanism is a nonlinear resonant interaction of b ound states eigenfunctions and radiation continuous sp ectral mo des, leading to energy transfer from the discrete to continuum mo des. This is in contrast to the KAM theory in which appropriate nonresonance conditions imply the p ersistence of invariant tori. A hyp othesis ensuring that such a resonance takes place is a nonlinear analogue of the Fermi golden rule, arising in the theory of resonances in quantum mechanics. The techniques used involve: i a time-dep endent metho d develop ed by the authors for the treatment of the quantum resonance problem and p erturbations of emb edded eigenvalues, ii a generalization of the Hamiltonian normal form appropriate for in nite dimensional disp ersive systems and iii ideas from scattering theory. The arguments are quite general and we exp ect them to apply to a large class of systems which can be viewed as the interaction of nite dimensional and in nite dimensional disp ersive dynamical systems, or as a system of particles coupled to a eld. Department of Mathematics, Rutgers University, New Brunswick, NJ y Department of Mathematics, UniversityofMichigan, Ann Arb or, MI Table of Contents Section 1: Intro duction and statement of main results - Smile of the Cheshire Cat Section 2: Linear analysis Section 3: Existence theory Section 4: Isolation of the key resonant terms and formulation as a coupled nite and in nite dimensional dynamical system Section 5: Disp ersive Hamiltonian normal form Section 6: Asymptotic b ehavior of solutions of p erturb ed normal form equations Section 7: Asymptotic b ehavior of solutions of the nonlinear Klein Gordon equation Section 8: Summary and discussion 1. Intro duction and statement of main results - Smile of the Cheshire Cat Structural stability and the p ersistence of coherent structures under p erturbations of a dynami- cal system are fundamental issues in dynamical systems theory with implications in many elds of application. In the context of discrete, nite dimensional Hamiltonian systems, this issue is addressed by the celebrated KAM theorem [3], which guarantees the p ersistence of most in- variant tori of the unp erturb ed dynamics under small Hamiltonian p erturbations. For in nite dimensional systems, de ned by Hamiltonian partial di erential equations PDEs, KAM typ e metho ds have recently b een used to obtain results on the p ersistence of p erio dic and quasi- p erio dic solutions, in the case where solutions are de ned on compact spatial domains with appropriate b oundary conditions [4], [17], [37]. Variational metho ds have also b een used to study this problem; see, for example, [6]. The compactness of the spatial domain ensures dis- creteness of the sp ectrum asso ciated with the unp erturb ed dynamics. Therefore this situation is the generalization of the nite dimensional case to systems with an in nite numb er of discrete oscillators and frequencies. In this pap er we consider these questions in the context of Hamiltonian systems for which the unp erturb ed dynamics has asso ciated with it discrete and continuous sp ectrum. This situation arises in the study of Hamiltonian PDEs governing functions de ned on unb ounded spatial domains or, more generally, extended systems. The physical picture is that of a system which can b e viewed as an interaction b etween one or more discrete oscillators and a eld or continuous medium. In contrast to the KAM theory, where nonresonance implies p ersistence, we nd here that resonant nonlinear interaction between discrete b ound state mo des and continuum 2 disp ersive radiation mo des leads to energy transfer from the discrete to continuum mo des. This mechanism is resp onsible for the eventual time-decay and nonp ersistence of trapp ed states. The rate of time-decay,however, is very slow and hence such a trapp ed state can b e thoughtofasa metastable state. The metho ds we develop are applicable to a large class of problems which can be viewed schematically in terms of "particle- eld interactions". In the present work, we have not at- tempted to present general results under weak hyp otheses but rather have endeavored to illus- trate, by way of example, this widely o ccurring phenomenon and clearly present the strategy for its analysis. A more general p oint of view will be taken up in future work. The approach we use was motivated and in part develop ed in the context of our study of a class of nonlinear Schrodinger equations with multiple nonlinear b ound states and the quantum resonance prob- lem [60], [61], [62], [63]. See also [11],[12] and [46]. Related problems are also considered in [32] and [44]. n We b egin with a linear disp ersive Hamiltonian PDE for a function ux; t, x 2 IR and t> 0. Supp ose that this system has spatially lo calized and time-p erio dic solutions. Such solutions are often called bound states. Atypical solution to such a linear system consists of i a non-decaying part, expressible as a linear combination of b ound states, plus ii a part which decays to zero in suitable norms disp ersion. This pap er is devoted to the study of the following questions: 1 Do small amplitude spatially lo calized and time-p erio dic solutions p ersist for typical non- linear and Hamiltonian p erturbations? 2 What is the character of general small amplitude solutions to the p erturb ed dynamics? 3 How are the structures of the unp erturb ed dynamics manifested in the p erturb ed dynamics? Representative of the class of equations of interest is the nonlinear Klein-Gordon equation: 2 2 @ u = V x m u + f u; 2 IR 1.1 t with f u real-valued, smo oth in a neighb orho o d of u = 0 and having an expansion: 3 4 f u=u + O u : 1.2 3 Here, u :x; t 7! ux; t 2 IR, x 2 IR , and t>0. We shall restrict our large time asymptotic 3 analysis to the case f u=u . The more general case 1.2 can be treated by the technique of s;p this pap er by making suitable mo di cation of W norms used. We consider 1.1 with Cauchy data: ux; 0 = u x; and @ ux; 0 = u x: 1.3 0 t 1 3 Equation 1.1 is a Hamiltonian system with energy: Z Z 1 2 2 2 2 2 @ u + jruj + m u + V xu dx + F u dx; 1.4 E [u; @ u] t t 2 0 where F u= f u and F 0=0. In the context of equations of typ e 1.1, we have found the following answers to questions 1, 2 and 3. A1 In a small op en neighb orho o d of the origin, there are no p erio dic or quasip erio dic solutions; Corollary 1.1. A2 All solutions in this neighb orho o d tend to zero radiate as t !1; Theorem 1.1. A3 The time decay of solutions is anomalously slow , i.e. a rate which is slower than the free disp ersive rate; Theorem 1.1. Dynamical systems of the typ e we analyze app ear in a numberofphysical settings. Consider a nonlinear medium in which waves can propagate. If the medium has lo cal inhomogeneities, defects or impurities, these arise in the mathematical mo del as a spatially dep endent co ecient in the equation e.g. lo calized p otential. Such p erturbations of the original homogeneous translation invariant dynamics intro duce new mo des into the system impurity modes which can trap some of the energy and a ect the time evolution of the system; see [41], [35], [73]. 1 2 2 Let hK i = 1 + jK j . For the nonlinear Klein-Gordon equation, 1.1, we prove the following result. Theorem 1.1. Let V x be real-valued and such that V1 For >5 and j j 2, j@ V xjC hxi . l 1 2 1 x r V x +1 is b ounded on L for jl j N with N 10. V2 +1 V3 Zero is not a resonance of the op erator +V ; see [31], [72]. Assume the op erator 2 2 B = +V x+m 1.5 2 has continuous sp ectrum, H =[m ; 1, and a unique strictly p ositive simple eigenvalue, cont 2 2 <m with asso ciated normalized eigenfunction, ': 2 2 B ' = ': 1.6 Corresp ondingly, the linear Klein-Gordon equation 1.1, with = 0, has a two-parameter family of spatially lo calized and time-p erio dic solutions of the form: ux; t = R cos t + 'x: 1.7 4 Assume the resonance condition 2 3 3 3 3 > 0: 1.8 P ' ;B 3 P ' F ' c c c 3 3 Here, P denotes the pro jection onto the continuous sp ectral part of B and F denotes the c c Fourier transform relativetothe continuous sp ectral part of B .
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