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 u x; 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! u x; 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:
u x; 0 = u x ; and @ u x; 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 x u 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 x jC 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
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: u x; 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 . 2;2 2;1 1;2 1;1 Assume that the initial data u , u are such that the norms ku k and ku k 0 1 0 1 W \W W \W are suciently small. Then, the solution of the initial value problem for 1.1 , with 6=0 and 3 f u =u decays as t !1. The solution u x; t has the following expansion as t !1: u x; t = R t cos t + t ' x + x; t ; 1.9 where 1 1 3 4 2 4 R t =O jtj ; t = O jtj ; and k ;t k = O jtj : 1.10 8 More precisely, 2 ~ ~ ~ R = R + O jR j ; jRj small 1 1 3 4 1 2 ~ ~ ~ 4 4 R jtj R 1 + 1+O jtj ; >0 R = 2 0 0 4 2 2 2 2 R 0 = R ; R = j '; u j + j '; u j 0 0 1 0 Corollary 1.1. Under the hyp otheses of Theorem 1.1, there are no p erio dic or quasip erio dic orbits of the ow t 7! u t ;@ u t de ned by 1.1 in a suciently small neighborhood of the t 2;1 2;2 1;2 1;1 origin in the space W \ W W \ W . It is interesting to contrast our results with those known for Hamiltonian partial di erential equations for a function u x; t , where x varies over a compact spatial domain, e.g. p erio dic or Dirichlet b oundary conditions [4], [17], [37]. For nonlinear wave equations of the form, 1.1 , with periodic boundary conditions in x, KAM typ e results have b een proved; invariant tori, asso ciated with a nonresonance condition p ersist under small p erturbations. The nonresonance hyp otheses of such results fail in the current context , a consequence of the continuous sp ectrum asso ciated with unb ounded spatial domains. In our situation, non-vanishing resonant coupling condition 1.8 provides the mechanism for the radiative decay and therefore nonp ersistence of lo calized p erio dic solutions. Remarks: 1 The condition 1.8 is a nonlinear variant of the Fermi golden rule arising in quantum mechanics; see, for example, [18], [62]. This condition holds generically in the space of p otentials 5 satisfying the hyp othesis of the theorem. Note that the condition 1.8 implies that 3 2 B 1.11 cont i.e. the frequency, 3 , generated by the cubic nonlinearity lies in the continuous sp ectrum of B . The hyp othesis 1.8 ensures a nonvanishing coupling to the continuous sp ectrum. 2 The regularity and decayhyp otheses on V x are related to the techniques we use to obtain 2 p suitable decay estimates on the linear evolution in L hxi dx and L ; see section 2. 3 Persistence of dynamical ly stable bound states for special perturbations: There are nonlinear p erturbations of the linear Klein-Gordon equation, 1.1 with = 0, for which there is a p ersis- tence of time-p erio dic and spatially lo calized solutions. Supp ose we extend our considerations n to the class of complex-valued solutions: u :IR IR ! C, and study the p erturb ed equation: 2 p 1 u = 0; 1.12 @ + V x + juj t n+2 where 1 < p < 1 for n = 1; 2 and 1 < p < for n 3. Unlike 1.1 , equation 1.12 n 2 i has the symmetry u 7! ue and it has b een shown [51] that 1.12 has time p erio dic and i! t 1 spatially lo calized solutions of the form e x; ! with 2 H , which bifurcate from the 2 zero solution at the p oint eigenvalue of +V x ! , by global variational [39], [65] and lo cal bifurcation metho ds [45]. There are numerous other examples of equations for which the p ersistence of coherent structures under p erturbations is linked to the p erturbation resp ecting a certain symmetry of the unp erturb ed problem. The stability of small amplitude bifurcating states of 1.12 can be proved using the metho ds of [24], [70]. An asymptotic stability and scattering theory has b een develop ed for nonlinear Schrodinger dynamics NLS in [60]. If the p otential V x supp orts more than one b ound state, the ab ove metho ds can be used to show the p ersistence of nonlinear excited states. However, it is shown in [63] that the NLS excited states are unstable, due to a resonant mechanism of the typ e studied here for 1.1 . 4 In contrast with the time-decay rates asso ciated with linear disp ersive equations, the time- decay rates of typical solutions describ ed by Theorem 1.1 is anomalously slow. See section 8 for a discussion of this p oint. Wenow present an outline of the ideas b ehind our analysis. For = 0, solutions of equation 1.1 are naturally decomp osed into their discrete and continuous sp ectral comp onents: u x; t = a t ' x + x; t ; '; ;t = 0; 1.13 6 2 where f; g denotes the usual complex inner pro duct on L . The functions a t and x; t satisfy system of decoupled equations: 00 2 a + a = 0; 1.14 2 2 @ + B = 0; 1.15 t with initial data 0 a 0 = '; u ; a 0 = '; u 0 1 x; 0 = P u ; @ x; 0 = P u 1.16 c 0 t c 1 For 6= 0, and for small amplitude initial conditions, we use the same decomp osition, 1.13 . Now the discrete and the continuum mo des are coupled and the dynamics are qualitatively describ ed by the following model system: 00 2 2 a + a = 3a ; ;t 1.17 2 2 3 @ + m = a : 1.18 t 3 Here, x is a lo calized function of x; in particular, = ' , and we assume see 1.8 3 >m. 2 In selecting the mo del problem 1.17-1.18 , we have replaced B , restricted to its continuous 2 2 sp ectral part, by B = +m , which intuitively should lead to the same qualtitative result. 0 The system 1.17-1.18 can be interpreted as a system governing the dynamics of discrete oscillator, with amplitude a t and natural frequency , coupled to a continuous medium in 1 which waves, of amplitude x; t , propagate, or as an oscillating particle coupled to a eld. The system 1.17-1.18 is a Hamiltonian system with conserved total energy functional: Z 1 1 2 0 0 2 2 2 2 2 2 ~ E [ ; @ ; a; a ] a @ + jr j + m dx + + a t t 2 2 Z 3 a x x; t dx 1.19 We now solve 1.18 and substitute the result into 1.17 . Note that, to leading order, 3 solutions of 1.17 oscillate with frequency . Therefore the a term in 1.18 acts as an external 2 2 driving force with a 3 frequency comp onent. Since 9 is larger than m , a nonlinear resonance of the oscillator with the continuum takes place. To calculate the e ect of this resonance requires a careful analysis involving i a study of singular limits of resolvents as an eigenvalue parameter 1 A related example is a mo del intro duced in 1900 by H. Lamb [38], governing the oscillations of mass-spring- string system. See also [7], [21], [53], [30]. 7 approaches the continuous sp ectrum see section 4 and ii and the derivation of a normal form which is natural for an in nite dimensional conservative system with disp ersion see section 5 . This leads to an equation of the following typ e for a t or rather some near-identity transform of it : 00 2 2 4 0 a + + O jaj a = r a ; t>0 1.20 where r = O jaj , and > 0 is the p ositivenumb er given in 1.8 . For the mo del system 1.18 , F in 1.8 is replaced by the usual Fourier transform. This is the equation of a nonlinearly c damp ed oscillator: d 4 0 2 0 2 2 2 2 = 2 r a < 0; for t>0 1.21 a +[ + O jaj ]a dt Therefore, nonlinear resonance is resp onsible for internal damping in the system; energy is lost or rather transferred from the discrete oscillator into the eld or continuous medium, where it 1=4 is propagated to in nity as disp ersive waves. Solutions of 1.20 decay with a rate O t , 3=4 and it follows that k ;t k = O t : Note, however that the total energy of the system, an 1 1 H typ e quantity, is conserved. In 1.20 , we have neglected higher order e ects coming from the continuous sp ectral part of H . These, it turns out, has a small e ect and can be treated p erturbatively. Asp ects of the analysis are related to our recent treatment of the quantum resonance problem [61], [62]. The situation with quantum resonances can b e summarized brie y as follows. Let H 0 be a self-adjoint op erator having an eigenvalue, , emb edded in its continuous sp ectrum, with 0 corresp onding normalized eigenfunction . Let W b e a small lo calized symmetric p erturbation, 0 satisfying the generically valid Fermi golden rule resonance condition: W ; H P W 6=0: 1.22 0 0 0 0 c 0 Then in a neighborhood of , we show that the sp ectrum of H = H + W is absolutely 0 0 continuous by proving that all solutions of the Schrodinger equation: i @ = H = H + W ; 1.23 t 0 with initial data which is sp ectrally lo calized with resp ect to H in a neighb orho o d of , decay 0 to zero in a lo cal energy norm as t !1. Such solutions are characterized by exp onential time-decay for an initial transient p erio d, and then algebraic disp ersive decay thereafter. During this transient p erio d, A t , the pro jection onto the mo de , is governed by the equation: 0 0 A t = i A t ; t>0 1.24 0 8 where + ;W , and by 1.22 , > 0. 0 0 0 0 In the class of nonlinear problems under consideration, if we express the amplitude, a t , as i t i t a t = A t e + A t e 1.25 then we nd after a near-identity transformation an equation of the form: 2 3 0 2 4 A = ic jAj A + ic jAj A; t>0: 1.26 21 32 4 1=4 From this, the O t b ehavior is evident. Equations 1.25-1.26 lead to 1.20 . As in 1.20 , we have in 1.24 and 1.26 neglected the higher order coupling to the continuous sp ectral radiation comp onent of the solution. These contributions are treated p erturbatively. Remarks: 1 Lamb shift: Note that in the nonlinear problem, asymptotically there is no Lamb shift typ e 2 1=2 correction to the frequency; the frequency shift is O jAj =O t as t !1. 2 Emergence of irreversible behavior from reversible dynamics; dissipation through dispersion: Being Hamiltonian, the underlying equation of motion, 1.1 , is time reversible. In particular, the equation has the invariance: u x; t 7! u x; t . Yet, the equation in 1.26 is clearly not time-reversible. This apparent paradox is related to the ""-prescription" discussed in section 4 and Prop osition 2.1; the singular limit p 2 1 2 +m t +m E i" 1.27 lim exp i "0 satis es a lo cal decay estimate as t ! 1. For t < 0, the corresp onding equation for A t would have replaced by + ; cf. [2], [12], [61], [62]. The nonp ersistence of small amplitude spatially lo calized and time-p erio dic or quasip erio dic solutions to 1.1 has b een studied in [56],[57]. In this work, the phenomenon of nonp ersistence is formulated as a question concerning the instability of emb edded eigenvalues of a suitable linear self-adjoint op erator. A result on structural instability is proved; under the hyp othesis 1.8 , time-p erio dic or quasip erio dic solutions of the linear problem equation = 0 do not continue to nearby solutions of the nonlinear problem 6= 0 . The question of nonp ersistence of p erio dic solutions has also b een considered extensively in the context of the sine-Gordon equation 2 2 @ u = @ u sin u: 1.28 t x 9 Spatially lo calized and time-p erio dic solutions of the sine-Gordon equation are called breathers and the question of their p ersistence under small Hamiltonian p erturbations in the dynamics has b een the sub ject of extensive investigations. See, for example, [54], [16], [13] [69], [8], [9], [19], [34], [47], [66], [42]. Analytical, formal asymptotic and numerical studies strongly suggest that for typical Hamil- 4 tonian p erturbations of the sine-Gordon equation, for example, the mo del: 2 2 3 @ u = @ u u + u ; 1.29 t x no small amplitude breathers exist and that solutions obtained from spatially lo calized initial 2 data radiate to zero very slowly as t tends to in nity. We b elieve this is related to the mechanism for slow radiative decay, as explained by Theorem 1.1 in the context of 1.1 . Finally,we wish to comment on the connection b etween our work and the approach taken in [56], [8], [19], and [34]. As in the continuation theory of p erio dic solutions of ordinary di erential equations [15], it is natural to seek a p erio dic solution of 1.1 for 6=0 which b ehaves, as the amplitude a tends to zero, like a solution 1.7 of the linear limit problem. The equation is 1 autonomous with resp ect to time, so we seek a 2 -p erio dic solution for 6= 0 with an a amplitude dep endent p erio d. Since we do not know the p erio d a priori, it is convenient to de ne u x; t = U x; s ; s = t 1.30 a a and require that U x; s be 2 p erio dic in s. Thus 1.1 b ecomes a 2 2 2 3 @ + B U = U : 1.31 a a s a We formally expand the solution and frequency: 3 U = aU + a U + ::: a 1 3 2 = + a + ::: : 1.32 a 2 Substitution of 1.32 into 1.31 and assembling terms according to their order in a, one gets a hierarchy of equations b eginning with 1 2 2 2 O a : @ + B U = 0 1.33 1 s 3 2 2 2 3 2 O a : @ + B U = U 2 @ U 1.34 3 2 1 s 1 s 2 2 In [40], breather typ e solutions have b een constructed for the discrete sine-Gordon equation, where @ , is x replace by its discretization on a suciently coarse lattice. Also, a generalization of the notion of breather has b een considered in the geometric context of wave maps [55]. 10 Equation 1.33 has a solution U x; s = cos s' x : 1.35 1 Substitution into 1.34 gives the following explicit equation for U : 3 ! 3 2 2 2 3 3 U = @ + B ' + 2 ' cos s + ' cos 3s 1.36 3 2 s 4 4 1 2 We now express U in the form U = U + U , where 3 3 3 3 ! 3 1 2 2 2 3 @ + B U = ' + 2 ' cos s 1.37 2 3 s 4 2 3 2 2 2 ' cos 3s: 1.38 U = @ + B 3 s 4 2 Since the inhomogeneous term in 1.37 is nonresonant with the continuous sp ectrum of B , one 1 3 2 1 can nd a 2 -p erio dic solution U provided the right hand side is L S IR - orthogonal to 3 2 the adjoint zero mo de: cos s' x . This latter condition uniquely determines . Note however 2 2 that the right hand side of 1.38 is resonant with the continuous sp ectrum of B if 3 >m. To 2 = cos 3s F . compute the obstruction to solvability, seek a solution of 1.38 of the form U 3 Then, 2 2 3 B 9 F = ' 1.39 4 2 3 and thus we exp ect to nd F 2 L if and only if the comp onent of ' in the direction of the generalized eigenfunction at frequency 3 of the op erator B vanishes. Therefore, if the nonlinear 3 2 1 Fermi golden rule 1.8 holds, our formal expansion in L S IR breaks down. The relation 2 with our work is that we prove that this obstruction to solvability, in fact, implies the radiative b ehavior of solutions describ ed in our main theorem. Acknowledgments: The problem studied in this pap er was raised by T.C. Sp encer in lectures and informal discussions in the late 1980's. M.I.W. learned of this problem from T.C. Sp encer and A.S. from I.M. Sigal. The authors wish to thank them for their insights and continued interest. The authors also wish to thank B. Birnir for stimulating discussions, and R. Pyke and J.B. Rauch for their careful reading of and thoughtful comments on the manuscript. This research was carried out while MIW was on sabbatical leave in the Program in Applied and Computational Mathematics at Princeton University. MIW would like to thank Phil Holmes for his hospitality and for providing a stimulating research environment. This research was supp orted in part by NSF grant DMS-9401777 and an FAS-Rutgers grant award AS and by NSF grant DMS-9500997 MIW . 11 2. Linear Analysis In this section we summarize the to ols of linear analysis employed in this pap er. 2.1. Estimates for the free Klein Gordon equation We b egin by considering the Cauchy problem for the linear Klein-Gordon equation for a function n u x; t ; x 2 IR ; t 6=0. 2 2 2 @ u = m u = B u 2.1 t 0 u x; 0 = u x ; @ u x; 0 = u x : 2.2 0 t 1 0 0 There exist op erators E t and E t such that 0 1 0 0 u x; t = E t u + E t u : 2.3 0 1 0 1 We write 0 E t f = cos B t f ; and 0 0 sin B t 0 0 g ; 2.4 E t g = 1 B 0 p 2 2 where ! k = m + k : Z 0 ik x ^ E t f = cos ! k t f k e dk 0 Z sin ! k t 0 ik x g^ k e dk : 2.5 E t g = 1 ! k The rst result we cite is proved using stationary phase metho ds; see [5]. See [36] for another n s;p approach to decay estimates for 2.1 . Let W IR denote the Sob olev space of functions with p derivatives of order s in L . 1 1 1 1 Theorem 2.1. Let 1 0 0 p p 2 p a if n +1+ + s, we have for =0; 1: 0 0 k E g k K t kg k ; t 0; where 2.6 p s;p n 1 K t = Ct ; 0 n 1+ = Ct ; t 1: 2.7 1 1 0 n 0 p 0 2 p b If s 1=2 1=p n +2 1, then for t 1 the Schrodinger-like L decay rate, t , holds in 2.7 . 12 In subsequent sections, we shall use some sp eci c consequences of this result. Corollary 2.1. Consider the linear Klein Gordon equation 2.1 in dimension n =3. Then, 9 0 8 8 t g jj C t jjE jjg jj t 1 2.8 8 1 1; 7 3 0 8 8 t g jj C t jjE 0 8 1 1; 7 1 0 2 4 t g jj C t jjE jjg jj ; t>0; = 0; 1: 2.10 4 1; 3 0 proof: Estimates 2.8 and 2.9 follow from the theorem with the choice of parameters: p =8, s =1,n = 3, and =1. Estimate 2.10 follows from the theorem with the choice of parameters: 0 p =4, s =1, n = 3 and =0. 2.2. Estimates for the Klein Gordon equation with a p otential We now consider the Cauchy problem for the linear Klein-Gordon equation with a p otential 2 2 2 @ u = m V x u = B u 2.11 t 2 where B is p ositive and self-adjoint. We write the solution to the initial value problem for 2.11 with initial data 2.2 as: u x; t = E t u + E t u ; where 2.12 0 0 1 1 E t f = cos Bt f ; and 2.13 0 sin Bt g : E t g = 1 B One exp ects that estimates of the typ e app earing in Theorem 2.1 and Corollary 2.1 will hold 2 2 as well for E t and E t restricted to P L , the continuous sp ectral part of H = B . One can 0 1 c 2 relate functions of the op erator H , on its continuous sp ectral part, to functions of H = B 0 0 using wave operators. Let itH itH 0 W = strong lim e e : 2.14 + t!1 2 Wave op erators relate H to H on P L via the intertwining prop erty: 0 c 2 H = W H W on P L : 2.15 + 0 c + n k;p K. Ya jima [72] has proved the W IR b oundedness of wave op erators for the Schrodinger equation. A consequence of this work is the following result for spatial dimension n =3: 13 3 Theorem 2.2. Let H = +V , where V x be real-valued function on IR satisfying the following hyp otheses, which are satis ed by smo oth and suciently rapidly decaying p otentials: For any j j ` there is a constant C such that @ V x C hxi ; >5 : @x Additionally, assume that 0 is neither an eigenvalue nor a resonance of H ; see [72], [31]. If 0 0 0 k; k l and 1 p; p 1, then there exists a constant C > 0 such that for any Borel 1 function f on R we have 1 0 0 0 0 0 0 C jjf H jj jjf H P H jj C jjf H jj : k;p k ;p k;p k ;p k;p k ;p 0 c 0 B W ;W B W ;W B W ;W Here, P H denotes the pro jection onto the continuous sp ectral part of the op erator H . c Remark: In our applications, we shall use Theorem 2.2 with jl j 2. Theorem 2.2 implies that the disp ersive estimates for the free Klein Gordon group carry over to the op erators E t and E t restricted to the continuous sp ectral part of B . 0 1 Theorem 2.3. If g 2 Range P H , then each of the estimates of Theorem 2.1 and Corollary c 0 2.1 hold with E replaced by E . j j 2.3. Singular resolvents and time decay As discussed in the intro duction, the damping term in the e ective nonlinear oscillator 1.26 , is related to the evaluation of a singular limit of the resolvent as the eigenvalue parameter approaches the continuous sp ectrum. To ensure that the correction terms to the nonlinear oscillator 1.26 can b e treated p erturbatively,we require lo cal decay estimates for the op erator iB t 1 e B i0 , where is a p oint in the interior of the continuous sp ectrum of B >m . Such estimates are analogous to those used by the authors in recent work on a time dep endent approach to the quantum resonance problem [62]. We b egin with a prop osition, which is essentially a restatement of Theorem 2.3 for E t , j and is proved the same way. s;p iB t Prop osition 2.1. W estimates for e Assume that V x satis es the hyp otheses of The- 0 orem 2.2 n =3 ; see [72] for hyp otheses for general space dimension, n 3. Let p and p be as 1 1 in Theorem 2.1 and let l s n +2 , where l is as in Theorem 2.2. Then, 0 2 p 1 1 n iB t 0 2 p 0 k e P k C jtj k k ; t 6=0; 2.16 c p s;p 14 The next prop osition is the "singular" resolvent estimate we shall require in section 7. Let n 2n n = maxf ; 2+ g. 2 n+2 Prop osition 2.2. Assume the hyp otheses of Prop osition 2.1. Additionally, assume that V x satis es hyp othesis V2 of Theorem 1.1. Also, let > n . Then, for any p oint >m in the continuous sp ectrum of B we have for l =0; 1; 2: 2n l iB t n+2 khxi e B +i0 P hxi k C hti k k ; t>0; c 2 1;2 2n l iB t n+2 khxi e B i0 P hxi k C hti k k ; t<0; 2.17 c 2 1;2 proof: We prove the estimate 2.17 for the case of the t > 0. For the case, t < 0, the same argument with simple mo di cations applies. Let g = g B denote a smo oth characteristic functions of an op en interval, which contains and is contained in the continuous sp ectrum of B . Also, let g 1 g . We then write, for l =1; 2: l l l iB t iB t iB t e B +i" P = e g B B +i" P + e g B B +i" P c c c " " S t + S t : 2.18 1 2 " We now estimate the op erators S lim S , j =1; 2 individually. j "!0 j Estimation of S t : 1 Consider the case l =2; the case l = 1 is simpler. First, we note that: " hxi S t hxi 1 Z Z 1 1 it i" i B +i" = e ds d hxi e g B P hxi 2.19 c t s Now we claim that the op erator in the integrand satis es the estimate: " e i B +i" 2 khxi e g B P hxi k C ; 2.20 c B L r h i " for r<. The desired estimate on S then follows by use of 2.20 in 2.19 and that > n . 1 It is simple to see that 2.20 is exp ected. For, supp ose that instead of B P we had B . Let c 0 p 1 2 2 m + k , the disp ersion relation of B . Then, setting L = i@ ! @ , and using ! k = 0 k k j j that jr ! j6=0 on the supp ort of g ,we get k Z iB t i! k t ik x 0 ^ e g B f = e e g k f k dk 0 Z r r i! k t ik x ^ = t L e e g k f k dk Z r r i! k t y ik x ^ = t e L e g k f k dk : 2.21 15 iB t 2 0 Estimation of hxi e g B f in L byFourier transform metho ds, using that jr! j is b ounded 0 n away from zero on the supp ort of g , we have that if >maxf ; 2g and r<, then 2 iB t r 0 khxi e g B f k C hti khxi f k : 2.22 0 2 2 " Use of this estimate, in the ab ove expression for S with B replaced by B gives, for l =0; 1; 2: 0 1 l iB t r +l 0 2 khxi e B +i0 hxi k C hti : 2.23 0 B L iB t Two approaches to proving an estimate of the typ e 2.22 and 2.23 for e g B P are as c follows: a One can "map" the estimate for B to that for B P using the wave op erator W . In this 0 c + 2 approachwewould need to derive estimates for W g B inweighted L spaces, or alternatively + 0 b One can use the approach based on the "Mourre estimate", develop ed in quantum scattering theory. This approach is more in the spirit of energy estimates for partial di erential equations and do es not require the use of wave op erators. Here, we shall follow the latter approach. Time decay estimates like 2.22 are a consequence of an approachtominimal velocity estimates of Sigal and So er [58], whichwere proved using the ideas of Mourre [43]; see also Perry, Sigal & Simon [48]. For our application to the the op erator iB t e g B , we shall refer to sp ecial cases of results stated in Skibsted [59] and Debi evre, Hislop & Sigal [20]. We rst intro duce some de nitions and assumptions:. A0 1 a Let N 2 and g 2 C , be a smo oth characteristic function which is equal to one on 0 the op en interval . b Let H and A denote self adjoint op erators on a Hilb ert space H . Assume H is b ounded b elow. Let D denote the domain of A and D H the domain of H . Assume D\D H is dense 1 2 2 . in D H , and let hAi I + jAj 0 A1 Let ad H = H . For 1 k N , de ne iteratively the commutator form A h i k k 1 k k 1 i ad H = i i ad H ; A 2.24 A A k k on D\D H . Assume that for 1 k N , i ad H extends to a symmetric op erator with A h i 3 domain D H , where N = N + +1. 2 iAs iAs A2 For jsj < 1, e : D H !D H and sup k He k < 1; for any 2D H . H jsj<1 16 A3 Mourre estimate: 2 g H i[H; A] g H g H 2.25 for some >0. Under the ab ove assumptions, we have, via Theorem 2.4 of [59], the following: Theorem 2.4. Assume conditions A0 - A3 . Then, for all " > 0 and t>0 1 N N A +" iH t 1 2 2 k F k C hti k k ; 2.26 < e g H hAi H H t where is as in 2.25 . To prove this theorem we set A = A b , where t +1. Then, for any b< , we have by 2.25 that the Heisenb erg derivative, DA @ A +i[H; A ], satis es t 2 g DA g = g i[H; A] b g b g : 2.27 The result now follows by an application of Theorem 2.4 of [59]. Remark: The hyp otheses of Theorem 2.4 can be relaxed [29] to the following two conditions: n i the op erator g H ad H g H can be extended to a b ounded op erator on H for n = A h i N 0; 1; 2:::; +1 +1, and ii the Mourre estimate, 2.25 . 2 q 2 +m + V x , A = x p + p x =2, We shall apply Theorem 2.4 for the case H = B = 2 p = ir and H = L . Before verifying the hyp otheses of Theorem 2.4, we show how this the- x iB t orem is used to derive the desired estimates on e g B . With the application of Theorem 2.4 iB t in mind, we estimate the norm of hxi e g B by decomp osing it into the sp ectral sets on which A=t is less than and greater than or equal to . Let denote an interval containing 1 and denote by C the op erator that asso ciates to a function f its complex conjugate f . Further- iB t iB t more, note that e = C e C and C g B = g B C , if g is real-valued on the sp ectrum of B . Then we nd, iB t iB t khxi e g B k = k hxi g B e g B k 2 2 1 iB t = k hxi g B C e C g B k 2 1 iB t = k hxi g B hAi hAi e g B k 2 1 iB t 2 k khxi g B hAi k k hAi e g B 2 B L 1 iB t k C k hAi e g B 2 A iB t C k F k < hAi e g B 1 2 t A iB t k 2.28 hAi e g B + C k F 2 2 t 17 2 We have used here that, with A de ned as ab ove, the L op erator norm of hxi g B hAi 1 is b ounded; see [48]. By Theorem 2.4, the rst term on the right hand side of 2.28 can be estimated from ab ove by: A iB t C k F < hAi e g B g B k 1 2 1 t N N N N +" 1 2 2 2 2 2 C C hti g B hxi k khxi k : 2.29 k hAi 1 2 B L 1 Since t > 0, the second term is b ounded as follows: A iB t C k F k C hti k k : 2.30 hAi e g B 2 2 2 t From 2.28 , 2.29 and 2.30 , we have N N iB t +" 1 2 2 khxi e g B k C hti + hti khxi k : 2.31 2 2 Use of 2.31 in a direct estimation of 2.19 gives: N " +2+" +2 1 2 2 khxi S t hxi k C hti + hti : 2.32 B L 1 " We make the nal choice of N and after estimation of S t . 2 " To complete the estimation of S t , it remains to verify hyp otheses A0 - A3 . These 1 2 2 hyp otheses are known to hold for the op erator H = B = +V + m ; see [18]. Hyp othesis n 1;2 A0 holds H = B with domain W IR . Clearly hyp othesis A2 holds for H = B since it 2 2 holds for H = B . Hyp othesis A1 can be reduced to its veri cation for B using the Kato 2 square ro ot formula [49]: for any 2D B : Z 1 1 1=2 2 2 1 B = w B B + w dw : 2.33 0 1 To verify A1 for the case k = 1 we must show that [A; B ] B is a b ounded op erator on 2 L . Using 2.33 we have: Z 1 1 2 1 2 1 2 1 1 1 2 B + w [A; B ] B B + w dw 2.34 [A; B ] B = w 0 Since h i 2 2 2 B ;iA = 2B x rV 2 V + m ; 2.35 we have that 2 1 1 2 1 [B ;iA] B = 2B x rV +2V B 2m B : 2.36 18 Substitution into 2.34 we get: Z 1 1 2 1 2 2 2 [A; B ] B = w B B + w dw 0 Z 1 1 2 1 2 1 2 1 B + w x rV +2V 2m B B + w dw 0 J + J : 2.37 1 2 The term J can b e rewritten as 1 Z 1 1 2 d 2 1 2 J = w B + w dw B = I; 2.38 1 dw 0 1 which follows from integration by parts and the formula 2.33 with replaced by B . Using 2 1 2 1 that k B + w k + w , 2 1 2 1 2 k x rV +2V 2m B k kx rV +2V 2m k kB k ; 2.39 1 B L and the hyp otheses on V we have that for some constant C dep endent on V , Z 1 1 2 2 2 w jJ j C + w dw < 1: 2.40 2 0 The higher order commutators are handled in a similar manner; they are even simpler b ecause 2 1 each successive commutator results in an extra factor of B + w . This leaves us with veri cation of the Mourre estimate A3 for H = B . Prop osition 2.2 of 2 [20] says that under hyp otheses on B , veri ed in [18], that A3 holds for B as well. Estimation of S : 2 k;s In S t , the energy is lo calized away from so we seek to use the W estimates of Prop o- 2 g lo calizes sition 2.1. Note that of the cases l = 1 and l = 2, the case l = 1 is "worse" b ecause the energy away from the singularity at , and the l = 1 term has slower decay for large energy. We therefore carry out the estimation for l =1. n q 1 0 1 0 1 1 , hxi 2 L and therefore: Let q + p =1=2 and p + p =1. Since > 2 " iB t 1 0 0 0 khxi S t hxi f k C khxi k ke B +i0 g B W hxi f k p 2 2 0 0 p 2 + B L ;L iB t 1 1 0 0 C ke B k k g B B +i0 B W hxi f k 1;p p 0 0 0 1;p 0 + B W ;L 1 iB t 1 0 0 g B B +i0 B B W hxi f k C ke B k k 1;p p 0 0 0 0 p + 0 B W ;L 1 iB t 1 0 0 p g B B +i0 B k kB W hxi f k 2.41 C ke B k k 1;p p 0 0 0 0 p B L + 0 B W ;L The three factors in 2.41 are estimated as follows: 19 i By part a of Theorem 2.1, 2n iB t 1 0 n+2 0 k C jtj ke B ; 2.42 1;p p 0 B W ;L 0 where p =2 8= n + 6 and p =2 n +2 = n 2 . 1 1 p g B B +i0 B k is b ounded b ecause g +i0 is a multiplier ii k 0 0 0 B L p on L [64]. n 2;p Using the b oundedness of W on W and that >maxf ; 2g,we have + 2 kB W hxi f k kW hxi f k 0 p 1;p + + 2;p kW k khxi f k 1;p B W + C kf k : 1;2 Therefore, 2n " n+2 khxi S hxi k C hti k k : 2.43 2 1;2 2 " We now combine the estimates of S t ; j = 1; 2 to complete the pro of. Combining 2.32 j and 2.43 and taking " 0 yields 2n N l +2 iB t +2+" 1 n+2 2 + hti + hti khxi e B +i0 P hxi k C hti k k : c 2 1;2 2.44 The estimates 2.17 now follows by taking N and such that 2n 2 + n +2 n > maxf ; 2g 2 N 2n 2 + + " 1 2 n +2 The constraints on are implied by the hyp othesis > n . The remaining constraint holds " if N > 8. Since we have applied Theorem 2.4 in our estimation of S ,we need that A1 hold 1 i h 3 +1 10. with N = N + 2 This completes the pro of of Prop osition 2.2. 3. Existence theory In this section we outline an existence theory for 1.1 with initial conditions u x; 0 = u 2 0 3 3 2;2 1;2 W IR and @ u x; 0 = u 2 W IR . We rst reformulate the initial value problem and t 1 20 then intro duce the hyp otheses on the op erator H . Regarding 1.1 as a p erturbation of a linear constant co ecient equation, we rst write the initial value problem as: 2 2 @ u + B u = Vu + f u : 3.1 t 0 u x; 0 = u x 0 @ u x; 0 = u x t 1 Let u cos B t sin B t =B 0 0 0 u = ; U t = ; v B sin B t cos B t 0 0 0 0 u 0 : ; F u = u = 0 Vu + f u u 1 Then, the initial value problem can be reformulated as a system of rst order equations: 0 1 @ u = u + F u : 3.2 t 2 B 0 0 We now follow the strategy of reformulating the problem of nding a solution of the initial value problem as the problem of nding a xed p oint u of an appropriate mapping. In particular, we seek u in the space 3 3 2;2 1;2 X = W IR W IR 3.3 0 satisfying A u = u; 3.4 where Z t Au t U t u + U t s F u ds: 3.5 0 0 For a discussion of the existence and low energy scattering for the case V 0, see [23], [65], [36] and references cited therein. Hyp otheses of H = +V 1;1 H1 V 2 W 2 H2 H = B is a p ositive and self-adjoint op erator 2 H3 The semi-in nite interval, [m ; 1 , consists of absolutely continuous sp ectrum of H . 2 2 2 H4 H has exactly one simple eigenvalue satisfying 0 < < m , with corresp onding 2 eigenfunction ' 2 L ; jj'jj =1: 2 21 3 4 N f u = u + f u ; f u = O u and f u is smo oth in a neighb orho o d of u =0. 4 4 These hyp otheses are by no means the least stringent, but are sucient for the present purp oses. Theorem 3.1. Lo cal existence theory Consider the Cauchy problem for the nonlinear Klein Gordon equation 1.1 with initial data, u of class X . 0 0 max a There exists strictly p ositive numb ers, T and T forward and backward maximal max times of existence which dep end on the X norm of the initial data, such that the initial value 0 0 max problem has a unique solution of class C T ; +T ; X in the sense of the integral max 0 equation 3.4 . max b For t 2 T ; +T conservation of energy holds: E [u ;t ;@ u ;t ] = E [u ;u ], where max t 0 1 E is de ned in equation 1.4 . c Either T is nite or T is in nite. If T is nite, then max max max ku ;t k = 1: 3.6 lim X 0 t"T max max max The analogous statement holds with T replaced by T and lim replaced by lim max t"T t T max . We now sketch a pro of of Theorem 3.1. We restrict our attention to the case t 0; the pro of is identical for t 0. Furthermore, we shall, for simplicity, consider the case of the cubic 3 nonlinearity: f u = u . Using hyp othesis H1 on V , it is simple to show, using the estimates of Corollary 2.1, that for T > 0 suciently small and dep ending essentially on the X norm of 0 0 u , the mapping A maps a closed ball in C [0;T ; X to itself, and is a strict contraction. This 0 0 0 ensures the existence of a unique xed p oint, which is a C [0;T ; X solution of the initial 0 value problem 3.1 in the sense of the integral equation 3.4 . That the X norm must blowup 0 if T is nite is a standard continuation argument based on the xed p oint pro of of part a . max In section 7, we study the asymptotic b ehavior of solutions as t !1. A consequence of this analysis and the argument b elow is that if u satis es the more stringent hyp othesis that 0 the norm ku k is suciently small, where 0 X 2;2 2;1 1;2 1;1 X W \ W W \ W ; 3.7 max then T = T = 1 and the solution decays to zero as dilineated in the statement of max Theorem 1.1. 22 We reason as follows. A unique solution, u t , exists lo cally in time and is continuous in t with 1;4 8 values in X . It follows that for jtj 0 1;2 solution satis es energy conservation and therefore ku t k + kv t k is b ounded uniformly 2 W 1;2 by a constant which is indep endent of T , and dep ends only on ku k + kv k . To ensure 0 0 2 W that the quantities estimated continue to be well de ned and satisfy the 'a priori estimates of section 7 it suces to show that ku t k remains b ounded as t " T . A direct and elementary X 0 estimation of the integral equation 3.4 yields the estimate: 2;2 1;2 ku t k C ku k + ku k X 0 1 W W 0 Z t 3 2 1;1 1;2 + C kV k ; ku s k + ku s k + ku s ru s k ds: 1;2 2 W W W 0 3.8 0 1 2 The main to ol used in obtaining 3.8 is the b ound kE t @ k m . i B L 1 A simple consequence of the 'a priori b ound 7.52 and the decomp osition of u t; is that ~ kru t k + ku t k C; jtj 4 8 ~ The constant C , dep ends on the norm ku k . The rst two terms in the integrand of 3.8 0 X are uniformly b ounded by conservation of energy. Furthermore, this energy b ound together with 3.9 implies a b ound on the last term in the integrand of the estimate 3.8 . It follows that ku t k remains b ounded as t " T . Therefore, given the estimates of section 7 we have X 0 T = 1, and the decay of solutions. max A further consequence of the pro of is: Corollary 3.1. Under the hyp otheses of Theorem 1.1, the solution of the initial value problem 3 2;2 exists globally in W IR and satis es the estimate: 2;2 ku t k C hti: 3.10 W 4. Isolation of the key resonances and formulation as coupled nite and in nite dimensional dynamical system Using the notation 1.5 , the initial value problem for 1.1 can be rewritten as 2 2 3 @ u + B u = f u ; f u = u 4.1 t u x; 0 = u x ; @ u x; 0 = u x : 0 t 1 23 For small amplitude solutions, it is natural to decomp ose the solution as follows: u x; t = a t ' x + x; t ; 4.2 ;t ;' = 0 for all t : 4.3 Substitution of 4.2 into 4.1 gives 00 2 2 2 a ' + @ + a' + B = f a' + 4.4 t We now implement 4.3 . Taking the inner pro duct of 4.4 with ' gives 00 2 a + a = '; f a' + : 4.5 2 Let P denote the pro jection onto the continuous sp ectral part of B , i.e. c P v v '; v ': 4.6 c Then, since = P ,we have c 2 2 + B = P f a' + 4.7 @ c t Equations 4.5 - 4.7 comprise a coupled dynamical system for the b ound state and contin- 2 uous sp ectral comp onents relativetoH = B of the solution u. The initial conditions for this system are given by 1.16 . Expansion of the cubic terms in 4.5-4.7 gives the system Z Z Z Z 00 2 3 4 2 3 2 2 3 a + a = a ' + 3a ' + 3a ' + ' 4.8 2 2 3 3 2 2 2 3 @ + B = P a ' + 3a ' + 3a' + ; 4.9 c t with initial conditions 0 a 0 = '; u ; a 0 = '; u 0 1 x; 0 = P u ; @ x; 0 = P u 4.10 c 0 t c 1 We now lo cate the source of the resonance. Equation 4.8 has a homogeneous solution which oscillates with frequencies . Thus, to leading order, solves a driven wave equation 2 2 containing the driving frequencies 3 . If 9 > m , then by H 3 , we exp ect a resonant interaction with radiation mo des of energy 3 2 H . cont 24 This resonant part of , has its dominant e ect on the a-oscillator in the term of 4.8 , which is linear in . Our goal is to derive from 4.8-4.9 an equivalent dynamical system which is of the typ e describ ed in the intro duction, but which is corrected by terms which decay suciently rapidly with time and which can therefore b e treated p erturbatively. We rst write = + + ; 4.11 1 2 3 where t satis es the linear dynamics with the given initial data: 1 2 2 @ + B = 0 ; x; 0 = P u ; @ x; 0 = P u ; 4.12 1 1 1 c 0 t 1 c 1 t and t is the leading order response 2 2 3 3 2 + B = a P ' ; x; 0 = 0; @ x; 0 = 0 : 4.13 @ 2 2 c 2 t 2 t Equation 4.8 can now b e written in an expanded form. Z Z Z Z 00 2 3 4 2 3 2 2 3 a + a = a ' +3a ' + + +3a ' + ' 1 2 3 F a; 4.14 We exp ect the function a t to consist of "fast oscillations", coming from the natural frequency and its nonlinearly generated harmonics, and slowvariations due to its small amplitude. We next extract from a t the dominant "fast" oscillations of frequency : i t i t a t = A e + A e : 4.15 We then substitute 4.15 into 4.14 and imp ose the constraint: 0 i t 0 i t A e = 0 4.16 A e + Equation 4.14 then is reduced to the rst order equation: 0 1 i t A = 2i e F a; ; 4.17 where F a; = F A; A; ; t . From 4.14 we have that F a; is the sum of the following terms: Z 3 4 F a = a ' 1 Z 2 3 F a; = 3a ' ; j =1; 2; 3 2 j j Z 2 2 F a; = 3a ' 3 Z 3 ' : 4.18 F = 4 25 The remainder of this section is primarily devoted to an involved expansion of F a; . 2 2 The terms of this expansion are of several typ es: a the resonant damping term, b terms of the same order whose net e ect is a nonlinear phase correction and c higher order terms which are to b e treated p erturbatively in the asymptotic analysis as t !1. Computation of F a; 2 2 r nr We rst break into a part containing the key resonance and nonresonant part . 2 2 2 Prop osition 4.1. r nr = + ; 2 2 2 where Z t r iB t is B 3 3 3 e = e A s ds P ' ; 4.19 c 2 2iB 0 and Z t iB t is B 2 nr e 3 e A s A s ds = 2 2iB 0 Z t 2 is B + + 3 e A s A s ds 0 Z t 3 is B +3 3 + e A s ds P ' c 0 Z t iB t is B +3 3 e e A s ds 2iB 0 Z Z t t 2 is B is B + 2 e A s ds +3 A s A s ds e A s + 3 0 0 Z t 3 is B 3 3 e + A s ds P ' c 0 7 X nr 2j j =1 4.20 The sup erscripts r and nr denote resp ectively a resonant contribution and nonresonant contri- bution. Proof: The solution of 4.13 can b e expressed, using the variation of constants formula, as: Z t sin B t s 3 3 = a s ds P ' ; 4.21 2 c B 0 Substitution of 4.15 for a s and using the expansion of sin B t s in terms of the op erators r exp iB t s leads to an expression for which is a sum of eight terms. The term , as 2 2 26 de ned ab ove, is anticipated to be the most imp ortant. The other seven terms are lump ed nr together in . 2 r r We now fo cus on . In order to study near the resonant p oint 3 in the continuous 2 2 r sp ectrum of B , we rst intro duce a regularization of . For ">0, let 2 Z t r iB t is B 3 +i" 3 3 e e A s ds P ' 4.22 c 2" 2iB 0 r r Note that = lim . The following result, proved using integration by parts, isolates the "!0 2 2" key lo cal in t resonant term. Prop osition 4.2. For " 0, h i 1 r 3i t 3 "t 3 = B B 3 + i" e A t e P ' c 2" 2 h i 1 iB t 3 3 e P ' A B B 3 + i" c 0 2 Z h i t 1 3 iB t is B 3 +i" 2 0 3 e e A A ds P ' B B 3 + i" c 2 0 r nr 1 nr 2 = + + 4.23 " " " Remark: The choice +i" in 4.22 is motivated by the fact that the op erator 1 iB t B 3 + i0 e 4.24 satis es appropriate decay estimates as t !1; see Prop osition 2.1. It follows that the limit as + 1 " ! 0 of the last two terms in 4.23 decay in time like hti >0 ; see section 7. Wenow use the ab ove computation to obtain an expression for F a; . First, from Prop o- 2 2 sition 4.1 we have r nr F a; = F a; +F a; 2 2 2 2 2 2 r nr 1 nr 2 nr = lim F a; + lim F a; + +F a; ; 2 2 2 " " " 2 "!0 "!0 r nr 1 nr 2 nr F a; + F a; + + F a; 4.25 2 2 2 2 where F a; is de ned in 4.18 . We b egin with the contribution to 4.17 coming from 2 r r nr F a; . What follows now is a detailed expansion of the term F a; and F a; . The 2 2 2 2 nr 1 nr 2 terms F a; + can b e treated p erturbatively by estimation of its magnitude; see section 2 5. r Computation of F a; 2 27 Let ! 1 B 3 3 3 lim P ' ; P ' 4.26 c c 2 2 "!0 B B 3 + " 1 1 3 3 = P ' ; P:V : P ' ; and c c B B 3 ! " 1 3 3 P ' lim P ' ; c c 2 2 "!0 B B 3 + " 3 3 = P ' ; B 3 P ' c c 3 2 3 F [' ] 3 : 4.27 = c 3 By hyp othesis 1.8 , > 0. r r We now substitute the expression for , given in 4.23 into the de nition of F a; in 2 " " 4.18 . Passage to the limit, " ! 0, and use of the distributional identity: 1 1 1 x i0 lim x i" = P:V : x i x 4.28 "!0 yields: Prop osition 4.3. r r F a; = lim F a; 2 2 " "!0 h i 3 2 4 i t 5 5i t 2 3 3i t = i jAj Ae + A e +2jAj A e 4.29 2 Wehave completed the evaluation of the rst term in 4.25 . To calculate the contribution 1 i t of 4.29 to the amplitude equation 4.17 we need only multiply 4.29 by 2i e . This gives Prop osition 4.4. 2 h i 3 4 5 4i t 2 3 2i t 1 i t r i+ jAj A + A e +2jAj A e 4.30 2i e F a; = 2 4 2 3 4 The term jAj A plays the role of a nonlinear damping; it drives the decay of A and, in 4 turn, that of ; see the discussion in the intro duction. nr Computation of F a; : 2 2 28 nr We now fo cus on F a; , the third term in 4.25 . This requires a rather extensive, 2 2 P nr nr ; see 4.20 . From Prop osition 4.3, we exp ect the dominant terms expansion of = j 2j 2 5 5 to be O jAj . Our approach is now to make explicit all terms which are formally O jAj 0 3 00 5 2 0 6 anticipating that A = O jAj , A = O jAj and jAj = O jAj and to treat the remainder as a p erturbation whichwe shall later estimate to b e of higher order. This expansion nr of is presented in the following prop osition which we prove using rep eated integration by 2 nr parts. As written, these expressions are formal. By the de nition of F a; we require that 2 2 3 they hold when integrating against a rapidly decaying function, i.e. ' . 0 Prop osition 4.5. The following expansions hold in S : 3 nr 2 it 3 it 2 0 3 = jAj Ae P ' + e jAj A P ' c c 21 2 2B B 2iB B iB t h i 3e 2 2 0 3 + jA j A + jAj A j P ' 0 0 c t=0 2B B Z t 3 iB t is B 2 00 3 e e jAj A ds P ' 4.31 c 2 2iB B 0 3 3 2 0 2 it 3 0 2 3 it nr jAj Ae P ' + A A +2A jAj P ' e = c c 22 2 2B B + 2iB B + i h 3 3 3 2 2 0 iB t A P ' jA j jAj A j + e c 0 t=0 2 2B B + 2B B + Z t 3 is B + 3 iB t 00 3 e jAj e A s ds P ' 4.32 c 2 2iB B + 0 3 A t 3 nr 3it 3 3 0 3 it3 = e P ' + A t P ' e c c 23 2 2B B + 3 2iB B + 3 iB t i h e 3 3 3 0 3 A + A j P ' c 0 t=0 2 B B + 3 Z t 3 iB t is B +3 2 00 3 e e A s ds P ' c 2 2 iB B + 3 0 4.33 3 nr 3 3it 3 3it 3 0 3 = A t e P ' e A P ' c c 24 2 2B B + 3 2iB B + 3 29 iB t h i 3 e 3 3 0 3 A + A j P ' c 0 t=0 2 B B + 3 Z t 3 is B +3 3 00 3 iB t e A s ds P ' e + c 2 2iB B + 3 0 4.34 3 3 nr 2 it 3 it 2 0 3 = jAj Ae P ' e jAj A P ' c c 25 2 2B B + 2iB B + iB t h i 3 e 2 2 0 3 jA j A + jAj A j P ' 0 0 c t=0 2 B B + Z t 3 iB t is B + 2 00 3 e e jAj A ds P ' + c 2 2iB B + 0 4.35 3 3 nr it 3 0 3 2 it 2 = Ae P ' A P ' jAj e jAj c c 26 2 2B B 2iB B iB t h i 3 e 2 0 2 3 A + jAj A j jA j P ' 0 0 c t=0 2 B B Z t 1 3 is B 2 iB t 00 3 e jAj e A ds P ' + c 2 2i B B 0 4.36 3 3it 3 A t e 3 3it nr 3 0 3 e = P ' A P ' c c 27 2 2B B 3 i0 2iB B 3 i0 iB t h i 3 e 3 3 0 3 + A A j P ' c 0 t=0 2 B B 3 i0 Z t 3 3 00 3 is B 3 iB t A ds P ' 4.37 e e + c 2 2iB B 3 i0 0 0 1 i t Recall that our goal is to elucidate the structure of the amplitude equation: A = 2i e F , 1 i t in 4.17 , where we rst fo cused on the contribution: 2i e F . From 4.25 and Prop o- 2 sition 4.4 we see now that we need to obtain convenient expressions for 7 X 1 i t nr 1 i t nr 2i e F a; 2i e F a; = 2 2 2j 2 j =1 Z 7 X 3 nr 1 2 i t ' : 4.38 = 3 2i a e 2j j =1 30 nr 5 Each of the seven terms contributing to is expressed as a part which is O jAj plus an 2 5 error term which is estimated in magnitude in section 5. The O jAj part and error terms are displayed in the following two prop ositions. Prop osition 4.6. Let 3 1 1 3 P ' ;B B P ' : 4.39 c c Then, 1 i t nr 2i e F a; 2 21 2 h i 3 3 3 2 3 2i t 4 4 2i t nr = jAj A e + jAj A + jAj Ae + E 4.40 21 4i 2i 4i 1 i t nr 2i e F a; 2 22 2 h i 9 9 9 3 4 4 2i t 2 4i t nr = jAj A + jAj Ae + jAj A e + E 4.41 22 4i 2i 4i 1 i t nr 2i e F a; 2 23 2 h i 3 3 3 3 5 4 2i t 2 4i t 6i t nr 3 jAj Ae + jAj A e + A e + E 4.42 = 23 4i 2 4 1 i t nr 2i e F a; 2 24 2 h i 3 3 3 5 4i t 3 2 2i t 4 nr = 3 A e + A jAj e + jAj A + E 4.43 24 4 4 4 1 i t nr 2i e F a; 2 25 2 h i 9 9 9 2 3 2i t 4 4 2i t nr jAj A e + jAj A + jAj Ae + E 4.44 = 25 4i 2i 4i 1 i t nr 2i e F a; 2 26 2 h i 9 9 9 3 4 4 2 nr 2i t 4i t jAj A + jAj jAj + E 4.45 Ae + A e = 26 4i 2i 4i 31 1 i t nr 2i e F a; 2 27 2 h i 3 3 3 5 4 2 nr 2i t 4i t 6i t 3 + i0 jAj jAj 4.46 + E Ae + A e + A e = 27 4i 2i In our analysis of the large time b ehavior t ! 1 , we shall require an upp er b ound of nr the error terms E given in: 2j Prop osition 4.7. n 2 0 1 iB t 3 2 2 nr jAj jA j + khxi B e P ' k j C jj jAj jE j c 2 ' 2j Z o t 2 3 iB t s 3 00 ; 4.47 + khxi B ds P ' k e O jAj j c 2 0 where = = ; = = ; = = 3 ; and =3 +i0. 1 6 2 5 3 4 7 In section 7, we shall estimate this expression using the decay estimates of section 2, in particular Prop osition 2.2. The desired form of the A-equation is now emerging. Use of Prop ositions 4.4 and 4.6 in 4.17 yields: Prop osition 4.8. The amplitude A t satis es the equation see Prop osition 4.6 for the de n- ition of : i 3 4 2 3 2i t 2 2i t 4i t 0 k'k 3jAj A + A e + 3jAj Ae + A e A = 4 2 2 h i 3 3i 4 2 4 jAj A 5 + 3 + 3 jAj A 4 4 2 h i 3 5 4i t A e i + 3 4 2 h i 3i 2 3 2i t jAj A e 2 + 2i + + i 3 + 3 4 2 h i 3i 4 2i t jAj A e 7 + 9 + 3 + 3 + i0 4 2 h i 3i 3 2 4i t jAj A e 3 2i 3 + 3 + i0 + 3 4 2 h i 4 3i 5 6i t A e i 3 i 3 + i0 + E 4 3 4.48 32 where Z 1 i t 2 E = 2i e 3a ' Z 1 i t 3 + 2i e ' 7 X nr + E 2j j =1 h i 1 i t + 2i e F a; +F a; 2 1 2 3 nr 1 nr 2 + F a; + 4.49 2 Here, F a; isgiven by 4.18 . 2 j In the next section we show how to rewrite 4.48 in a manner which makes explicit which terms determine the large time b ehavior of the amplitude and phase of A t . 5. Disp ersive Hamiltonian normal form To analyze the asymptotic b ehavior of A t or equivalently a t and t; x as t ! 1 it is useful to use the idea of normal forms [3], [26], [52] from dynamical systems theory. We derivea p erturb ed normal form which makes the anticipated large time b ehavior of solutions transparent. ~ Prop osition 5.1. There exists a smo oth near-identity change of variables, A 7! A with the following prop erties: ~ A = A + h A; t 3 h A; t = O jAj ; jAj!0 1 h A; t = h A; t +2 ; 5.1 ~ and such that in terms of A equation 4.48 b ecomes: 0 2 2 4 2 4 ~ ~ ~ ~ ~ ~ ~ A = ic jAj A + d jAj A + i c jAj A 21 32 32 7 ~ ~ + O jAj +E ; 5.2 2 3 2 where d = < 0. The constants c and c are real numb ers, explicitly calculable in 32 21 32 4 ~ terms of the co ecients app earing in 4.48 . The remainder term jEj is estimable in terms of ~ jEj for jAj < 1 equivalently jAj < 1 . 33 Remarks: The p oint of this prop osition is that in the new variables the dynamics evidently have a dissi- pative asp ect. Sp eci cally, neglecting the p erturbation to the normal form one has: 2 3 2 6 ~ ~ @ jAj = jAj < 0 5.3 t 8 We therefore refer to 5.2 as a dispersive Hamiltonian normal form. In nite dimensional Hamiltonian systems, the normal form asso ciated with a one degree of freedom Hamiltonian 2m system is an equation like 5.2 , but with all the co ecients of the terms jAj A b eing purely imaginary. Here we nd that resonant coupling to an in nite dimensional disp ersive wave eld can lead to a normal form with general complex co ecients, which in our context implies the internal damping e ect describ ed ab ove. See section 8 for further discussion. We now present an elementary derivation of the change of variables 5.1 leading to 5.2 . Equation 4.48 is of the form: X X l i k l 1 t 0 k A e + E; 5.4 A = A kl k +l =j j 2f3;5g where the co ecients can be read o 4.48 . The pro of we present is quite general and kl shows, in particular, that the normal form for equations like 5.4 is 5.2 Prop osition 5.2. There is a change of variables, as in 5.1 , such that equation 5.4 is mapp ed to: 0 2 4 ~ ~ ~ ~ ~ A = k jAj A + k jAj A 21 32 7 ~ ~ + O jAj +E ; where 5.5 ~ E = E I + h ; k = 21 21 3 2 2 1 j j 2 + 2j j 5.6 k = + 2i 03 30 12 12 32 32 2 The conclusion in Prop osition 5.1 concerning the "damping co ecient" d dep ends on the 32 particular prop erties of the co ecients in 4.17 . In particular we have from 4.17 and 5.4 that: 3 4 = = k'k ; 21 12 4 2i 4 k'k ; = 03 4 2i 2 2 3i 3 [ 5 + 3 + 3 ] : 5.7 = 32 4 4 34 2 From these formulae, wehave d is given by the real part of , the "singular" Fermi golden 32 32 rule contribution. ~ Remark: The construction of the map A 7! A can b e applied as well to equations of the form 5.4 where the right hand side is an arbitrary expansion in p owers of A and A. To make the structure clear we write out the equation with a particular ordering of terms: 0 2 4 A = jAj A + jAj A + O A + O A + E; 5.8 21 32 3 5 where 2 3 3 2i t 2i t 4i t O A = A e + AA e + A e ; and 5.9 3 30 12 03 3 5 4i t 4 2i t 2 2i t O A = A e + A A e + A A e 5 50 41 23 4 5 4i t 6i t A e + A e 5.10 + A 05 14 Note that each term in O A and O A is of the form: oscillatory function of t times order 3 5 3 5 O jAj or O jAj . The computations that follow, though elementary, are rather lengthyso we rst outline the strategy of our pro of. Integration of 5.8 gives Z t 2 4 A = A + jAj A + jAj A + O A + O A + E ds: 5.11 0 21 32 3 5 0 The idea is that terms with explicit p erio dic oscillations average to zero and can b e neglected in determining the large time b ehavior of the solution. Our strategy is now to expand the explicitly oscillatory terms using rep eated integrations by parts and to make explicit all terms up to and 5 including order jAj . The computation has two stages. In stage one, after rep eated integration by parts the equation 5.11 is expressed in the equivalent form: A t H A t ;t = A H A ; 0 1 0 1 0 Z t 2 4 + "resonant" terms like jAj A; jAj A ds 0 Z t + terms of typ e O A ds + higher order corrections 5.12 5 0 This suggests the change of variables A 7! A = A t H A t ;t , giving 1 1 Z t 2 4 A t = A + terms like jA j A ; jA j A ds 1 10 1 1 1 1 0 + terms of typ e O A + higher order corrections: 5.13 5 1 The latter equation is equivalent to a di erential equation of the form: 0 2 4 A t = terms like jA j A ; jA j A 1 1 1 1 1 + terms of typ e O A + higher order corrections: 5.14 5 1 35 which is a step closer to the form of the equation we seek. A second iteration of this pro cess yields the result. We now embark on the details of the pro of. R t Expansion of O A ds: 3 0 Integration of the expression in 5.9 gives Z t O A ds = h A t ;t h A ; 0 3 3 3 0 0 Z Z t t 0 3 2 30 12 2i s 2 0 2i s ds A e A A ds + e A 2i 2i 0 0 Z t 0 3 03 4i s e + A ds; 5.15 4i 0 where 2 3 12 30 03 3 2i t 2i t 4i t A e A h A; t = A e A e 5.16 3 2i 2i 4i 0 We now replace A in 5.15 by its abbreviated expression given in 5.8 . Thus we have: Z t O A ds = h A t ;t h A ; 0 3 3 3 0 0 Z h i t 3 30 2i s 2 2 5 e A jAj A + O A + O jAj + E ds 21 3 2i 0 Z h i t 2 12 2i s 2 5 e A jAj A + O A + O jAj + E ds + 21 3 2i 0 Z h i t 12 2i s 2 2 5 + e 2jAj jAj A + O A + O jAj + E ds 21 3 2i 0 Z h i t 3 2 03 2 5 4i s + A jAj A + O A + O jAj + E ds 5.17 e 21 3 4i 0 Substitution of 5.9 into 5.17 and integrating by parts, we arriveat the following expres- sion: Z t O A ds = h A t ;t + h A t ;t h A ; 0 h A ; 0 3 3 3a 3 0 3a 0 0 Z t 3 1 4 2 2 jAj A ds j j 2 + 2j j + 03 30 12 12 2i 2 0 Z t 2 5 + O jAj jAj + jEj ds; 5.18 0 where 2 3 3 30 21 03 12 30 4 2i t 5 4i t A + Ae A e h A; t = 3a 2 2 2 2 4 8 36 3 1 3 2 2i t + A + 3 + 2 A e 21 12 03 12 30 03 12 21 2 4 2 1 3 4 4i t 2 + A + 2 A e + 21 12 30 03 12 2 8 2 1 1 1 5 6i t + + A e : 5.19 03 12 03 30 2 4 3 2 Referring back to 5.8 , we see we must now obtain an R t Expansion of O A ds: 5 0 From 5.10 we have, after integration by parts: Z t O A ds = h A t ;t h A ; 0 5 3b 3b 0 0 Z t 4 3 + jAj ds; 5.20 O jAj + jEj 0 where 3 41 23 50 5 4i t 4 2i t 2 2i t A e + A Ae A A e h A; t = 3b 4i 2i 2i 4 5 14 05 4i t 6i t AA e A e : 5.21 4i 6i Therefore from 5.11 and our computations we have: A h A; t h A; t h A; t 3 3a 3b = A h A ; 0 h A ; 0 h A ; 0 0 3 0 3a 0 3b 0 Z t 2 4 jAj A + jAj A ds + 21 32 0 Z Z t t 1 3 4 2 2 E ds + jAj A ds + j j 2 + 2j j 03 30 12 12 2i 2 0 0 Z t 4 3 2 5 + O jAj jAj + jEj + O jAj jAj + jEj ds: 5.22 0 This suggests the change of variables: A A H A; t ; where 1 1 H A; t = h A; t + h A; t + h A; t ; 5.23 1 3 3a 3b which for small jAj, is a near-identity change of variables. Using this change of variables we have that 5.22 b ecomes Z t 2 4 A = A + jA j A + jA j A ds 1 10 21 1 1 32 1 1 0 37 Z t 1 3 2 2 4 + j j 2 + 2j j jA j A ds 03 30 12 12 1 1 2i 2 0 Z Z t t 2 2 H A ;s ds + E ds + 2 jA j H A ;s + A 1 1 1 21 1 1 1 1 0 0 Z t 4 3 2 5 + O jA j jA j + jE j + O jA j jA j + jE j ds: 5.24 1 1 1 1 1 1 0 We expand the terms involving H A; t . Using integration by parts, as ab ove, we obtain: 1 Z Z t t 2 4 3 jA j H ds = h A ;t h A ; 0 + O jA j jA j + jE j ds; 5.25 2 1 1 5a 1 5a 10 1 1 1 21 0 0 and Z Z t t 2 4 3 A H ds = h A ;t h A ; 0 + O jA j jA j + jE j ds; 5.26 21 1 5b 1 5b 10 1 1 1 1 0 0 where 3 21 12 21 30 2i t 4 2i t 2 e e A A A A h A; t = 1 1 5a 1 1 2 2 2 2 4 21 03 4i t e A A 5.27 1 1 2 8 3 30 12 21 21 2i t 2 2i t 4 A A e A e A h A; t = 1 1 5b 1 1 2 2 4 4 21 30 4i t 5 e A 1 2 16 H A; t = h A; t + h A; t 5.28 5 5a 5b Thus A H A t ;t = A H A ; 0 1 5 1 10 5 10 Z t 2 4 + jA j A + jA j A ds 21 1 1 32 1 1 0 Z t 3 1 4 2 2 jA j A ds j j 2 + 2j j + 1 1 03 30 12 12 2i 2 0 Z t 7 2 O jA j + jA j jE j + jE j ds: + 1 1 1 1 0 5.29 Now de ne ~ A A H A ;t : 5.30 1 5 1 In terms of this new variable we have: Z t 2 4 ~ ~ ~ ~ ~ ~ jAj A + jAj A ds A = A + 21 32 0 0 Z t 1 3 4 2 2 ~ ~ + jAj A ds j j 2 + 2j j 03 30 12 12 2i 2 0 Z t 7 ~ ~ + O jAj + jEj ds; 5.31 0 38 ~ where jEj is estimable in terms of jEj for jAj < 1. This completes the pro of. 6. Large t b ehavior of solutions to the p erturb ed normal form equa- tions Wenow consider the large time b ehavior of solutions to the ordinary di erential equations of the form 5.2 . In particular, we compare the b ehavior of solutions of 5.2 to those of the equation ~ with E set equal to zero. Thus we consider the equations: 0 2 4 = ic j j + ic j j + Q t ; 6.1 21 32 0 2 4 = ic j j + ic j j ; 6.2 21 32 where > 0. We rst consider the unp erturb ed equation, 6.2 . Multiplication of 6.2 by and taking the real part of the resulting equation yields the equation: 0 3 2 r = 2 r ; r = j j : 6.3 Integration of 6.3 yields: 1 2 2 2 : 6.4 r t = r 1 + 4 r t 0 0 To prove the ab ove lemma, we b egin by multiplying 6.1 by and taking the real part of the resulting equation. This gives: 0 3 t + Q t t ; 6.5 r t = 2 r t + Q t which implies the di erential inequality: 1 0 3 2 t : 6.6 r t 2 r t + 2 jQ t j r We now prove 2 Lemma 6.1. Supp ose r t = j t j satis es 6.6 with: 5 4 ; 0: 6.7 jQ t j Q hti 0 Then, 0 1 8 2 5 1 m Q 0 4 4 4 @ A j t j 1+4j j t 2j j + ; 6.8 0 0 3 8 5 3 hti 4 where m = maxf1; 4 j j g. 0 39 proof: Use of 6.7 in 6.6 gives the inequality 5 1 0 3 4 2 r t 2 r t + 2Q hti r t : 6.9 0 Multiplication by r t gives: 5 3 0 2 2 4 4 z t 4 z t + 4Q hti z t ; where z t = r t : 6.10 0 Note that the equation 0 2 4 t = 4 t ; 0 = z = j j 6.11 0 0 has solutions: 1 t = z 1 + 4 z t : 6.12 0 0 Anticipating this as the dominant b ehavior for large t, we de ne: z t t R t : 6.13 Substitution into 6.10 and simplifying gives: 1 3 5 4 z Q 0 0 0 4 4 4 R t R R 1 + C 1+4z t R hti t : 6.14 0 1 1 + 4 z t 4 jz j 0 0 We now consider the last term in 6.14 . Noting that 1 1 1 + t m 1+4 z t ; m = maxf1; 4 z g; 6.15 0 0 we have for any "> 0: 1 1 4 4 3 3 1+4z t Q m Q 0 0 0 4 4 R t C C R 1 5 1 1+ + hti 4 4 4 jz j hti jz j 0 0 5 3 8 4 Q m "R t 0 C 1 5 3 + 4 8 8 " jz j hti 1 + 4z t 0 0 3 8 8 2 5 5 3 R t m Q " 0 : C + C 1 2 2 8 8 1+4z t 1+ 5 0 5 5 " z hti 0 1 p 1 q 1 1 The last inequality follows from the inequality: ab p "a + q b=" ; p + q = 1, for 8 8 the choice p = and q = . 3 5 40 This last estimate can now be used in 6.14 and implies: 8 3 8 5 5 3 Q 4 z + C " 4 z m 0 2 0 0 0 2 R R t + R t + C 1 2 8 8 1+4z t 1+4z t 1+ 5 0 0 5 5 z hti " 0 8 3 which when setting C " =2 z ; is 2 0 8 5 2 z Q m 0 0 R R 2 + C : 6.16 3 8 1+ 1+4z t 5 3 z hti 0 0 2 Now set R = 2 + S . Since the term prop ortional to S is negative, we obtain 8 5 4 z Q m 0 0 0 S S + C : 6.17 8 3 1+ 1 + 4 z t 5 5 z hti 0 0 Multiplication by 1+4 z t yields: 0 8 5 Q 1 1+4 z t m 0 0 @ [ 1 + 4 z t S t ] C t 0 3 8 1 + t 5 5 hti z 0 8 2 5 m 1 Q 0 : 6.18 C 3 8 5 5 z hti 0 Integration of 6.18 from 0 to t implies: 8 2 5 Q 1 m 0 S t C : 6.19 3 8 5 5 hti z 0 The Lemma now follows from 6.19 and the relation: 4 j t j z t t R t t 2 + S t : 6.20 7. Asymptotic b ehavior of solutions to the nonlinear Klein- Gordon equation 0 In x2 we proved that lo cal in time solutions, u t , exist in the space C T ;T ; X , for some 0 T ;T > 0. We now study obtain the required 'a priori b ounds to ensure a p ersistence of the solution, u t , as a continuous function with values in X T = T = 1 and b the decay 0 of solutions as t !1 in suitable norms. Due to the linear estimates of section 2, we require 41 1;4=3 more stringenthyp otheses on u and u . These linear estimates require niteness of W and 0 1 1;8=7 W norms which, by interp olation, are controlled under the assumption u 2 X see section 0 2;2 2;1 1;2 1;1 2 . Sp eci cally, u 2 W \ W and u 2 W \ W : 0 1 Using the results of the previous section, the original dynamical systems 4.1 can now be rewritten as: u x; t = a t ' x + x; t ; x; t = x; t + x; t + x; t ; 1 2 3 i t i t a t = A t e + A t e ; ~ ~ A t = A t +h A t ;t ; 7.1 ~ ~ ~ Here, h A; t is a smo oth p erio dic function of t, cubic in A for A ! 0, and are de ned by 1 2 ~ 4.12 - 4.13 , and A t satis es the p erturb ed normal form equation: 7 2 4 4 0 2 ~ ~ ~ ~ ~ ~ ~ ~ ~ A + O jAj + E t; a; : 7.2 A + i c jAj A jAj A = ic jAj 32 21 ~ Here E = E I + h , with E given by 4.49 . In 7.2 , 2 3 3 3 = ; P ' ; B 3 P ' > 0: 7.3 c c 4 3 The constants c and c are real numb ers which are computable by the algorithm presented 21 32 in section 5. ~ The ab ove de nitions of A, A, and h, together with the estimates proved b elow, can b e used to verify in a straightforward manner the assertions of Theorem 1.1 concerning R t jA t j, and t arg A t . To pro ceed with a study of the t !1 b ehavior, recall that: 2 2 @ + B =0; x; 0 = P u ; @ x; 0 = P u ; 7.4 1 1 c 0 t 1 c 1 t 2 2 3 3 @ + B = a P ' ; x; 0 = 0; @ x; 0 = 0; 7.5 2 c 2 t 2 t 2 2 2 2 2 3 @ + B = P 3a ' +3a' + ; x; 0 = @ x; 0 = 0: 7.6 3 c 3 t 3 t ~ To motivate the strategy, we rst argue heuristically. Since > 0, if E is negligible, then 1=4 1=4 ~ jAj hti as t ! 1, and therefore by 1.25 , a hti , as t ! 1. It then follows from 3=4 1+ 7.5 that, in appropriate norms, that hti and hti for some >0. 2 3 42 To make all this precise requires a priori estimates on the system the ab ove equations. We now recall Lemma 6.1 concerning equations of the form 7.2 . This result gives conditions ~ ~ ensuring that A b ehaves like the solution of the equation obtained by setting E to zero. ~ Our next task is to obtain an upp er b ound on E t in 4.49 of the form 6.7 . Since the ~ ~ equation for A t is coupled to that of , the factor Q in 6.7 will dep end on and A . The 0 next prop osition, together with Prop osition 4.7, provides estimates for the individual terms in ~ E. It is convenient to intro duce the notation: [A] T = sup hti jA t j 7.7 0tT [ ] T = sup hti jj t jj : 7.8 p; p 0tT To avoid cumb ersome notation, where it should cause no confusion, we shall abbreviate ex- pressions like [ ] T by [], until it is necessary to make the dep endence on T explicit. In the estimates b elow, we shall often use the notation C to denote a constant dep ending on ' k;p some W norm of the b ound state, '. Under our hyp otheses, ' is a suciently smo oth and exp onentially decaying function for which these norms are nite; see [1]. Because of Prop osition 5.1, we need only estimate the terms of E, given in 4.49 . The following prop osition is the main step toward the estimate on E. Prop osition 7.1. Estimates on the terms in E t : For 0 s t, the terms of E s ,as de ned in 4.49 , are estimated as follows: Z 2 2 7=4 i a ' jj C [A] [ ] hti ; 7.9 ' 1=4 8;3=4 Z 3 3 9=4 ii ' jj C [ ] hti ; 7.10 ' 8;3=4 2 13=8 iii jF a; j jj [A] ku k hti ; 7.11 2 1 0 X 1=4 2 5=4 0 iv jF a; j C jj [A] C [A] ; [ ] ; [B ] ; ku k hti ; 2 3 ' 3 0 X 1=4 8;3=4 4;1=4+ 1=4 0 7.12 nr 1 2 2 2 2 v F a; C jj jA j [A] hti ; 7.13 2 ' 0 1=4 nr 2 2 2 2 0 7=4 vi F a; C jj [A] [jAj jA j] hti ; 7.14 2 ' 5=4 1=4 13=8 4 0 2 2 3 00 nr hti [jAj jA j] + [A] + [jAj j A j] E jj C vii ' 7=4 7=4 1=4 2j : 7.15 43 P P jr j tends to zero; jr j b ounded and tends to zero as In iv , C r ;r ;r ;r is b ounded for j j 1 2 3 4 j j see Prop osition 7.4. Wenowembark on the pro of of this prop osition. Parts i and ii followbyHolder's inequality. To prove part iii , apply Holder's inequality and the linear propagator estimates of Theorem 2.3. We now fo cus on iv - vi . Estimation of F a; 2 3 Recall see 4.18 that Z 2 3 F a; = 3a ' : 7.16 2 3 3 We start by estimating jj jj . We rst express , de ned in 7.6 , as 3 8 3 Z 3 X 2 2 2 3 E t s P t = 3a ' + 3a' + ds; 7.17 1 c 3 I j j =1 where I =[0;t=2]; I =[t=2;t 1]; and I =[t 1;t]: 7.18 1 2 3 We estimate each integral separately. Z 3 X 2 2 2 3 jj t jj jj 3a ' + 3a' + jj ds: 7.19 jjE t s P 3 8 8 1 c I j j =1 The integrands are estimated as follows using the linear estimates of Corollary 2.1: Z Z t=2 9=8 jjE t s P f gjj ds C jt sj jjf gjj ds; 1 c 8 1;8=7 I 0 1 Z Z t 1 9=8 jjE t s P f gjj ds C jt sj jjf gjj ds; 1 c 8 1;8=7 t=2 I 2 Z Z t 3=8 jjE t s P f gjj ds C jt sj jjf gjj ds; 7.20 1 c 8 1;8=7 I t 1 3 where 2 2 2 3 jjf gjj C jAj jj' jj + jAjjj' jj + jj jj : 7.21 1;8=7 1;8=7 1;8=7 1;8=7 Prop osition 7.2. 2 2 2 i jAj jj' jj C jAj jj jj + jjB jj ; 1;8=7 ' 8 4 2 2 ; ii jAjjj' jj C jAj jj jj + jjB jj jj jj 1;8=7 ' 4 8 8 5=3 3 iii jj jj C jj jj ;' jj jj 7.22 1;8=7 2;1 8 44 proof: We prove part i . The estimates ii and iii follows similarly. 2 2 2 jj' jj C jj' jj + jj'@ ' jj + jj' @ jj 1;8=7 8=7 8=7 8=7 2 2 C jj' jj jj jj + jj'@ 'jj jj jj + jj' jj jj@ jj ; 7.23 4=3 8 4=3 8 8=5 4 byHolder's inequality. To express the right hand side of 7.23 in terms of jjB jj , and thereby 4 completing the pro of of part i , it suces to show that: jj@ jj C jjB jj : 7.24 4 4 1 p This follows if we show that the op erator @ B is b ounded on L with p = 4 . i p 1 To prove the L b oundedness of @ B , we can apply the results on the wave op erator, W i p 1;p in x2.2. Indeed, for any g 2 L , we have using the b oundedness of the wave op erators on W , for p 1, that 1 1 k@ B g k = k@ W B W g k i p i + p 0 + 1 1;p C kW B W g k + W 0 + 1 1;p C kB W g k W 0 + C kW g k C kg k : p p + Remark: We o er here an alternative pro of which do es not make use of the wave op erators, and therefore applies under weaker hyp otheses on V . We b egin with the square ro ot formula see [49] and 2.33 : 2 For any 2D B : Z 1 1 1 1=2 2 1 B = w B + w dw : 7.25 0 By 7.25 and the second resolvent formula we have: Z 1 1 1 1 1=2 2 1 2 1 @ B = @ B + w @ B + w V B + w dw : 7.26 i i i 0 0 0 1 p 2 2 1=2 The b oundedness in L p 1 of the op erator @ B holds b ecause j j + m is a i i 0 p 2 1 p multiplier on L ; see [64]. Similarly, @ B + w is b ounded on L for any w > 0 and i 0 estimation of the second term in 7.26 is reduced to estimation of the norm of the op erator 2 1 V B + w . Note that: Z 1 2 2 1 t B +w B + w = e dt 7.27 0 2 2 Since inf B = > 0, 2 2 t B =2 p ke k C 7.28 B L 45 and therefore Z 1 2 2 1 t =2+w 0 1 p k B + w k C e C 1 + w : 7.29 B L 0 p 1 Boundedness in L of @ B now follows. Namely, i Z 1 1=2 1 1 0 2 1 1 p p p w 1 + w dw < 1: + w k kV k k + C k@ B k@ B k k@ B 1 i i i B L B L B L 0 0 0 7.30 From 7.22 we see that jjB jj must be estimated. Recall that 4 B = B + B + B : 7.31 1 2 3 By 7.4 B = E t B P u + E t B P u ; 7.32 1 0 c 0 1 c 1 and so 1=2 jjB t jj C hti ku k ; 7.33 1 4 0 X see 3.7 for de nition of ku k . 0 X By 7.5 Z t 3 3 E t s a s B P ' ds; 7.34 B = 1 c 2 0 which can by Theorem 2.3 b e estimated as Z t 3=4 3 3 3 1=2 jt sj jA s j kP ' k ds C jj [A] hti : 7.35 jjB t jj C jj c ' 2 4 4=3 1=4 0 The estimation of jjB t jj is more involved. By 7.6 , 3 4 Z h i t 2 2 2 3 B = sin B t s P 3a ' + 3a' + ds: 7.36 3 c 0 By Theorem 2.3, Z h i t 1=2 2 2 2 3 jjB t jj C jj jAj jt sj jj' jj + jAjjj' jj + jj jj ds: 7.37 3 4 1;4=3 1;4=3 1;4=3 0 The following prop osition provides estimates for the integrand. It is proved in the same manner as Prop osition 7.2. Prop osition 7.3. 2 2 2 jAj jj' jj C jAj jj jj + jjB jj + jjB jj + jjB jj ; ' 8 1 4 2 4 3 4 1;4=3 2 2 jAjjj' jj C jAj jj jj + jj jj jjB jj ' 8 4 1;4=3 8 3 2 jj jj 3 jj jj jj@ jj : 7.38 1;4=3 2 8 46 Anticipating the b ehavior 1=4 3=4 jA t j t ; jj t jj t ; 8 1 and using 7.33 , we nd that B is driven by terms which are formally of order hti . Esti- 3 4 1 1=2 1=2+ 0 mation in L will lead to convolution of hti with hti giving a rate of hti , for any > 0. 0 For 0 t T , with T xed and arbitrary,we have: 2 2 2 5=4 1 jAj jj' jj C [A] [ ] hti + [B ] hti 4=3;1 ' 8;3=4 1 4;1=2 1=4 1 1+ 0 ; + [B ] hti + [B ] hti 2 4;1=2 3 4;1=2 0 where is p ositive and arbitrary. Also, 0 2 2 7=4 1 jAjjj' jj C [A] [ ] hti + [ ] [B ] hti ' 1 4;0 1;4=3 1=4 8;3=4 8;3=4 3 2 3=2 jj jj jj jj [ ] hti : 1;2 1;4=3 8;3=4 It follows that for 0 t T Z t 1=2 1+ 0 jjB jj C jj jt sj dshti G A; ; T ; 7.39 3 4 ' 0 where 2 G A; ; T [A] [ ] + [B ] + [B ] + [B ] 1 2 3 8;3=4 4;1=2 4;1=2 4;1=2 1=4 0 2 2 + jj jj [ ] : [ ] + [ ] [B ] + [A] 1;2 8;3=4 1 4;0 1=4 8;3=4 8;3=4 Therefore, for 0 t T : 1=2+ 0 ; 7.40 jjB t jj C jj G A; ; T hti 3 4 ' for any > 0. Wenow use the estimate 7.40 in 7.22 in order to obtain a b ound on jj t jj . 0 3 8 First, a consequence of 7.22 , 7.33 , 7.35 and 7.40 is that for 0 t T : 2 2 2 3 1+ 0 jAj jj' jj C [A] [ ] + ku k + [A] + [B ] hti ; ' 0 X 3 1;8=7 8;3=4 4;1=2 1=4 1=4 0 2 2 3 1+ 0 jAjjj' jj C [A] [ ] + fku k + [B ] + [A] g [ ] hti ; 1;8=7 ' 1=4 0 X 3 4;1=2 8;3=4 0 8;3=4 1=4 5=3 3 5=4 jj jj C jj jj [ ] hti : 7.41 1;8=7 ' 1;2 8;3=4 47 Substitution of 7.41 into 7.17-7.20 leads to an estimate for jj t jj . For 0 t T : 3 8 1+ 2 3 0 jj t jj jj C hti [A] f[ ] + ku k + [A] + [B ] g 3 8 ' 8;3=4 0 X 3 4;1=2 1=4 1=4 0 h i 2 3 + [A] f [ ] + ku k + [B ] + [A] [ ] g 0 X 3 1=4 4;1=2 8;3=4 8;3=4 0 1=4 5=3 + C jj jj ;' [ ] ; 7.42 1;2 8;3=4 where is arbitary. 0 Finally we can now estimate F a; using 4.18 and 7.42 . Cho osing so that 2 3 0 1+ = 3=4 ; with > 0; 0 0 0 we have Prop osition 7.4. 2 2 3 jF a; j C jj [A] [A] f [ ] + ku j + [A] + [B ] g 2 3 ' 0 X 3 8;3=4 4;1=4+ 1=4 1=4 1=4 0 h i 2 3 + [A] f [ ] + ku k + [A] + [B ] [ ] g 1=4 0 X 3 4;1=4+ 8;3=4 8;3=4 1=4 0 5=3 5=4 0 + C jj jj ;' [ ] hti : 7.43 1;2 8;3=4 nr 1 nr 2 Estimation of F a; and F a; 2 2 By 4.23 nr 1 3 1 1 iB t 3 = A B B 3 + i0 e P ' ; and c 0 2 Z t 3 1 1 iB t s i3 s 2 0 3 nr 2 B B 3 + i0 e e A A dsP ' = c 2 0 Estimation using Prop osition 2.2 gives: Z nr 1 3 nr 1 2 F a; ' = 3 jjjaj 2 Z 1 3 iB t 3 2 3 2 ' [B B 3 + i0 ] e P ' = C jj jA j jAj c 0 2 3 2 3 1 iB t 1 3 C jj jA j jAj khxi ' k khxi B 3 + i0 e P B ' k 0 2 c 2 5 2 2 2 + 4 C jj jA j [A] hti : 7.44 ' 0 1=4 Here, we have used that 1=2+6=5 = 5=4+ , for some > 0. The constant C dep ends ' 1 1 2 on khxi B hxi k . It is easy to check that this norm is b ounded if we replace B by B L 48 2 B . The estimate of interest is reduced to this case using the Kato square ro ot formula 7.25 . Similarly,we have Z nr 2 2 3 nr 2 F a; = 3jjjaj ' 2 Z Z t 1 2 2 3 i t s B 3 2 0 3 C jj jAj ' [B B 3 + i" ] e A A ds P ' c 0 Z t 3 2 2 + 2 0 2 ht si C jj jAj jAj jA j ds ' 0 7 2 2 2 0 4 C jj [A] [jAj jA j] hti ; 7.45 ' 5=4 1=4 where wehave taken suciently small so that the last integral t integral is convolution with 5 7 1 4 4 , which exceeds hti an L function, which then preserves the decay rate , hti nr Estimation of E : 2j To estimate 4.47 we use Prop ositions 2.1 and 2.2. Due to the singularity in the resolvent at frequency 3 , the case j =7, is most dicult and we fo cus on it. To treat this singularity, we use Prop osition 2.2, for n =3: 6 2 iB t s 5 khxi B e hxi k C ht si k k : 7.46 7 2 1;2 Use of this estimate in 4.47 yields nr jE j 2j 2 4 0 7=4 2 13=8 4 00 3 0 2 13=8 C jj [jAj jA j] hti + [A] hti + [jAj jA j + jAj jA j ] hti ; ' 7=4 9=4 1=4 from which estimate vii of Prop osition 7.1 follows. This nally completes the pro of of Prop o- sition 7.1. ~ Prop osition 7.1 can now b e used to obtain the desired estimate for E t and therefore A t , 0 00 for jAj suciently small. Note that the right hand side of the estimates dep end on jA j and jA j. These can each be estimated directly from the equation for A, 4.48 , and its derivative. The 3 0 00 3 4 , the only e ect b eing a change in result is that jA j and jA j may be estimated by [A] hti 1=4 the multiplicative constants which is indep endent of A and . This observation together with Prop osition 7.1 yields: Prop osition 7.5. There is p ositivenumber such that 5 4 jE t j Q A; hti 7.47 0 49 where 3 Q A; = [ ] 7.48 1 + C [A] ; [ ] ; [B ] ; ku k + [A] 0 1=4 8;3=4 3 4;1=4+ 0 X 1=4 0 8;3=4 It now remains to estimate jj jj . 8 jj jj jj jj + jj jj + jj jj 8 1 8 2 8 3 8 Z t 3 3 jjE t P u jj + jjE t P u jj + jj jjE t s a s P ' jj ds + jj jj : 0 c 0 8 1 c 1 8 1 c 8 3 8 0 Using Theorem 2.3 and the b ound 7.42 we get Prop osition 7.6. 3 ku k + [A] jj jj C 0 X 8 ' 1=4 2 3 + [A] f [ ] + ku k + [A] +[B ] g 8;3=4 0 X 3 4;1=4+ 1=4 1=4 0 h i 2 3 + [A] f [ ] + ku k + [A] + [B ] [ ] g 0 X 3 1=4 4;1=4+ 8;3=4 8;3=4 1=4 0 5=3 3=4 + C jj jj ;' [ ] hti : 7.49 1;2 8;3=4 The following prop osition summarizes our lab ors. Prop osition 7.7. For any T > 0: 3 2 ku k + [A] [ ] T C T + [B ] T 0 X 8;3=4 ' 3 1=4 4;1=4+ 0 5=3 T + C jj jj [ ] ' 1;2 8;3=4 8 4 4 5 [A] T jA j + Q A; 0 0 1=4 2 2 2 2 [B ] T [A] T + [ ] T + [B ] T + [B ] T ; 3 4;1=4+ 2 3 0 1=4 8;3=4 4;1=2 4;1=4+ 0 sup jj t jj C E u ;u C ku ;u k : 1;2 0 1 0 1 1;2 0tT The rst three estimates are proved ab ove while the last follows from conservation of energy see section 1 and the decomp osition of the solution. Now de ne M T [ ] T + [A] T + [B ] T : 7.50 3 8;3=4 1=4 4;1=4+ 0 Then, combining the estimates of the previous prop osition we have, for some >0: M T 1 M T C ku k : 7.51 ' 0 X 50 If M 0 and ku k are suciently small, we have by the continuity of M T , that there is 0 X a constant M , which is indep endent of T , such that for all T M T M : 7.52 This completes the pro of of Theorem 1.1. 8. Summary and discussion Wehave considered a class of nonlinear Klein-Gordon equations, 1.1 , which are p erturbations of a linear disp ersive equation which has a time p erio dic and spatially lo calized b ound state solution. The unp erturb ed and p erturb ed dynamical systems are Hamiltonian. Wehave shown that if a nonlinear resonance condition 1.8 holds, then solutions with suciently small initial data tend to zero as t !1. This resonance condition is a nonlinear variantofthe Fermi golden rule 1.22 . A consequence of our result is that time-p erio dic and spatially lo calized solutions do not p ersist under small Hamiltonian p erturbations. Time-decay of small amplitude solutions is also a prop erty of the translation invariant V 0 nonlinear Klein-Gordon equation. However, the presence of a b ound state of the unp erturb ed problem, causes a nonlinear resonance leading to the anomalously slow radiative decay of solutions. Wenow conclude with some further remarks on the results of this pap er, mention directions currently under investigation and some op en problems. 1. Anomalously slow time-decay rates: It is natural to compare the time-decay rate of solutions describ ed by Theorem 1.1 with those of related problems. 1a Translation invariant linear Klein-Gordon equation V 0 and = 0 : We shall refer to free dispersive rates of decay as those asso ciated with the constant co ecient equation: 2 2 @ u u + m u = 0: 8.1 t Results on this are presented in x2. Roughly sp eaking, if the initial data has a sucientnumber 1 1 0 n p p 0 2 p in L . Here, of derivatives in L ; 1 < p 2, then the solution decays at a rate O t 1 0 1 p + p =1. 1b Translation invariant nonlinear Klein Gordon equation V 0 and 6= 0 : For small initial conditions it has b een shown that solutions decay at free disp ersive rates; see [65] and references cited therein. 51 1c Linear Klein-Gordon equation with a potential having a bound state, as hypothesized V 6= 0 and = 0 : By the sp ectral theorem, a typical solution will decomp ose into a linear sup erp osition of i a b ound state part, of the form: R cos t + ' x , and ii a part which 0 0 disp erses to zero at free disp ersive rates. 1d Nonlinear Klein-Gordon equation with a potential having a bound state, as hypothesized V 6= 0 and 6= 0 : Whereas the decaying part of the solution in the ab ove cases tends to zero at a free disp ersive rate, Theorem 1.1 implies that the decay rate is anomalously slow. In 1 9 8 8 4 8 particular, the decay rate obtained in L is O t , while the free disp ersive rate in L is O t . This slow rate of decay, due to the nonlinear resonant interactions give rise to a long-lived or metastable states. 2. Disp ersive Hamiltonian normal form In a nite dimensional Hamiltonian system of one degree of freedom, the general normal form is: 0 2 4 2n A = i c + c jAj + c jAj + ::: + c jAj + ::: A; 8.2 10 21 32 n+1;n where c are real numb ers. In the present context wehave the dispersive Hamiltonian normal n;n+1 form 0 2 4 2n A = k + k jAj + k jAj + ::: + k jAj + ::: A; 8.3 10 21 32 n+1;n where k = d + ic n+1;n n+1;n n+1;n are, in general, numb ers with real and imaginary part. Contributions to the real parts of co ecients come from resonances with the continuous sp ectrum. If 1.8 fails there are two p ossibilities; either a 3 do es not lie in the continuous sp ectrum of B it lies in the gap 0;m , or b 3 lies in the continuous sp ectrum of B but we are in the non-generic situation 3 where the pro jection of ' onto the generalized continuum eigenmo de of frequency 3 is zero. In either case, we exp ect that, typically,internal dissipation would arise in the normal form at higher order. More precisely,we conjecture that the leading order nonzero d , whichwould n +1;n corresp ond to a resonance with the continuum of a higher harmonic: q 2 B , q > 3 cont is always negative. If the sign of d were p ositive, this would seem to be in violation n +1;n of the conservation of energy and the implied Lyapunov stability of the zero solution. The 1 2n corresp onding decay rate would then be slower, sp eci cally O jtj . From this p ersp ective one may exp ect the existence of breather solutions of integrable non- linear ows like the sine-Gordon equation or the extremely long-lived breather -like states of 4 the - mo del as corresp onding to the case where to all orders the co ecients d are zero. n+1;n 52 Do es the vanishing of all such co ecients haveaninterpretation in terms of the in nitely many time-invariants for the integrable ow? 3. Multiple b ound state problems: Of interest are problems where the underlying p otential, V x , supp orts more than one b ound state. What is the large time b ehavior of such systems? Our analysis and the ab ove remarks suggest that a more general normal form could b e develop ed, and the corresp onding in general slower decay rates anticipated. Is it p ossible that a resonance among the multiple "discrete oscillators" can be arranged so that one gets a p ersistence of nondecaying solutions, or do es radiation to the continuum always win out? 4. Relation to center manifold theory: Our program of decomp osing the original conserva- tive dynamical system into a nite dimensional dynamical system 1.20 whichisweakly coupled to an in nite dimensional dynamical system is one commonly used in dissipative dynamical sys- tems. There, it is often p ossible using the center manifold approach [14], [67], to construct an invariant center-stable manifold. The dynamics in a neighborhood of an equilibrium p oint are characterized by an exp onentially fast contraction on to the stable-center manifold. The con- traction is exp onential b ecause the part of the linearized sp ectrum asso ciated with the in nite dimensional part of the dynamics is contained in the left half plane. In the current context of conservative dynamical systems, the linearization ab out the equilibrium p oint here u 0 lies on the imaginary axis; in particular, two complex conjugate eigenvalues i and continous sp ec- trum from im to i1. The analogue of dissipation is the mechanism of disp ersive radiation of energy, related to the continuous sp ectrum, and the asso ciated algebraic time-decay. There is recentwork on the application of center manifold metho ds to certain sp ecial conser- vative dynamical systems of nonlinear Schrodinger typ e; see the center manifold analysis of [46] applied to problem studied by the authors in [60]. It would b e of interest to understand whether geometric insight on the structure of the phase space for problems of the typ e considered in this pap er can b e obtained using the ideas of invariant manifold theory. 5. Systems with disorder: In this pap er, we have seen the e ect of a single lo calized defect on the wave propagation dynamics in a nonlinear system. Of great interest would be an understanding of the e ects of a spatially random distribution of defects mo deled, for example, by random p otential V x on the lo calization of energy in nonlinear systems such as 1.1 . Related questions are studied in [22], [25], [10]. 53 References [1] S. Agmon, Lectures on Exp onential Decay of Solutions of Second-Order Elliptic Equations: Bounds on Eigenfunctions of N-Bo dy Schrodinger Op erators, Princeton University Press 1982 [2] P.C. Aichelburg & R. Beig, Radiation damping as an initial value problem, Ann. Phys. 98 1976 264{283. [3] V.I. Arnol'd, Geometric Metho ds in the Theory of Ordinary Di erential Equations, Springer-Verlag, New York 1983 [4] J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrodinger equations, preprint 1994. [5] P. Brenner, On the existence of global smooth solutions of certain semilinear hyperbolic equations, Math. Z. 167 1979 99-135; On scattering and everywhere de ned scattering operators for nonlinear Klein-Gordon equations,J. Di . Eqns. 56 1985 310-344. [6] H. Brezis, Periodic solutions of nonlinear vibrating strings and duality principles, Bull. Amer. Math. So c. N.S. 8 1983 409-426. [7] L. Bonilla & J.B. Keller, Irreversibility and nonrecurrence, J. Stat. Phys. 42 1986 1115- 1125. [8] B. Birnir, H.P. McKean & A. Weinstein, The rigidity of sine-Gordon breathers, Comm. Pure Appl. Math. 47 1994 , 1043{1051. [9] B. Birnir, Qualitative analysis of radiating breathers, Comm. Pure Appl. Math. 47 1994 , 103{119. [10] J.C. Bronski, A numerical study of solitary wave propagation in a disordered medium, preprint. [11] V.S. Buslaev & G.S. Perel'man, Scattering for the nonlinear Schrodinger equation: states close to a soliton, St. Petersburg Math. J. 4 1993 1111{1142. [12] V.S. Buslaev & G.S. Perel'man, On the stability of solitary waves for nonlinear Schrodinger equation, preprint. 54 [13] D.K. Campb ell & M. Peyrard, Solitary wave col lisions revisited,Physica D 18 1986 47 [14] J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York 1981 [15] E. Co ddington & N. Levinson, Theory of Ordinary Di erential Equations, McGraw-Hill, New York, 1955. [16] J.M. Coron, Periode minimale pour une corde vibrante de longeur in nie, C.R. Acad. Sci. A294 1982 127-129 [17] W. Craig & C.E. Wayne, Newton's method and periodic solutions of nonlinear wave equa- tions, Comm. Pure Appl. Math. 46 1993 , 1409{1498. [18] H.L. Cycon, R.G. Fro ese, W. Kirsch&B. Simon, Schrodinger Op erators, Springer-Verlag, Berlin Heidelb erg New York, 1987 [19] J. Denzler, Nonpersistence of breather families for the perturbed Sine-Gordon equation, Commun. Math. Phys. 158 1993 397{430 [20] S. Debi evre, P.D. Hislop & I.M. Sigal, Scattering theory for the wave equation on non- compact manifolds, Rev. Math. Phys. 4 1992 575{618. [21] G. Ford, M. Kac & P. Mazur, Statistical properties of assemblies of coupled oscil lators J. Math. Phys. 6 1965 504{515 [22] J. Frohlich, T. Sp encer & C.E. Wayne, Localization in disordered nonlinear dynamical systems,J. Stat. Phys. 42 1986 247{274. [23] J. Ginibre & G. Velo, Scattering theory in the energy space for a class of nonlinear wave equations, Comm. Math. Phys. 123 1989 535{573. [24] M. Grillakis, J. Shatah & W. Strauss, Stability theory of solitary waves in the presence of symmetry, I,J.Func. Anal. 74 1987 160{197. [25] S.A. Gredeskul, Y.S. Kivshar, A. Sanchez & L. Vazquez, Localization decay induced by strong nonlinearity in disordered systems,Phys. Rev. Lett. 64 1990 1693{1696. [26] J. Guckenheimer & P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag 1983 55 [27] C. G erard & I.M. Sigal, Space-time picture of semiclassical resonances, Commun. Math. Phys. 145 1992 281{328. [28] W. Hunziker & I.M. Sigal, Time dependent scattering theory for N-body quantum systems, ETH-Zuric h preprint 1997 [29] W. Hunziker, I.M. Sigal & A. So er, Minimal escape velocities, ETH-Zuric h preprint 1997 [30] V. Jaksi c & C.-A. Pillet, On a model for quantum friction. I. Fermi's golden rule and dynamics at zero temperature, Ann. Inst. H. Poincar e Phys. Th eor. 62 1995 47{68. [31] A. Jensen & T. Kato, Spectral properties of Schrodinger operators and time-decay of wave functions, Duke Math. J. 46 1979 , 583{611. [32] G. Jona-Lasinio, C. Presilla & J. Sjostrand, On Schrodinger equations with concentrated nonlinearities, Ann. Phys. 240 1995 [33] J.-L. Journ e, A. So er & C.D. Sogge, Decay estimates for Schrodinger operators, Comm. Pure Appl. Math. 44 1991 573{604. [34] S. Kichenassamy, Breather solutions of nonlinear wave equations, Commun. Pure Appl. Math. 44 1991 789 [35] Yu.S. Kivshar & B.A. Malomed, Rev. Mo d. Phys. 61 1989 763 n+1 [36] Klainerman, S. Remark on the asymptotic behavior of the Klein-Gordon equation in IR , Comm. Pure Appl. Math. 46 1993 137{144. [37] S.B. Kuksin, Nearly integrable in nite-dimensional Hamiltonian systems, Lecture Notes in Math. 1556, Springer Verlag, Berlin - Heidelb erg - New York, 1993. [38] H. Lamb, On a peculiarity of the wave-system due to the free vibrations of a nucleusin an extended medium, Pro c. London Math. So c. 32 1900 208-211 [39] P.-L. Lions, The concentration compactness principle in the calculus of variations, I. The local ly compact case, Ann. Inst. Henri Poincar e, Analyse nonlin eaire, 1 1984 109{145. [40] R.S. MacKay & S. Aubry, Proof of existenceofbreathers for time-reversible or Hamiltonian networks of weakly coupled oscil lators, Nonlinearity 7 1994 1623-1643. 56 [41] D.W. McLaughlin & A.C. Scott, Perturbation analysis of uxon dynamics,Phys. Rev. A18 1978 1652 [42] D.W. McLaughlin & J. Shatah, Homoclinic orbits for pde's, preprint. [43] E. Mourre, Absence of singular continuous spectrum for certain self-adjoint operators, Com- mun. Math. Phys. 78 1981 391 [44] F. Nier, The dynamics of some open quantum systems with short-range nonlinearities, preprint 1997. [45] L. Nirenb erg, Topics in Nonlinear Functional Analysis, Courant Institute Lecture Notes, 1974. [46] C.-A. Pillet & C.E. Wayne, Invariant manifolds for a class of dispersive , Hamiltonian, partial di erential equations,to app ear in Journal of Di erential Equations. [47] R.M. Pyke & I.M. Sigal, Nonlinear wave equations: Constraints on periods and exponential bounds for periodic solutions , Duke Math. J. 88 1997 [48] P. Perry, I.M. Sigal & B. Simon, Spectral analysis of N-body Schrodinger operators, Ann. Math 114 1981 519{567 [49] M. Reed & B. Simon, Mo dern Metho ds of Mathematical Physics, Volume 1, Functional Analysis, Academic Press 1972 [50] M. Reed & B. Simon, Mo dern Metho ds of Mathematical Physics, Volume 4, Analysis of Op erators, Academic Press 1978 [51] H.A. Rose & M.I. Weinstein, On the bound states of the nonlinear Schrodinger equation with a linear potential, Physica D 30 1988 207{218. [52] J.A. Sanders & F. Verhulst, Averaging Metho ds in Nonlinear Dynamical Systems, Applied Mathematical Sciences 59 Springer-Verlag, New York 1985 [53] L.S. Schulman, C.R. Do ering & B. Gaveau, Linear decay in multilevel quantum systems,J. Phys. A: Gen. 24 1991 2053{2060. 4 [54] H. Segur & M.D. Kruskal, Nonexistence of smal l amplitude breather solutions in theory, Phys. Rev. Lett., 58 1987 747 57 [55] J. Shatah & W.A. Strauss, Breathers as homoclinic geometric wave maps,Physica D 1996 [56] I.M. Sigal, Nonlinear wave and Schrodinger equations I. Instability of time-periodic and quasiperiodic solutions, Commun. Math. Phys. 153 1993 297 [57] I.M. Sigal, General characteristics of nonlinear dynamics, in Sp ectral and Scattering The- ory; Pro ceedings of the Taniguchi international workshop, ed. M. Ikawa, Marcel Dekker, Inc. New York - Basel - Hong Kong 1994 [58] I.M. Sigal & A. So er, Local decay and velocity bounds for quantum propagation, preprint, Princeton 1988 ftp://www.math.rutgers.edu/pub/so er [59] E. Skibsted, Propagation estimates for N-body Schrodinger operators, Commun. Math. Phys. 142 1991 67{98 [60] A. So er & M.I. Weinstein, Multichannel nonlinear scattering theory for nonintegrable equa- tions I& II, Commun. Math. Phys. 133 1990 , 119-146; J. Di . Eqns. 98, 1992 , 376-390. [61] A. So er & M.I. Weinstein, Dynamic theory of quantum resonances and perturbation theory of embedded eigenvalues, in Pro ceedings of Conference on Partial Di erential Equations and Applications, University of Toronto, June, 1995, CRM Lecture Notes, Eds. P. Greiner, V. Ivrii, L.Seco, & C. Sulem [62] A. So er & M.I. Weinstein, Time dependent resonance theory, to app ear in Geometric and Functional Analysis. [63] A. So er & M.I. Weinstein, The large time behavior of the nonlinear Schrodinger equation; Selection of the ground state and instability of excited states, in preparation. [64] E.M. Stein, Singular Integrals and Di erentiability Prop erties of Functions, Princeton Uni- versity Press 1970 [65] W.A. Strauss, Nonlinear Wave Equations, CBMS Regional Conference Series in Mathemat- ics 7 AMS 1985 [66] D.M. Stuart, Minimal periods for solutions of some classical eld equations, SIAM Journal of Mathematical Analysis, to app ear. [67] A. Vanderbauwhede & G. Io oss Center manifold theory in in nite dimensions, Dynamics Rep orted 2 1990 58 [68] C.E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys. 127 1990 479 [69] A. Weinstein, Periodic nonlinear waves on a half-line Commun. Math. Phys. 99 1985 383; Erratum 107 1986 177. [70] M.I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equa- tions, Commun. Pure Appl. Math. 39 1986 51{68. [71] V. Weisskopf & E. Wigner, Berechnung der naturlichen Linienbreite auf Grund der Dirac- schen Lichttheorie,Z.Phys. 63 1930 54{73. k;p [72] K. Ya jima, W -continuity of wave operators for Schrodinger operators, J. Math. So c. Japan 47 1995 551-581. [73] F. Zhang, Y.S. Kivshar, B.A. Malomed & L. V azquez, Kink capture by a local impurity in the sine Gordon model,Phys. Lett. A159 1991 318{322. 59