American. Mathematical Society (Providence, Rhode Island 1999). 6] W. Craig, C.E. Wayne: Newton's Method and Periodic Solutions of Nonlinear Wave Equa-.
EXPONENTIAL STABILITY IN A NONLINEAR STRING EQUATION SIMONE PALEARI
DARIO BAMBUSI
SERGIO CACCIATORI
Abstract. We study the nonlinear wave equation
utt ? c2 uxx = (u) u(0; t) = 0 = u(; t) with an analytic nonlinearity of the type (u) = u3 + k4 k uk . On each
P
small{energy surface we consider a solution of the linearized system with initial datum having the pro le of an elliptic sinus: we show that solutions starting close to the corresponding phase space trajectory remain close to it for times growing exponentially with the inverse of the energy. To obtain the result we have to compute the resonant normal form of 0.1, and we think this could be interesting in itself.
1. Introduction and statement In this paper we study small amplitude solutions of the equation 0.1 concentrating on the periodic behaviour. Consider the solution of the linear string equation utt ? c2 uxx = 0 with the initial datum: u(x; 0) = Vmsn(!xjm) u_ (x; 0) = 0 ; (1.1) where sn is the elliptic sine [1], m, ! and Vm are constants whose values will be xed later in the paper (see 4.1,4.4), and is a small parameter; its phase space trajectory is a closed curve that will be denoted by ?. The main result of the present paper is the forthcoming theorem 1.3, it ensures that solutions starting close to ? remain close to it for times exponentially long with ?1 . In the corresponding statement we will make use of the distance d(:; :) induced in the phase space L2 (0; ) H01 (0; ) 3 (u;_ u) by the norm Z (1.2) k(u;_ u)k2 := 21 (u_ (x)2 + c2 ux(x)2 )dx : 0 . Theorem 1.3 Consider the nonlinear wave equation 0.1 with analytic in a neighbourhood of the origin, assume (0) = 0 (0) = 00 (0) = 0 and 000 (0) 6= 0. Then there exist strictly positive constants ; C1 ; :::; C4 such that the following holds true: x < and consider an initial datum z0 = (u_ 0 ; u0) 2 L2 (0; ) H01 (0; ) close to ? , precisely such that d(z0 ; ? ) C1 2 ; then the corresponding solution z (t) remains close to it for exponentially long times, precisely one has d(z (t); ? ) C2 2 ? for times jtj C3 2 exp C4 . This result is obtained by applying the general theory developed in [4], and recalled in the next section, which requires the veri cation of a non degeneracy condition. From the technical point of view the veri cation of the non degeneracy condition is quite complicate, indeed as pointed out by Moser [10] in a similar 1
(0.1)
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SIMONE PALEARI
DARIO BAMBUSI
SERGIO CACCIATORI
context, non degeneracy is "a property which can be rarely veri ed"; this is a remarkable case where it is possible. We think that besides its interest for the dynamics of the string equation, our result could be interesting as a rst natural application of the theory of [4]. We recall the work [9] where existence of some periodic solutions for the equation utt ? uxx + u3 = 0 was obtained, but as far as we know no stability result are known for these solutions. Finally we recall that the results usually obtained by KAM theory (see [8, 6, 5, 12]) or Nekhoroshev theory [5, 3] do not apply to 0.1 since it is a perturbation of a completely resonant system, and even the result of [2] does not apply since it requires the existence of an integrable rst order normal form. 2. General setting First we rescale u by introducing a variable u0 de ned by u0 = u. We recall that the equation for the rescaled variable is hamiltonian with hamiltonian H(p0 ; u0) = h! (p0 ; u0) + 2 f (u0 ) + 3 f1 (u0 ) ; (2.1) where Z 1 0 0 p0 2 (x) + c2 u0x2 (x) dx ; h! (p ; u ) = 2 0 the main part of the perturbation given by Z (2.2) f (u0 ) = 41 u0 (x)4 dx ; 0 and higher order corrections given by X k k?4 Z 0 0 u k+1 (x)dx : f1 (u ) = k + 1 0 k4 To x ideas we will always consider the case where the sign in front of the integral in 2.2 is plus. In the minus case nothing changes. The idea of [4] is to use averaging theory to transform 2.1 into a system of the form H(p0 ; u0 ) = h! (p0 ; u0 ) + 2 hf i(p0 ; u0 ) + 3 Z (p0 ; u0 ) + exp ? C R (p0 ; u0) ; (2.3)
where hf i is the average of f with respect the unperturbed ow (for a precise de nition see equation 3.1), Z is a part of the Hamiltonian which commutes with h! , and R is an exponentially small remainder. Consider then the simpli ed system obtained from 2.3 by neglecting R, namely Hs (p0 ; u0 ) = h! (p0 ; u0 ) + 2 hf i(p0 ; u0 ) + 3 Z(p0 ; u0 ) : (2.4) Then h! is an integral of motion independent of the energy for 2.4. The critical points of Hs constrained to the surface of constant h! = 1 are invariant sets. It easy to realize that if they are non degenerate in the transversal directions they are trajectories of periodic solutions of 2.4. Moreover, if such non degenerate critical points are also extrema, the corresponding periodic orbits are stable. Taking into account the remainder R one obtains that the above extrema are no more invariant sets, but, due to the very slow change in time of both Hs and h! for the complete dynamics, it turns out that solutions starting close to extrema remain close to them for exponentially long times.
EXPONENTIAL STABILITY IN A NONLINEAR STRING EQUATION
3
Finally, remark that usually one can hope to compute hf i but not the exact expression of Hs , so we will look for non degenerate critical points of Hs jh?! 1 (1) as perturbations of non degenerate critical points of hf ijh?! 1 (1) . Actually, applying the theory of [4] (see theorem 6.1), in the improved formulation by [11] one obtains the following theorem (stated in non rescaled variables), which gives a more precise description than theorem 1.3. Theorem 2.5 Consider the nonlinear wave equation 0.1 with analytic in a neighbourhood of the origin, assume (0) = 0 (0) = 00 (0) = 0 and 000 (0) 6= 0. There exist strictly positive constants ; C4 ; :::; C9 and a family of curves f g0 0, also (V; 2 ; 2 E; 2 m; 3 ) is a solution of the same system.
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SIMONE PALEARI
DARIO BAMBUSI
SERGIO CACCIATORI
So, we will remove the condition 4.9 and look for a solution of the system 4.8, 4.10, 4.6, 4.7, with the unknowns (V; ; m; ). Remark that it is possible to nd by quadratures the solution of 4.8: de ne (s) := ?s + 21 s2 + 321 s4 , then V (x) is obtained by Z V ds q x= ; (4.12) 2 0 0 ( E ? ( s )) m 0 where E is a parameter representing the mechanical energy of the nonlinear oscillator 4.8. We state now a lemma which will be useful in the following. Its proof is deferred to the appendix. Lemma 4.13 The equation 4.8 with the condition 4.7 has solutions only if = 0. In such case condition 4.7 is implied by 4.6. So we are left with the system 4.8, 4.6, 4.10, with = 0, for the unknowns (V; ; m). From Z V dV q ; x= 2 0 ? (V )) 0 ( E m with := 0 , one gets p 2 sin ' = VV ; x = p 4 m 2 F ' ? 16V+m V 2 16 + Vm m m where F is the incomplete elliptic integral of rst kind, and E 0 = (Vm). Using the de nition of elliptic sine, which is the inverse of such elliptic integral, we get the solution of 4.8 (with = 0) in the form V (x) = Vm sn(! xjm) ; (4.14) with p 2 ! = 164m+ Vm ; (4.15) Vm2 : m=? 16 + Vm2 Since the natural period of the elliptic sine is 4K, (4.6) gives: (4.16) ! 2n = 4K ) !n = 2K n : Now consider condition 4.10: using properties of elliptic functions we have ? EV 2 ; = 38 KmK (4.17) m inserting in the second of 4.15, we obtain 4.1 which, as stated before, is the equation xing the value of the parameter m. To complete the picture we give the value of the other parameters involved. From 4.9 we obtain a relation between E and m ?1 1E: 4 (4.18) mn = 9128 2 5 (m + 14m + 1)K n4 c2 From this last equation and using the 4.15 we have 2 ? 2 (4.19) Vm n = 92 (m2 + 14mm + 1)K3 n12 cE2 ;
EXPONENTIAL STABILITY IN A NONLINEAR STRING EQUATION
and than also
?(1 + m) 1E 2 n = 932 (m2 + 14m + 1)K3 n2 c2 :
7
(4.20)
5. Extrema In this section we will show that, among all the critical point found, there's only one extremum, while all the other points are saddle points. Proposition 5.1 hf i constrained to SE has only one extremum, V1 which is a maximum; the other critical points Vn ; n 6= 1 are saddle points. Proof. We will show that all the critical points but the rst are saddle points. Then we will ensure the presence of a maximum by variational methods. First of all, we calculate the second di erential of hf ijSE at the critical points: it is given by the following formula d2 (hf ijSE ) (Vn )(h; h) = d2 h! (Vn )(h; h) + d2 hf i(Vn )(h; h) ; where is the lagrange multiplier at the critical point and h belongs to the tangent space to SE . Thus we have Z Z 3 2 2 2 d (hf ijSE ) (Vn )(h; h) = c hx(x)dx + 8 Vn2 (x)h2 (x)dx+ ? ?
Z 2 Z Z Vn (x)h(x)dx ; Vn2 (x)dx h2 (x)dx + 83 + 163 ? ? ? which can be expressed as an inner product hh; An hiL2 , with An h =Bn h + 43 [Vn h] Vn ; (5.2) ? Bn h =mn hxx + 83 Vn2 + Vn2 h: We must evaluate this second di erential along directions lying on the tangent space to the energy surface. In the forthcoming computation we use the remark that if i is an even integer, and j is an odd integer then Vi is L2 and H 1 orthogonal to Vj . So, having xed n, consider a value of i with a di erent parity. One has
Z Z 1 3 3 2 2 hVi ; An Vi iL2 = hVi ; mn Vi xx + 8 Vn + 2 Vn Vi + 8 Vn Vn Vi iL2 Z Z 2 Z Z n 2 + 3 V 2 V 2 ? 1 mn V 4 + 3 V V V = n ? m i n i n i 8 mi i 8 i 8 mi Z 2 2 2 = c24 Ec4 n14 3(1 + m)2 n i?2 i + 4(1 + 5m + m2 ) + 38 Vn2 Vi2 ; with 2 c4 := 38 (m2 + 141m + 1)K3 :
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SIMONE PALEARI
DARIO BAMBUSI
SERGIO CACCIATORI
When i is suciently large the above expression is negative, indeed 1 + 5m + m2 is easily seen to be negative (using the value of m), and all the positive terms vanish as i ! 1. Concerning the non existence of maxima it is easy to see that for any n 2 there are directions h making the value of the second di erential positive. For even n take i = 1, and for odd n 6= 1 take i = 2 using 3(1 + m)2 + 4(1 + 5m + m2 ) > 0 one obtains that the square bracket is positive. Up to now we showed that, among all the critical points, only V1 can be an extremum, in particular a maximum; by variational techniques we now prove that a maximum actually exists, and therefore it must coincide with V1 . We exploit the fact that hf i is positive and homogeneous, and can be extended to a continuous functional on L1 . Introduce the 0 degree homogeneous functional V 2 P n f0g: F (V ) := E 23 ?Rhf i(V)32 ; 2 Vx We have sup F = sup F jSE = sup hf ijSE : 0g V 2Pnf
From Sobolev embedding theorem H 1 ( ) C ( ) it follows sup F sup f jSE < 1, while from positivity of hf i we have sup F > 0. Let fVn g P be a maximizing sequence, and vn 2 SE be given by: p vn := E qRVn ; Vnx2 which is also maximizing, since F is homogeneous. fvn g is bounded in H 1 , and thus there exists a subsequence (that we call again fvn g) weakly converging to a point v: vn * v. L1 v . By compact embedding we have vn ?! Now, by construction, hf i(vn ) = F (vn ) ! sup F ; by continuity of hf i in L1 , hf i(vn ) ! hf i(v), and so hf i(v) = sup F > 0, which gives v 6= 0. It remains to prove that v satis es the constraint. From weak lower semicontinuity of the H 1 norm it follows Z
23 2 v
x
lim inf
Z
Since v 6= 0 we also have
v2
nx
3 2
= E 23 :
E 32 lim sup E 23 = 1 ; 3 ?R 32 ?R vx2 vx2 n 2 and remembering the de nition of F F (v) = E 32 ?Rf (v) 23 f (v) = sup F vx2 which implies F (v) = sup F , and
qR
vx2 = E .
EXPONENTIAL STABILITY IN A NONLINEAR STRING EQUATION
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6. Non degeneration Finally, to apply the theory of [4] we have to verify that the second di erential of hf ijSE is non degenerate, in the directions transversal to the orbit of the linearized system, at V1 V . We know that V1 (x + s) is again a critical point at the same level, so V1x is a null direction of the second di erential of the functional (remark that A1 V1x = 0 because V and Vx are L2 orthogonal and B1 V1x = 0 being the derivative of the equation de ning V ). We have to prove that this is the only null direction. So we have to nd all solutions h 2 TV1 SE of the equation (6.1) 0 = d2 (hf ijSE ) (V )(h; h) hh; Ahi ; with A := A1 , and A1 de ned by 5.2 with n = 1. It is easy to see that 6.1 is equivalent to D E 0 = d2 (hf ijSE ) (V )(h~ ; h) h~ ; Ah ; 8h~ 2 TV1 SE ; (6.2) which in turn is equivalent to the following equation for h 2 TV1 SE and 2 R Ah = ? Vxx : (6.3) We study equation 6.3 in the space P adding the constrain that the solutions must belong to TV1 SE , i.e. that its H 1 scalar product with V must vanish or, more explicitly that [hx Vx ] = 0 : (6.4) We will prove that the only solution of this problem is h / Vx . To nd the solutions of 6.3 we introduce a new variable g de ned by h = Vx g : Remark that g could be discontinuous, but this will not cause any problem. The equations we will write will be intended to hold at the points x such that g is continuous and di erentiable (x 6= =2). In terms of g, 6.3 takes the form ? m@x Vx2 gx = @x ? 83 [V Vx g]V 2 ? 2 Vx2 ; from which mVx2 gx = ? 38 [V Vx g]V 2 ? 2 Vx2 + ; (6.5) with a real . In terms of g, eq. 6.4 takes the form [Vx2 gx ] = 0. So, averaging 6.5, we get 2 (6.6) ? 38 [V Vx g] = 2 [[VVx2 ]] ? [V 2 ] ; from which, inserting in 6.5 we obtain 2 2 2 mgx = 2 [[VVx2 ]] VV 2 ? 1 + V12 ? [V V2 ]V 2 : x x x Remembering that V = Vmsn(!xjm) and using known expressions for the integrals of elliptic functions one obtains the general solution of the above equation. This is given by 2 m+1 (1 + 7m)A ? m(7 + m)B mg(x) = + 2 !(3m ++m14 A ? B ? x +
)(1 ? m)2 Vm2 !3(1 + m)(1 ? m)2 (6.7)
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SIMONE PALEARI
DARIO BAMBUSI
SERGIO CACCIATORI
with
jm)dn(!xjm) ; A ?E(!x) + (1 ? m)!x + sn(!xcn( !xjm) jm)cn(!xjm) ; B E(!x) ? m sn(!xdn( !xjm) and is a real parameter. Imposing now the conditions of periodicity of g and that of orthogonality (see 6.4 and 6.6) we get a linear homogeneous system in the variables and ; the determinant of such system is given by (1 + m)(255 ? 3444m ? 32661m2 ? 27448m3 ? 6147m4 ? 180m5 ? 7m6) ; K4 216 and this quantity is di erent from zero (remember that m, and hence K, are real numbers xed by 4.1). The conclusion is than that and must be zero, and the only null direction is h = Vx .
Remark 6.8 It is possible to prove the non degeneracy of all the other critical
points Vn exactly in the same way, obtaining for the determinant precisely the same expression, which turns out to be independent from n.
7. Appendix Proof of lemma 4.13. The idea is to show that, when 6= 0, due to the broken symmetry of the potential , the average of the function V is no more zero. We can write Z Vmax V dV q [V ] = n 2 0 Vmin m (E ? (V )) (7.1) Z sM sds p / ; p(s; E 0 ; c; ) sm where we changed the variable of integration from x to V and where we denoted p(s; E 0 ; ; ) = E 0 ? (s; ; ), and again p(sm ) = p(sM ) = 0. Call this last integral I (), and observe that I (?) = ?I (), so we restrict to the case > 0. We restrict also to the case E 0 > 0, because otherwise the inversion points are both positive and the proof is nished. Let's write the polynomial p in the following way 2 1 s4 = E 0 + s ? 2 s2 ? 32 (7.2) 0 s + s 1 E m M 2 =(s ? sm )(s ? sM ) s s ? 32 s ? 32 s ; m M and comparing the coecient of s and s2 deduce the following relations: 0 E s s m M = ?(sm + sM ) s s + 32 ; (7.3) m M 2
2
0
? 2 ? (sm +32sM ) = s Es ? sm32sM : m M
(7.4)
EXPONENTIAL STABILITY IN A NONLINEAR STRING EQUATION
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From 7.3 you see that when > 0, then sM > jsm j. Now look at the other two roots of p q E0 (sm + sM )2 + s128 s + s m sM m M ; (7.5) s3;4 = ? 2 2 which are not real (use 7.4). Now de ne sm > 0 ( > 0)
:= sM + 2 ! := sM ?2 sm > 0 (7.6) 0 E := (sm + sM )2 + 128 s s
