On the Korteweg-De Vries Equation: Frequencies and Initial Value Problem

pacific journal of mathematics Vol. 181, No. 1, 1997

ON THE KORTEWEG-DE VRIES EQUATION: FREQUENCIES AND INITIAL VALUE PROBLEM

D. Battig,¨ T. Kappeler and B. Mityagin

The Korteweg-de Vries equation (KdV)

3 2 1 ∂tv(x, t)+∂xv(x, t) − 3∂xv(x, t) =0 (x∈S ,t∈R)

is a completely integrable system with phase space L2(S1). R Although the Hamiltonian H(q):= 1 ∂q(x)2+q(x)3 dx S1 2 x is defined only on the dense subspace H1(S1), we prove that ∂H the frequencies ωj = can be defined on the whole space ∂Jj 2 1 L (S ), where (Jj)j≥1 denote the action variables which are globally defined on L2(S1). These frequencies are real analytic functionals and can be used to analyze Bourgain’s weak solutions of KdV with initial data in L2(S1). The same method can be used for any equation in the KdV −hierarchy.

1. Introduction and summary of the results. It is well known that the Korteweg-de Vries equation (KdV ) on the circle

3 (1.1) ∂tv(x, t)=−∂xv(x, t)+6v(x, t)∂xv(x, t) can be viewed as a completely integrable Hamiltonian system of inﬁnite dimension. We choose as its phase space L2(S1)=L2(S1;R) where S1 is the circle of length 1. The Poisson structure is the one proposed by Gardner, Z ∂F d ∂G (1.2) {F, G} = dx S1 ∂q(x) dx ∂q(x) where F, G are C1- functionals on L2(S1), and ∂F denotes the L2-gradient ∂q(x) of the functional F . The Hamiltonian H corresponding to KdV is then given by

Z 1 2 3 (1.3) H(q):= (∂xq) +q dx. S1 2

1 2D.BATTIG,¨ T. KAPPELER AND B. MITYAGIN

Note that the Poisson structure (1.2) is degenerate and admits the average as a Casimir function Z [q]= q(x)dx. S1 Moreover, the Poisson structure is regular and induces a trivial foliation whose leaves are given by

2 1 2 1 Lc(S )={q∈L (S ); [q]=c}.

2 1 Consider the leaf L0(S ) and denote by ωG the symplectic structure on L2(S1) induced by the Poisson structure (1.2). 0 2 1 In previous work [Ka], [BBGK] the phase space L0(S ),ωG was analyzed and it was proved that KdV admits globally defined (generalized) action-angle variables. Recall that for a finite dimensional, completely integrable Hamiltonian system action-angle variables linearize the flow and, therefore, solutions can be found by quadrature. When trying to apply this procedure to find L2-solutions of KdV one notices that the Hamiltonian H is only defined on the dense subspace H1(S1) and so are the frequencies ∂H ωj = , given by the partial derivatives of H with respect to the action ∂Jj variables Jj. In Section 3, using auxilary results proved in Section 2,we show the following main result of this paper:

Theorem 1. Let q ∈ L2(S 1). Then there exists a neighborhood U of q in 0 0 q0 0 L2(S1; C) so that the frequencies are deﬁned and analytic on U . Moreover, 0 q0 they satisfy, uniformly on U , q0

3 (1.4) ωj =(2πj) + o(1).

Our starting point of the proof of Theorem 1 is the Its-Matveev formula which, at least for ﬁnite gap potentials, provides a formula for the frequencies

ωj (j ≥ 1) in terms of periods of Abelian diﬀerentials of the second kind on the hyperelliptic Riemann surface Σq, associated to the periodic spectrum of the Schr¨odinger operator − d2 + q where q is the initial data of (1.1) (cf. dx2 e.g. [DKN], [FFM], [MT1, 2]). Using Riemann bilinear relations (cf e.g.

[EM]) one sees that the frequencies ωj (j ≥ 1) can be expressed in terms of the zeroes of a conveniently normalized basis of holomorphic diﬀerentials on

Σq. Our strategy for the proof of Theorem 1 is to show that for elements q merely in L2(S1), such a basis of holomorphic diﬀerentials still exists with the property that their zeroes are real analytic functions of q and that these zeroes can be rather precisely located (cf. Section 2 and Appendix). As an application of Theorem 1 we investigate the initial value problem for 2 1 2 2 KdV on the circle. Denote by Ω : L0(S ) → `1/2(R ) the symplectomorphism ON THE KORTEWEG-DE VRIES EQUATION 3

constructed in [BBGK] and by (Jj,αj)j≥1 the symplectic polar coordinates 2 2 in `1/2(R ), q q (1.5) Ω(q)=(x, y)=(xj,yj)j≥1 = 2Jj cos αj, 2Jj sin αj . j≥1

2 1 2 1 For q in L (S ), introduce p = q − [q] ∈ L0(S ) and deﬁne q q (1.6) (x(t),y(t)) = 2Jj cos ωj(q)t + αj , 2Jj sin ωj(q)t + αj j≥1 where

(1.7) ωj(q)=ωj(p) + 12[q]πj. Introduce the solution operator S(1) of KdV, S(1) :L2(S1) → C(R; L2(S1)) where (1.8) S(1)(q)(t)=[q]+Ω−1 x(t),y(t) .

Recently, Bourgain [Bo1] has found weak solutions of KdV, which can be analyzed further by using Theorem 1:

Theorem 2. Let c ∈ R. Then (1) (i) S (q) coincides with the weak solution SB(q) constructed by Bourgain 2 1 for any initial data q in Lc(S ). 2 1 (ii) Given q1,q2 in Lc(S ), there exists M>0so that for any t ∈ R (1) (1) (1.9) S (q1) (·,t)− S (q2) (·,t) ≤M(1 + |t|) kq1 − q2kL2(S1). L2(S1)

(1) 2 1 2 1 (iii) For any 0

Remark 1. The estimate (1.9) strengthens a result of Bourgain [Bo1], Theorem 5. Instead of M(1 + |t|), Bourgain obtains eC|t| where C>0is some constant depending on q1 and q2. Remark 2. Recently, we have learnt that property (iii) has also been obtained by Zhang [Zh] using methods very diﬀerent from ours. Remark 3. McKean-Trubowitz [MT1] have constructed classical solutions of (1.1), using theta functions. However, with their approach, one has to ﬁx an isospectral set of potentials in advance and therefore, properties of well posedness cannot be proved. Remark 4. Note that S(1) is not real analytic on all of L2(S1): for generic initial data q with Im[q] 6= 0 one sees from (1.6) and (1.7) that KdV is ill posed. 4D.BATTIG,¨ T. KAPPELER AND B. MITYAGIN

Similar results as the ones presented for KdV hold for any of the equations in the KdV hierarchy. As an example we consider KdV2, i.e., the fifth order equation in the hierarchy. 1 5 15 (1.10) ∂ v = ∂5v − v∂3v − 5∂ v∂2v + v2∂ v. t 4 x 2 x x x 2 x In Section 4 we analyze the frequencies corresponding to the Hamiltonian of KdV and use the results to construct weak solutions S(2)(q)of(1.10) for 2 R initial data in H1 (S1):={q∈H1(S1)|[q]=c;5 q(x)2dx = c }, with c1,c3 3 8 S1 1 c ,c ∈R, 1 3 S(2) : H1 (S1) → C R,H1 (S1) c1,c3 c1,c3 and prove that S(2) has properties similar to those of S(1). We point out that, according to [Bo2], Bourgain’s method cannot be used 1 1 to obtain weak solutions of (1.10) for initial data in Hc (S ). Results similar to the ones presented in this paper hold most likely also for the defocusing nonlinear Schrödinger equation 1 ∂ ψ = ∂2ψ −|ψ|2ψ. However we have not i t x verified the details. This paper is related to [BKM] where we present further applications of the existence of action-angle variables for KdV . Among other things, we prove that in a neighborhood in H1(S1) of the elliptic fix point q ≡ 0, the Hamiltonian H admits a convergent Birkhoff normal form. The notation in this paper is standard and coincides with the one used in [Ka] and [BBGK]. Acknowledgement. It is a pleasure to thank J. Bourgain, G. Forest, J.C. Guillot, A. Kostyuchenko, D. McLaughlin and J. Moser for helpful discussions. T. Kappeler would like to thank IHES and Université Paris- Nord and B. Mityagin the Center for Research in Mathematics, Guanajuato, Mexico for the kind hospitality.

2. Auxilary results. In this section we prove auxilary results needed in the following section to 2 1 show that the frequencies ωj can be deﬁned on the whole space L0(S ). Given a potential q in L2(S1; C), denote by (λ = λ (q)) the union of 0 j j j≥0 the periodic and antiperiodic spectrum of − d2 + q (with multiplicities). dx2 The eigenvalues are ordered so that Re λ0 ≤ Re λ1 ≤ Re λ3 ≤ ... and, if Re λn =Reλn+1, then Im λn ≤ Im λn+1. Introduce τj =(λ2j +λ2j−1)/2, γj =(λ2j −λ2j−1) and the gap interval Ij = {tλ2j +(1−t)λ2j−1;0 ≤ t ≤ 1}⊆C. Note that τj, but not γj are real 2 1 analytic functions on L0(S ). Denote by ∆(λ)=∆(λ, q) the discriminant d ∆(λ):=y (1,λ)+ y (1,λ) 1 dx 2 ON THE KORTEWEG-DE VRIES EQUATION 5 where y (x, λ) and y (x, λ) denote the fundamental solutions for − d2 + q. 1 p 2 dx2 2 Introduce ∆(µp) − 4 with the sign determined so that, in the case where 2 q is real valued, ∆(µ) − 4 > 0 for µ ∈ (−∞,λ0).

2 1 Theorem 2.1. Let q0 be in L0(S ). Then there exists a (suﬃciently small) neighborhood U of q in L2(S1; C) with the following properties: q0 0 0 For any j ≥ 1 and q ∈ U there exists a unique sequence q0 (j) (µk )k∈N\{j} satisfying uniformly for j ≥ 1,k ∈N\{j} and q in U , q0

1 (2.1) |µ(j) − τ |≤|γ |2O , k k k k so that, for all k in N \{j}, Z λ2k ϕ (λ)dλ (2.2) p j =0 2 λ2k−1 ∆(λ) − 4 where ϕj(λ) is the entire function 1 Yµ(j) −λ (2.3) ϕ (λ)= k . j j2π2 k2π2 k6=j Moreover, the µ(j)’s are analytic functions on U . In case where q is real k q0 (j) valued, the µk are real valued and satisfy (j) λ2k−1 ≤ µk ≤ λ2k (∀j 6= `).

Rephrasing the above results, Theorem 2.1 states that there exist 1-forms ϕj(λ)dλ Ωj(λ)=√ in the Hilbert space of quadratically integrable holomor- ∆(λ)2−4 p phic 1-forms on the (open) Riemann surface y = ∆(λ)2 − 4 which satisfy

(2.2) and that the zeroes of these 1-forms Ωj(λ) depend analytically on q. Actually, the Ω0 s are a basis in the space of these 1-forms. If q ∈ U is j q0 real valued, the existence of such a basis of 1-forms Ωj(λ) with a product representation (2.3) has been proved by [MT2] (cf also [FKT]). A proof of Theorem 2.1 is provided in the Appendix. The rest of this section is devoted to a reﬁnement of the asymptotics (2.1). (j) First note that, in the case where γk =0,µk =τk. In case γk 6=0,we make a change of coordinates in (2.2), λ(t)=τ +γkt, which leads to k 2 Z 1 ϕ λ(t) γk dt j 2 (2.4) q =0. −1 ∆ λ(t) 2 − 4 6D.BATTIG,¨ T. KAPPELER AND B. MITYAGIN

It is convenient to introduce

(2.5) ∆α(t)=(λ2α −λ(t)) (λ2α−1 − λ(t)) (α ≥ 1)

∆0(t)=(λ(t)−λ0). Notice that γ γ γ 2 ∆ (t)= τ + α −λ(t) τ − α −λ(t) =(τ −λ(t))2 − α . α α 2 α 2 α 2

Therefore, with τ − λ(t)=τ −τ −γkt, k k k 2 γ 2 −∆ (t)=(λ −λ(t)) (λ(t) − λ )= k (1 − t2) k 2k 2k−1 2 and (2.6) Y (λ − λ(t)) (λ − λ(t)) ∆(λ(t))2 − 4=4(λ −λ(t)) 2α 2α−1 0 (α2π2)2 α≥1 γ 2 (1 − t2) Y ∆ (t) =4∆ (t) k α 0 2 k4π4 (α2π2)2 α6=k γ 2 1−t2 ∆(t) Y (τ − λ(t))2 =4∆ (t) k j α 0 2 k4π4 (j2π2)2 (α2π2)2 α∈N\{j,k} ! Y ( γα )2 2 · 1 − 2 . α∈N \{j,k} (τα − λ(t)) Further we introduce

(j) (j) (2.7) ξα := µα − τα. This leads to (2.8) 1 Y τ − λ(t)+ξ(j) ϕ (λ(t)) = α α j j2π2 α2π2 α6=j ! (j) 1 ξ − γk t Y τ − λ(t) ξ(j) = k 2 α 1+ α . j2π2 k2π2 α2π2 τ −λ(t) α∈N \{j,k} α Combining (2.6) and (2.8) we obtain (2.9) ! (j) ϕ (λ(t)) ξ − γk t 1 Y ξ(j) q j = k q2 √ 1+ α p 2 2 2 γk ∆ (t) ∆ (t) 1 − t τα −λ(t) ∆(λ(t)) − 4 2 0 j α∈N \{j,k} ON THE KORTEWEG-DE VRIES EQUATION 7

!−1/2 Y ( γα )2 2 · 1 − 2 . α∈N \{j,k} τα − λ(t) The Equation (2.4) then takes the form (2.10) Z 1 (j) dt ξk p q √ −1 2 ∆0(t) ∆j(t) 1 − t ! !−1/2 Y ξ(j) (γα )2 · 1+ α 1− 2 τ −λ(t) 2 α∈N \{j,k} α τα −λ(t) Z 1 γk tdt = p q √ 2 −1 2 ∆0(t) ∆j(t) 1 − t ! !−1/2 Y ξ(j) (γα )2 · 1+ α 1− 2 . τ −λ(t) 2 α∈N \{j,k} α τα −λ(t) Now introduce ! Y ξ(j) A (t) ≡ A (t, q):= 1+ α jk jk τ −λ(t) α∈N\{j,k} α and

!−1/2 Y ( γα )2 2 Bjk(t) ≡ Bjk(t, q):= 1 − 2 . α∈N\{j,k} τα − λ(t) After multiplying left and right hand side of (2.10) with the t-independent nonvanishing quantity p q ∆0(0) ∆j(0)

Ajk(0)Bjk(0) the Equation (2.10) reads q q R 1 ∆ (0) ∆ (0) A (t) B (t) √tdt 0 j jk jk γ −1 2 (2.11) ξ(j) = k 1−t q ∆0(t) q ∆j (t) Ajk(0) Bjk(0) . k R 1 ∆ (0) ∆ (0) A (t) B (t) 2 √ dt 0 j jk jk 2 −1 1−t ∆0(t) ∆j (t) Ajk(0) Bjk(0) It will turn out that for k ≥ N, with N sufficiently large, the denominator in (2.11) is different from 0. Thus the quotient in (2.11) is well defined at least for k sufficiently large. To simplify (2.11), we write s s ∆0(0) ∆j(0) Ajk(t) Bjk(t) Cjk(t) ≡ Cjk(t, q):= ∆0(t) ∆j(t) Ajk(0) Bjk(0) 8D.BATTIG,¨ T. KAPPELER AND B. MITYAGIN and obtain

R 1 √tdt C (t) − C (−t) (j) γk 0 1−t2 jk jk (2.12) ξk = R 1 . 2 √ dt C (t)+C (−t) 0 1−t2 jk jk

(j) To obtain asymptotics for ξk we estimate s s ∆ (0) ∆ (0) A (t) B (t) j − 1, 0 − 1, jk − 1 and jk − 1. ∆j(t) ∆0(t) Ajk(0) Bjk(0) P 2 1 2 Recall that for q ∈ L0(S ), k |γk| < ∞. Lemma 2.2. Let q ∈ L2(S 1). Then there exist a neighborhood U of q 0 0 q0 0 in L2(S1; C) and C>0, independent of j and k, so that, for q ∈ U and 0 q0 j 6= k,  

A (t) |γ | X 1 (2.13) sup jk − 1 ≤ C k  |γ |2 +  . A (0) k3 α k 0≤t≤1 jk k |α−k|≤ 2

Proof. Notice that

A (t) ∂ A (t) jk ∂t jk sup − 1 ≤ sup . 0≤t≤1 Ajk(0) |t|≤1 Ajk(0) Q ξ(j) Recall that A (t)= 1+ α and λ(t)=τ +γktand there- jk τ −λ(t) k 2 α∈N\{j,k} α fore,   1 ∂ A (t) X 1 (−1)ξ(j) γ A (t)= jk  α − k  . jk (j) 2 A (0) ∂t A (0) ξα 2 jk jk α6=j,k 1+ τα − λ(t) τα−λ(t) This leads to,  

1 ∂ X |ξ(j)| (2.14) A (t) ≤ Cb|γ | α  jk k 2 Ajk(0) ∂t |τα − λ(t)| |α−k|≥ k  2  X |ξ(j)| + Cb|γ |  α  . k |τ − λ(t)|2 k α |α−k|≤ 2 Notice that, for any α with |α − k|≥ k, 2 1 C 1 ≤ =O . 2 2 2 2 4 |τα −λ(t)| (α −k ) k ON THE KORTEWEG-DE VRIES EQUATION 9

Therefore, X |ξ(j)| C X 1 (2.15) α ≤ |ξ(j)| = O . |τ − λ(t)|2 k4 α k4 k α α |α−k|≥ 2

For α with |α − k|≤ k we use the estimate ξ(j) = γ2 O( 1 ) to conclude 2 α α k that X |ξ(j)| C X (2.16) α ≤ |γ |2. |τ − λ(t)|2 k3 α k α k |α−k|≤ 2 |α−k|≤ 2 Combining (2.14)-(2.16) yields (2.13).

Lemma 2.3. Let q ∈ L2(S 1). Then there exist a neighborhood U of q 0 0 q0 0 in L2(S1; C) and C>0, independent of j and k, so that for q ∈ U and 0 q0 j 6= k,  

B (t) C|γ | X 1 (2.17) sup jk − 1 ≤ k  |γ |2 +  . B (0) k3 α k3 0≤t≤1 jk k |α−k|≤ 2

Proof. Notice that

B (t) ∂ B (t) jk ∂t jk sup − 1 ≤ sup . 0≤t≤1 Bjk(0) |t|≤1 Bjk(0)

!−1/2 Q γα 2 ( 2 ) Recall that Bjk(t)= α6=j,k 1 − 2 to conclude that τα−λ(t)       ∂  X −1/2 2( γα )2 −γ  B (t)=B (t) ! · 2 k . jk jk  3  ∂t  ( γα )2 τ − λ(t) 2  α∈N \{j,k} 2 α   1 − 2  τα−λ(t)

This leads to  

1 ∂ |B (t)|  X 1  B (t) ≤ jk C|γ | |γ |2 |B (0)| ∂t jk |B (0)|  k |τ − λ(t)|3 α  jk jk α∈N \{j,k} α   X |γ |2 X |γ |2 ≤ C|γ |  α + α  k |τ − λ(t)|3 |τ − λ(t)|3 k α k α |α−k|≥ 2 |α−k|≤ 2 10 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

  C 1 X ≤ C|γ |  + |γ |2 . k k6 k3 α k |α−k|≤ 2

Lemma 2.4. Let q ∈ L2(S 1). Then there exist a neighborhood U in 0 0 q0 L2(S1; C) and C>0, independent of k, such that for any q in U 0 q0

1/2 2 τk − λ0 γk 1 |γk| (2.18) sup − 1 − t ≤ C 4 |t|≤1 λ(t) − λ0 4 τk − λ0 k and

1/2 τk − λ0 |γk| (2.19) sup − 1 ≤ C . 2 0≤t≤1 λ(t) − λ0 k

Proof. Notice that d τ − λ 1/2 τ − λ 1/2 (−1/2) γ k 0 = k 0 k , dt λ(t) − λ0 λ(t) − λ0 λ(t) − λ0 2 (2.20) 2 1/2 1/2 2 d τk − λ0 τk − λ0 3/4 γk = , dt2 λ(t) − λ λ(t) − λ 2 2 0 0 λ(t) − λ0

1/2 1/2 τk − λ0 d τk − λ0 1 sup − 1 ≤ sup ≤ C |γ |. 2 k 0≤t≤1 λ(t) − λ0 |t|≤1 dt λ(t) − λ0 k This proves (2.19). Taylor’s formula and (2.20) lead to (2.18). q The diﬀerence ∆j (0) − 1 needs to be treated with more care as ∆j (t) s ∆j(0) C|γk| 1 sup − 1 ≤ = |γk|O . |t|≤1 ∆j(t) |τk − τj| k

Lemma 2.5. Let q ∈ L2(S 1). Then there exist a neighborhood U of q 0 0 q0 0 in L2(S1; C) and C>0, independent of j and k, such that for any q in U 0 q0 and j 6= k,

(2.21) s 2 2 2 ∆j(0) 1γk(τj −τk) −1 γ 3 γ (τj − τk) − 1+ t + k + k t2 2 ∆j(t) 2 ∆j(0) 8 ∆j(0) 8 ∆j(0) ON THE KORTEWEG-DE VRIES EQUATION 11

|γ |3 ≤ C k . 3 (τk − τj)

Proof. Recall that γ 2 γ 2 γ γ 2 ∆ (t)= τ −λ(t) 2− j =(τ −τ )2− k −2 k(τ −τ )t+ k t2 j j 2 j k 2 2 j k 2 and therefore   1/2 ∆ (0) 1 j =   γ (τ −τ )t+( γk )2t2 ∆j(t) 1+ k k j 2 ∆j (0)     3/2   γk 2  1 1 γk(τk − τj)+2( ) t =1+   − 2 t γk 2 2  γk(τk−τj)t+( ) t   1+ 2 2 ∆j(0)  ∆ (0) j t=0    5/2 !2  1 1 3 γ (τ −τ )+2(γk )2t +   − − k k j 2 γk 2 2  γk(τk−τj)t+( ) t  1+ 2 2 2 ∆j(0) ∆ (0) j   3/2 γ 2  1 1 2( k ) t2 γ3 +   − 2 + k O(1) γk 2 2 γk(τk−τj)t+( ) t  3 1+ 2 2 ∆j(0)  2 (τk − τj) ∆j (0) t=0 which leads to (2.21).

Lemma 2.6. Let q ∈ L2(S 1). Then there exist a neighborhood U of q 0 0 q0 0 in L2(S1; C) and C>0, independent of j and k, such that for any q in U , 0 q0 and j 6= k, q q R 1 ∆ (0) ∆ (0) √tdt j − j 2 γ 2 1 (−γ ) (2.22) k 0 1−t q ∆j (t) q ∆j (−t) − k R 1 ∆ (0) ∆ (0) 2 √ dt j + j 8 τk − τj 0 1−t2 ∆j (t) ∆j (−t) C ≤ |γ |2 |γ |2 + |γ | . 3 k j k (τk − τj)

Proof. Using Lemma 2.5, we obtain (2.23) Z s s ! 1 tdt ∆ (0) ∆ (0) √ j − j 2 0 1 − t ∆j(t) ∆j(−t) Z γ (τ − τ ) 1 t2dt γ3 = k j k √ + k O(1) 2 3 ∆j(0) 0 1 − t (τk − τj) 12 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN and

(2.24) Z s s ! 1 dt ∆ (0) ∆ (0) √ j + j 2 0 1 − t ∆j(t) ∆j(−t) Z Z 1 dt −1 γ2 3 γ2(τ − τ )2 1 t2dt =2 √ +2 k + k j k √ 2 2 2 0 1 − t 8 ∆j(0) 8 ∆j(0) 0 1 − t γ3 + k O(1). 3 (τk − τj)

Notice that Z Z 1 dt π 1 t2dt π √ = ; √ = . 2 2 0 1 − t 2 0 1 − t 4 Therefore (2.25) q q R 1 ∆ (0) ∆ (0) √tdt j − j γ 0 2 k 1−t q ∆j (t) q ∆j (−t) R 1 ∆ (0) ∆ (0) 2 √ dt j + j 0 1−t2 ∆ (t) ∆ (−t) j j 2 γk(τj −τk) π γk γ 1+ 3O(1) = k ∆j (0) 4 (τk−τj) γ2 γ2(τ −τ )2 γ3 2 1 k 3 k j k k π 1 − − 2 + 3 O(1) 16 ∆j (0) 16 ∆j (0) (τk−τj ) γ2 τ − τ γ3 = k j k + k O(1). 3 8 ∆j(0) (τk − τj)

Recall that ∆ (0)=(τ −τ )2 −(γj)2.Therefore j j k 2 2 τj − τk 1 1 1 (γj/2) (2.26) = 2 = + O(1). ∆ (0) (τ − τ ) (γj /2) τ − τ (τ − τ )3 j j k 1 − 2 j k j k (τj −τk) Combining (2.25) and (2.26) we obtain q q R 1 ∆ (0) ∆ (0) √tdt j − j 2 2 γ 0 2 γ 1 (γ + γ ) k 1−t q ∆j (t) q ∆j (−t) = − k + γ2 k j O(1). R 1 ∆ (0) ∆ (0) k 3 2 √ dt j + j 8 τk − τj (τk − τj) 0 1−t2 ∆j (t) ∆j (−t)

Theorem 2.7. Let q ∈ L2(S 1). Then there exists a neighborhood U of 0 0 q0 q in L2(S1; C),k ≥ 1 and C>0such that ∀q ∈ U , ∀j and k ≥ k with 0 0 0 q0 0 k 6= j, 1 (γ /2)2 γ2 1 (j) k k (2.27) ξk − − − 2 τk − τj 16 τk − λ0 ON THE KORTEWEG-DE VRIES EQUATION 13

  C|γ |2 X 1 ≤ k |γ | + |γ |2 + |γ |2 + O  . k3 k j α k k |α−k|≤ 2

Proof. Recall that

Z 1 (j) dt ξ √ (Cjk(t)+Cjk(−t)) k 2 0 1 − t Z 1 γk tdt = √ (Cjk(t) − Cjk(−t)). 2 2 0 1 − t Write s ∆j(0) Cjk(t)= ∆j(t) s s ! ∆ (0) ∆ (0) + j 0 − 1 ∆j(t) ∆0(t) s s ∆ (0) ∆ (0) A (t) + j 0 jk − 1 ∆j(t) ∆0(t) Ajk(0) s s ∆ (0) ∆ (0) A (t) B (t) + j 0 jk jk − 1 . ∆j(t) ∆0(t) Ajk(0) Bjk(0)

Notice that, by Lemma 2.5 s s ! ∆ (0) ∆ (0) j 0 − 1 ∆j(t) ∆0(t) 1γ (τ −τ ) 1 γ2 3 γ2(τ − τ )2 |γ |3 = 1+ k j k t + − k + k j k t2 + O k 2 3 2 ∆j(0) 8 ∆j(0) 8 ∆j(0) k γ 1 |γ |2 · − k t + O k 4 τ − λ k4 k 0 γ 1 1 γ (τ − τ ) γ 1 |γ |2 = − k t + k j k − k t2 + O k . 4 4 τk − λ0 2 ∆j(0) 4 τk − λ0 k Therefore Z ( s s ! γ 1 tdt ∆ (0) ∆ (0) (2.28) k √ j 0 − 1 2 2 0 1 − t ∆j(t) ∆0(t) s s !!) ∆ (0) ∆ (0) − j 0 − 1 ∆j(−t) ∆0(−t) 14 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

Z γ 1 t2dt γ 1 |γ |3 = k √ 2 − k +0+O k 2 4 2 0 1 − t 4 τk − λ0 k γ2 π γ3 =− k + O k 4 16 τk − λ0 k and (2.29) Z (s s ! s s !) 1 dt ∆ (0) ∆ (0) ∆ (0) ∆ (0) √ j 0 − 1 + j j − 1 2 0 1 − t ∆j(t) ∆0(t) ∆j(−t) ∆j(−t) Z γ2 (τ −τ) 1 1 t2dt |γ |2 =+ k k j √ + O k . 2 4 4 ∆j(0) (τk − λ0) 0 1 − t k By Lemma 2.2, we conclude (s s Z 1 γk tdt ∆j(0) ∆0(0) Ajk(t) (2.30) √ − 1 2 2 0 1 − t ∆j(t) ∆0(t) Ajk(0) s s ) ∆j(0) ∆0(0) Ajk(−t) − − 1 ∆j(−t) ∆0(−t) Ajk(0)   |γ |2 X 1 ≤ C k  |γ |2 + O  k3 α k k |α−k|≤ 2 and (s s Z 1 dt ∆j(0) ∆0(0) Ajk(t) (2.31) √ − 1 2 0 1 − t ∆j(t) ∆0(t) Ajk(0) s s ) ∆j(0) ∆0(0) Ajk(−t) + − 1 ∆j(−t) ∆0(−t) Ajk(0)   |γ | X 1 ≤ C k  |γ |2 + O  . k3 α k k |α−k|≤ 2 Finally, using Lemma 2.3, we obtain

(2.32) (s s Z 1 γk tdt ∆j(0) ∆0(0) Ajk(t) Bjk(t) √ − 1 2 2 0 1 − t ∆j(t) ∆0(t) Ajk(0) Bjk(0) s s ) ∆j(0) ∆0(0) Ajk(−t) Bjk(−t) − − 1 ∆j(−t) ∆0(−t) Ajk(0) Bjk(0) ON THE KORTEWEG-DE VRIES EQUATION 15

  |γ |2 X 1 ≤ C k  |γ |2 + O  k3 α k3 k |α−k|≤ 2 and

(2.33) (s s Z 1 dt ∆j(0) ∆0(0) Ajk(t) Bjk(t) √ − 1 2 0 1 − t ∆j(t) ∆0(t) Ajk(0) Bjk(0) s s ) ∆j(0) ∆0(0) Ajk(−t) Bjk(−t) + − 1 ∆j(−t) ∆0(−t) Ajk(0) Bjk(0)   |γ | X 1 ≤ C k  |γ |2 + O  . k3 α k3 k |α−k|≤ 2 We combine the above estimates to conclude (2.34) Z 1 γk tdt √ (Cjk(t) − Cjk(−t)) 2 2 0 1 − t Z s s ! γ 1 tdt ∆ (0) ∆ (0) = k √ j − j 2 2 0 1 − t ∆j(t) ∆j(−t)   γ2 π 1 X 1 − k + |γ |2 O  |γ |2 + O  16 τ − λ k k3 α k k 0 k |α−k|≤ 2 and

(2.35) Z 1 dt √ (Cjk(t)+Cjk(−t)) 2 0 1 − t Z s s ! 1 dt ∆ (0) ∆ (0) |γ |2 = √ j + j + O k 2 2 0 1 − t ∆j(t) ∆j(−t) k   γ X 1 + O k  |γ |2 + O  k3 α k |α−k|≤k/2 1 γ2 3 γ2(τ − τ )2 γ3 γ2 = π 1 − k − k j k + k O(1) + O k 16 ∆ (0) 16 ∆ (0)2 k3 k2  j j  γ X 1 + O k  |γ |2 + O  . k3 α k |α−k|≤k/2 16 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

Therefore, there exists k ≥ 1 and a neighborhood U of q in L2(S1; C) such 0 q0 0 0 R 1 that for q in U , for any k ≥ k and j ≥ 1, √ dt C (t)+C (−t) 6=0 q0 0 0 1−t2 jk jk and its inverse satisﬁes the estimate (2.36) Z 1 dt −1 √ (Cjk(t)+Cjk(−t)) 2 0 1 − t Z s s !!−1 1 dt ∆ (0) ∆ (0) = √ j + j 2 0 1 − t ∆j(t) ∆j(−t)    |γ |2 γ X 1 · 1+O k +O k  |γ |2 + O  . k2 k3 α k |α−k|≤k/2

Combining (2.34) and (2.36)

(2.37) R 1 γk √tdt (C (t) − C (−t)) 2 0 1−t2 jk jk R 1 √ dt (C (t)+C (−t)) 0 2 jk jk 1−t q q R ∆ (0) ∆ (0) γ2 2 P γk 1 tdt j j k π |γk| 2 1 √ − − +O |γα| +O( ) 2 2 ∆ (t) ∆ (−t) 16 τ −λ0 k3 |α−k|≤ k k = 0 1−t j jq qk 2 R 1 ∆ (0) ∆ (0) 2 √ dt j + j 1+O |γk| 2 ∆ (t) ∆ (−t) k2  0 1−t j j  1 γ2 1 γ2 |γ |2 = − k − k + k O(|γ |)+O(|γ |2) 3 k j  8 τk − τj 16 τk − λ0 k   |γ |2 X 1  |γ |2 +O k  |γ |2 + O  1+O k k3 α k  k2 |α−k|≤ k 2   1 γ2 1 γ2 |γ |2  X  = − k − k + O k γ | + |γ |2 + |γ |2 3 k j α 8 τk − τj 16 τk − λ0 k   |α−k|≤ k 2 γ4 1 1 + k O + γ2O . 2 k 4 τk − τj k k

3. Frequencies of KdV. The aim of this section is to prove Theorem 1, as stated in the introduction and to apply it to the initial value problem of KdV. In the case where 1 1 ∂H q0 ∈ H (S ), the frequencies ωj = are well deﬁned and can be obtained, 0 ∂Jj ON THE KORTEWEG-DE VRIES EQUATION 17

using Riemann bilinear relations, from the expansion at inﬁnity of the L2− ϕj (λ)dλ integrable holomorphic diﬀerentials Ωj = √ constructed in Theorem ∆(λ)2−4

2.1. More explicitly, ωj is given by (cf. e.g. [EM], [BKM])   Z !−1 ϕ(µ)dµ X λ (3.1) ω =8π p j τ − (µ(j) − τ )+ 0 j ∆(µ)2 − 4 j k k 2 Γj k∈N\{j} where Γj is a counterclockwise oriented circle with center τj and sufficiently small radius ρj > |γj|/2 so that all eigenvalues (λk)k∈N\{2j,2j−1} lie outside Γj. Due to Theorem 2.1, the right hand side of formula (3.1) is well defined 2 1 even if q0 ∈ L0(S ) and can be extended to an analytic function in a sufficiently small complex neighborhood U of q in L2(S1; C). For q in U we q0 0 0 q0 therefore define the frequency ωj by formula (3.1). In view of Theorem 2.1 we obtain the following

Theorem 3.1. Let q ∈ L2(S 1). Then there exists a neighborhood U of 0 0 q0 2 1 q0 in L0(S ; C) so that, for any j ≥ 1, the frequency ωj, given by (3.1), is well deﬁned and analytic.

It remains to prove the asymptotics for ω as j →∞as stated in Theorem R j ϕj (µ)dµ 1. We start by analyzing √ . Note that in the case where γj =0 Γj ∆(u)2−4 we have by the residue theorem, taking into account the determination of the root,

Z ϕ (µ)dµ ϕ(τ) (3.2) p j =2πq j j . 2 ∆(µ)2−4 Γj ∆(µ) − 4 (τj −µ)(µ−τj ) µ=τj

In the case where γ 6= 0, introduce µ(t)=τ +γjt(−1≤t≤1) and j j 2 ϕ µ(t) (3.3) ψ (t):=s j . j 2 ∆ µ(t) −4 λ2j−µ(t) µ(t)−λ2j−1 Then, with λ − µ(t) µ(t) − λ = γj (1 − t) γj (t +1)=(γj)2(1 − t2) 2j 2j−1 2 2 2 Z Z 1 γjdt ϕj(µ)dµ 2 (3.4) p =2 ψj(t) √ 2 γj 2 Γj ∆(µ) − 4 −1 1 − t Z 2 1 dt =2 ψj(t)+ψj(−t) √ . 2 0 1 − t 18 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

Summarizing, we obtain

Z ( ϕ (µ)dµ 2πψ (0) for γ =0 (3.5) p j = R j j 2 1 dt Γ ∆(µ) − 4 2 ψ(t)+ψ (−t) √ for γ 6=0. j 0 j j 1−t2 j

The function ψj(t) has a product representation (3.6)  !−1/2 Y µ(j) − µ(t) 1 Y λ − µ(t) λ − µ(t)  k  (µ(t) − λ )−1/2 2k 2k−1 . k2π2 2 0 (k2π2)2 k6=j k6=j

Notice that γ 2 λ − µ(t) λ − µ(t) = τ − µ(t) 2 − k 2k 2k−1 k 2 ! 2 2 (γk/2) = τk − µ(t) 1 − 2 τk − µ(t)

(j) (j) and, with ξk = µk − τk, ! (j) (j) ξk µk − µ(t)= τk −µ(t) 1+ . τk −µ(t)

Therefore, (3.7) ! !−1/2 Y ξ(j) Y (γ /2)2 2ψ (t)= µ(t)−λ −1/2 1+ k 1− k . j 0 τ −µ(t) 2 k6=j k k6=j τk − µ(t)

Similarly, as in Section 2, it is convenient to introduce

∆0(t):=µ(t)−λ0; ! Y ξ(j) A(t):= 1+ k j τ −µ(t) k6=j k !−1/2 Y 2 (γk/2) Bj(t):= 1− 2 . k6=j τk − µ(t)

This allows us to write ψj(t) as follows

−1/2 (3.8) 2ψj(t)=∆0(t) Aj(t)Bj(t). ON THE KORTEWEG-DE VRIES EQUATION 19

Further, for some θ(t) with −t<θ−(t)<θ+(t)0such that, for q in U and j ≥ 1, q0 (3.10) 

 X ξ(j) 2ψ (0) −1 − (τ − λ )1/2 1 − k j j 0  τ − τ k6=j k j    2 !2 X 2 X (j) X (j)  1 (γk/2) 1 ξ 1 ξ − +  k  + k 2 (τ − τ )2 2 τ − τ 2 τ − τ  k6=j k j k6=j k j k6=j k j 

 2 C X C ≤  |γ |2 + . j3 k j5 k≥j/2

Proof. Notice that µ(0) = τj and therefore (3.11) −1 2ψj(0) !−1 Y ξ(j) Y (γ /2)2 1/2 =(τ −λ )1/2 1+ k 1− k . j 0 (τ −τ ) (τ − τ )2 k6=j k j k6=j k j Further (3.12) !−1 Y ξ(j) X ξ(j) X ξ(j) ξ(j) 1+ k =1− k + k1 k2 (τ −τ ) τ −τ τ − τ τ − τ k6=j k j k6=j k j k1 j k2 j k1,k26=j k1

(j) ξ 1 (3.13) k ≤|γ |2O . k 2 τk − τj j Similarly Y (γ /2)2 1/2 (3.14) 1 − k (τ − τ )2 k6=j k j 1X (γ /2)2 1 X (γ /2)4 =1− k − k 2 (τ − τ )2 8 (τ − τ )4 k6=j k j k6=j k j 1 X (γ /2)2(γ /2)2 1 + k1 k2 + O 4 (τ − τ )2(τ − τ )2 j6 k1 j k2 j k1,k26=j k

−1 (3.15) 2ψj(0)   X ξ(j) 1X (γ /2)2 =(τ −λ )1/2 · 1− k − k j 0  τ −τ 2 (τ − τ )2 k6=j k j k6=j k j    2 !2  1 X ξ(j) 1 X ξ(j)  +  k  + k + R 2 τ − τ 2 τ − τ j k6=j k j k6=j k j  P ξ(j) P 2 P 2 k (γk/2) 1 2 1 where, with 2 = O( 4 ) |γ | + O( 6 ), k6=j τk−τj k6=j (τk−τj ) j k≥j/2 k j we have X X 1 4 1 2 2 1 (3.16) Rj := O γ + O γ γ + O . j4 k j4 k1 k2 j6 k≥j/2 k1≥j/2 k2≥j/2 Finally use that 1 (τ − λ )1/2 = jπ + O( 1 ) to obtain (3.10). 2 j 0 2 j2 Lemma 3.3. | d2 ψ (t)| |γ |2 (3.17) sup dt2 j ≤ C j . 4 0≤|t|≤1 ψj(0) j ON THE KORTEWEG-DE VRIES EQUATION 21

Proof. In a straightforward fashion one computes   X (j) d γj −1/2 1 (−1)ξk ψj(t)= ψj(t) γ + (−1) dt 2 τ −λ + jt ξ(j) 2  j 0 2 k6=j 1+ k τk − µ(t) τ −µ(t) k   X (−1/2) (−1)(−2)(γ /2)2  + k (−1) 2 3 (γk/2)  k6=j 1 − 2 τk − µ(t)  τk−µ(t)    X (j) γj −1/2 1 ξk = ψj(t) γ + 2 τ − λ + j t ξ(j) 2  j 0 2 k6=j 1+ k τk −µ(t) τ −µ(t)  k  X 1 (γ /2)2  + k . 2 3 (γk/2)  k6=j 1− 2 τk − µ(t)  τk−µ(t)

(3.18) d2 1 d 2 γ 2 ψ (t)= ψ (t) + j ψ (t) dt2 j ψ(t) dt j 2 j j   !2  1/2 X −1 ξ(j) · + k γj 2 2 2 (τj − λ0 + t) ξ(j) (τ −µ(t)  2 k6=j 1+ k k τk−µ(t) X 1 2ξ(j) + k ξ(j) 3 k τ −µ(t) k6=j 1+ 2 k τk−µ(t) !2 X 2 (−1) (γk/2) + !2 3 (−1) k6=j 2 τk − µ(t) (γk/2) 1 − 2 τk−µ(t)   X 1 (−3)(γ /2)2  + k (−1) . 2 4 (γk/2)  k6=j 1 − 2 τk − µ(t)  τk−µ(t)

d ψ (t) We begin by estimating | dt j |, ψj (t)   d  X (j) X 2  ψj(t) 1 ξ 1 |γk| dt ≤ C|γ | + k + ψ (t) j j2 τ − µ(t) |τ − µ(t)| |τ − µ(t)|3  j k6=j k k k6=j k 22 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

|γ | ≤ C j . j2 Therefore, d 2 2 ψj(t) |γj| (3.19) sup dt ≤ C . 4 |t|≤1 ψj(0)ψj(t) j All the other terms are treated similarly. Theorem 3.4. Let q ∈ L2(S 1). Then there exists a neighborhood U of 0 0 q0 q in L2(S1; C) and C>0so that for any q in U and j ≥ 1, 0 0 q0

Z !−1 ϕ (µ)dµ |γ |2 (3.20) p j − 2πψ (0) −1 ≤ C j 2 j 3 Γj ∆(µ) − 4 j and  

 1 λ X ξ(j)  (3.21) 2ψ (0) −1 − τ 1/2 − 0 − τ 1/2 k j  j 2 1/2 j τ − τ  τj k≤j/2 k j   C X 1 ≤  |γ |2 + . j2 k j k≥j/2

R 1 Proof. Notice that, according to (3.5), (3.9) and as √ dt = π 0 1−t2 2 Z !−1 ϕ (µ)dµ p j 2 Γj ∆(µ) − 4 !−1 Z 1 00 00 2 −1 ψj θ+(t) − ψj θ−(t) t dt = 2πψj(0) 1+ √ . 2 0 4πψj(0) 1 − t Thus, by Lemma 3.3 ! Z −1 00 ϕj(µ)dµ −1 C ψ (t) p − 2πψ (0) ≤ sup j 2 j Γj ∆(µ) − 4 |ψj(0)| |t|≤1 ψj(0) |γ |2 |γ |2 ≤ Cj j ≤ C j . j4 j3 This proves (3.20). To prove the estimates (3.21) use (3.10) and notice that

  X 1(γ /2)2 X 1 1 (3.22) (τ − λ )1/2 2 k ≤ Cj  |γ |2 = O j 0 (τ − τ )2 k j4 j3 k≤j/2 k j k≤j/2 ON THE KORTEWEG-DE VRIES EQUATION 23 and, in view of (3.13),   2 !2 X (j) X 2 ξ |γk| 1 (3.23) (τ − λ )1/2  k  ≤ Cj = O . j 0 τ − τ j2 j3 k6=j k j k

According to Theorem 2.7

! X ξ(j) 1(γ /2)2 (3.24) (τ − λ )1/2 k + 2 k j 0 τ − τ (τ − τ )2 k≥j/2 k j k j X |γ |2 X |γ |2 ≤ Cj k +Cj k |τ − τ ||τ −λ | j3|τ − τ | k≥j/2 k j k 0 k≥j/2 k j C X 1 = |γ |2 + O . j2 k j3 k≥j/2

Finally

(3.25) λ 1/2 1 λ 1 (τ − λ )1/2 = τ 1/2 1 − 0 = τ 1/2 1 − 0 + O j 0 j τ j 2 τ j4 j j 1 λ 1 = τ 1/2 − 0 + O . j 2 1/2 j3 τj Combining these estimates we obtain   ! 1 λ 1 X ξ(j) 2ψ (0) −1 = τ 1/2 − 0 +0 ·1− k  j j 2 1/2 j3 τ − τ τj k≤j/2 k j   1 X 1 + O  |γ |2 +  j2 k j k≥j/2 1 λ X ξ(j) 1 = τ1/2 − 0 − τ1/2 k + O j 2 1/2 j τ − τ j3 τj k≤j/2 k j   1 X 1 + O  |γ |2 +  j2 k j k≥j/2 which proves (3.21).

Combining Theorem 3.4 and Theorem 2.7 we obtain the following result which completes the proof of Theorem 1: 24 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

Theorem 3.5. Let q ∈ L2(S 1). Then there exists a neighborhood U of 0 0 q0 q in L2(S1; C) and C>0so that for any q in U and j ≥ 1, 0 0 q0   X √ 1 (3.26) |ω − (2πj)3|≤C |γ |2 +( τ −jπ)j2π2 + O . j k j j k≥j/2

Proof. According to (3.1) and using (3.20)   Z !−1 ϕ(µ)dµ X λ ω =8π p j τ − (µ(j) − τ )+ 0 j ∆(µ)2 − 4 j k k 2 Γj k∈N\{j}      X λ  X =8π (2πψ (0))−1 τ − (µ(j) − τ )+ 0 +O |γ |2 j  j k k 2  k k≤j/2 k≥j 1 +O . j By (3.21),

(3.27)    8π 1 λ X ξ(j) X λ ω = τ 1/2 − 0 − τ 1/2 k τ − ξ(j) + 0  j π j 2 1/2 j τ − τ j k 2 τj k≤j/2 k j k≤j/2   X 1 + O(1)  |γ |2 +  k j k≥j    1 X ξ(j) X 1  =8 τ3/2 − λ τ1/2 −τ3/2 k − τ1/2 ξ(j) + λ τ1/2  j 2 0 j j τ − τ j k 2 0 j  k≤j/2 k j k≤j/2   X 1 + O(1)  |γ |2 +  . k j k≥j Notice that

X ξ(j) X (3.28) −τ 3/2 k − τ1/2 ξ(j) j τ − τ j k k≤j/2 k j k≤j/2

X τ = τ 1/2 ξ(j) j − 1 j k τ − τ k≤j/2 j k

X τ ≤ C|j| |ξ(j)| k k τ − τ k≤j/2 j k ON THE KORTEWEG-DE VRIES EQUATION 25

1 X 1 ≤ C k2|ξ(j)| = O j k j k≤j/2 P 2 (j) where we used that, by Theorem 2.7, k≤j/2 k |ξk |≤C. √ ∧ ∧ Finally, write τ = jπ + τj , to obtain (with τ ∈ l2(j)) j (jπ)2 j ! ! τ∧ 1 τ∧ (3.29) τ 3/2 = (jπ)2 +2 j + O jπ + j j (jπ) j4 (jπ)2 1 =(jπ)3 +3τ∧ +O . j j3

Combining (3.27)-(3.29) leads to the claimed estimate.

The above results can be applied to the initial value problem for KdV,

3 2 1 (3.30) ∂tv :=−∂xv+3∂x(v )(t>0; x ∈ S ) (3.31) v(x, 0) = q(x).

Recently, using a ﬁxed point argument, Bourgain [Bo1] has constructed 2 1 weak solutions vB(x, t)=SB(q)(x, t)of(3.30)-(3.31) for initial data in L (S ) globally in time and proved that these solutions are unique within a certain space of functions f : R × [−T,T] → R (and T>0) and that they can be approximated within this space by smooth solutions corresponding to a smooth approximation of the initial data.

We obtain Bourgain’s weak solutions SB(q)of(3.30)-(3.31) by using action-angle coordinates as follows: Given q in L2(S1), deﬁne p := q − [q] ∈ R 2 1 L0(S ) where [q]= S1 q(x)dx and introduce (j ≥ 1), with ωj(p) given by Theorem 1,

(3.32) ωj(q)=ωj(p) + 12[q]πj.

In view of Theorem 3.1, the ω ’s are real analytic functions on L2(S1). De- j note by Ω : L2(S1) → `2 (R2) the symplectomorphism Ω(p)= x(p),y(p) = 0 1/2 x (p),y (p) constructed in [BBGK] and denote by J ,α the symplectic j j j≥1 j j polar coordinates corresponding to xj,yj, i.e. q q (3.33) (xj,yj)= 2Jjcos αj, 2Jj sin αj . Deﬁne x(t),y(t) ≡ x(t, q),y(t, q) by setting q q (3.34) x(t),y(t) = 2Jj cos ωj(q)t + αj , 2Jj sin ωj(q)t + αj j≥1 26 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN and introduce the solution operator S(1) S(1) : L2(S1) → C R; L2(S1) deﬁned by (3.35) S(1)(q)(t)=[q]+Ω−1 x(t),y(t) .

To prove that (3.35) deﬁnes a weak solution of KdV , we approximate the initial data q by ﬁnite gap potentials (qN )N≥1 with −1 (3.36) qN := Ω ΠN Ω(p) +[q] where ΠN denotes the projection 2 2 2 2 (3.37) ΠN : `1/2(R ) → `1/2(R ), (xj,yj)j≥1 7→ (xj,yj)1≤j≤N,0 .

2 1 Note that q = limN→∞ qN in L (S ) and that for any j ≥ 1,ωj(q)= limN→∞ ωj(qN ) uniformly in j. Therefore x(·,q),y(·,q) = lim x(·,qN),y(·,qN) N→∞ 2 2 in C [−T,T],`1/2(R ) (for any T>0), which in turn implies that

(1) (1) 2 1 (3.38) S (q) = lim S (qN )(in C([−T,T]; L (S ))). N→∞

(1) Notice that for arbitrary N ≥ 1, S (qN )(t)isanN-gap potential for any t and therefore a classical solution of KdV with initial data q .Forc∈R,we RN 2 1 2 1 have introduced the symplectic leaf Lc(S )={q∈L (S ); S1 q(x)dx = c}. Theorem 3.6. Let c ∈ R. Then (1) 2 1 (i) S (q)=SB(q)for q in Lc(S ). 2 1 (ii) Given q1,q2 in Lc(S ) there exists M>0so that for any t ∈ R, (1) (1) (3.39) S q1 (·,t)− S q2 (·,t) ≤M(1 + |t|)kq1 − q2kL2(S1). L2(S1) (1) 2 1 2 1 (iii) For any T>0,S : Lc(S ) → C [−T,T]; Lc(S ) is real analytic. For remarks concerning results related to Theorem 3.6 we refer the reader to the introduction.

2 1 Proof of Theorem 3.6. Let q be in L (S ) and qN (N ≥ 1) be deﬁned as in (1) (3.36). As classical solutions of KdV are unique we conclude that S (qN )= SB(qN) for any N ≥ 1. Moreover Bourgain proved that limN→∞ SB(qN )= ON THE KORTEWEG-DE VRIES EQUATION 27

2 1 (1) SB(q)inC [−T,T]; L (S ) . Together with (3.38) we conclude that S (q)= SB(q). Statements (ii) and (iii) rely both on Theorem 3.5 which, combined with 2 1 (3.32) leads to the following result: Let c ∈ R and q0 ∈ Lc(S ). Then there exists a neighborhood U of q in L2(S1; C) so that q0 0 c (3.40) ω (q) can be analytically extended to U j q0 and

3 (3.41) ωj(q)=(2πj) +12cπj + o(1) (as j →∞) uniformly for q in U . q0 Fix 0 0, depending on q1 and q2, such that for any qj ∈ Iso(qj) ∼ ∼ ∼ ∼ (3.42) Ω q1 − c − Ω q2 − c ≤ M1 q1 − q2 `2 (R2) L2(S1) 1/2

−1 where c =[q1]=[q2]. Using Ω instead of Ω, one concludes that there exists M2 > 0 such that 0 ∼ ∼ ∼ ∼ (3.42 ) q1 − q2 ≤ M2 Ω q1 − c − Ω q2 − c . L2(S1) `2 (R2) 1/2 28 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

Therefore it suﬃces to show that, for any t, (1) (1) Ω S (q1)(·,t)−c −Ω S (q2)(·,t)−c `2 1/2

(3.43) ≤Mkq1 −q2kL2(1 + |t|).

To simplify notation, write for k =1,2 (k) (k) (1) (k) x (t),y (t) =Ω S (qk)(·,t)−c ; ωj =ωj(qk).

Recall that for k =1,2 and j ≥ 1 ! ! (k) (k) d xj (t) (k) yj (t) (k) = ωj (k) . dt yj (t) −xj (t)

(j) ∼ (k) 3 Therefore, with ωj := ωj − (2πj) − 12[c]πj,

(3.44) ! ! ! (1) (2) (1) (2) d x (t) − x (t) (1) y (t) (2) y (t) j j = ω∼ j − ω∼ j dt y(1)(t) − y(2)(t) j −x(1)(t) j −x(2)(t) j j j j ! (1) (2) 3 yj (t) −yj (t) + (2πj) + 12[c]πj (1) (2) . − xj (t) −xj (t) ! (1) (2) xj (t) −xj (t) Scalar multiply (3.44)by (1) (2) to conclude that yj (t) −yj (t)

(3.45)

(1) (2) 2 1 d x (t) − x (t) j j 2 dt y(1)(t) − y(2)(t) j j ! ! (1) (2) (1) (2) ∼(1) y (t) ∼(2) y (t) x (t) − x (t) ≤ ω j − ω j j j . j (1) j (2) (1) (2) −xj (t) −xj (t) yj (t) − yj (t) ∼ 2 2 Notice that ωj yj(t), −xj(t) is analytic on U with values in `1/2(C ) j≥1 ∼ due to the fact that ωj = 0(1) uniformly on U (cf. Theorem 1). Therefore ! !! (1) (2) (1) (2) ∼ yj (t) ∼ yj (t) (3.46) ωj (1) − ωj (2) ≤ Mkq1 − q2kL2 −xj (t) −xj (t) j≥1 `2 (C2) 1/2 where M>0 is uniform for t ∈ R. (Use compactness of the isospectral sets.) ON THE KORTEWEG-DE VRIES EQUATION 29

Combining (3.45) and (3.46) we conclude that d (1) (1) (2) (2) x (t),y (t) − x (t),y (t) ≤ Mkq − q k 2 . `2 1 2 L dt 1/2 This proves estimate (3.43).

Using again properties of the map Ω one obtains in a straight forward way a result due to McKean-Trubowitz [MT1], and in its more general version, due to Bourgain [Bo1], concerning the almost periodicity of weak solutions of KdV . Recall (cf. [MT1], [Bo1]) that a function u ∈ C R; L2(S1) is called almost periodic if for any >0 there exists 0

ku(·,t)−u(·,t+τ0)kL2 < ∀t∈R.

Proposition 3.7. Let q ∈ L2(S 1). The weak solution S(1)(q)(·,t) of KdV is almost periodic.

Proof. For ease of exposition we assume that [q] = 0 and write v(·,t)= S(1)(q)(·,t). Note that v(·,t) ∈ Iso(q) and Iso(q) is compact. Therefore Ω Iso(q) is compact as well and Ω−1 is uniformly continuous on Ω Iso(q) . In particular, given >0, there exists δ = δ() > 0 such that for z(1) = (1) (1) (2) (2) (2) (1) (2) (x ,y ), z =(x ,y )inΩ Iso(q) with kz − z k`2 <δone has 1/2 −1 (1) −1 (2) kΩ (z ) − Ω (z )k 2 <. It therefore suﬃces to prove that, given any L0 δ>0, there exists 0 0, there exists 0

4. Frequencies of KdV2. In this section we investigate the frequencies of the second Hamiltonian in the KdV hierarchy and apply the results to the initial value problem of KdV2, (cf. e.g. [MM]) 1 5 15 (4.1) ∂ v = ∂5v − v∂3v − 5∂ v∂2v + v2∂ v t 4 x 2 x x x 2 x (4.2) v(x;0)=q(x). We want to construct weak solutions of (4.1)-(4.2)inC R;H1(S1) for initial data q in H1(S1). The construction of the solution map S(2) is very similar as in the case of KdV . Recall that the Hamiltonian H(2) corresponding to

KdV2 is given by (cf. e.g. [MM]) Z (2) 1 2 2 5 2 5 4 (4.3) H (q)= (∂xq) + q(∂xq) + q dx. S1 8 4 8 We point out that H(2) is only deﬁned for q in H2(S1). Nevertheless we will show that one can deﬁne the frequencies ∂H(2) corresponding to H(2). ∂Jj First notice that, with p := q − [q],

(4.4) H(2)(q)=H(2)(p +[q]) Z 5 15 1 5 = H(2)(p)+ [q]·H(1)(p)+ [q]2 p2dx + [q]4. 2 2 2 S1 8

1 1 Therefore it suﬃces to deﬁne the frequencies for q ∈ H0 (S ). Recall that for√ the construction of the L2-integrable, holomorphic 1-forms Ωj on y = 2 ϕj(λ) ∆ − 4 we have introduced the functions fj(λ):=Cj√ with ∆(λ)2−4

Z !−1 1 Yµ(j) −λ φ (u)dµ (4.5) φ (λ):= k ;C := p j j j2π2 j2π2 j ∆(µ)2 − 4 k6=j Γj where Γj is the counterclock oriented circle with center τj and suﬃciently small radius ρj >γj/2 and Cj is a normalizing constant. The frequencies (2) ωj are then deﬁned by πi (4.6) ω(2) := f(iv)(∞) j 3 j

(iv) where fj (∞) denotes the fourth deriviative of fj(λ) evaluated at inﬁnity with respect to local coordinates τ 2 = 1 near λ = ∞. A straightforward λ ON THE KORTEWEG-DE VRIES EQUATION 31 calculation shows that (4.7)   1 λ X 3 λ 2 i · C−1 f (iv)(∞)=τ2 +τ  0 − ξ(j)+ 0 j 4! j j j 2 k 2 2 k6=j   λ X 1 X 1 X X − 0 ξ(j) + γ2 +  ξ(j)ξ(j) − τ ξ(j) 2 k 8 k 2 k l k k k6=j k k,l6=j,k6=l k6=j

(j) (j) where ξk := µk − τk (as in Section 2). In view of Theorem 2.1, given q ∈ H1(S1), there exists a neighborhood U of q in H1(S1; C) so that the 0 0 q0 0 0 right hand side of (4.7) is well deﬁned and analytic for any j ≥ 1. In order to (2) derive an asymptotic expansion of the ωj ’s we need a number of auxilary results. From Theorem 2.7 we deduce the following

Lemma 4.1. Let q ∈ L2(S 1). Then there exist a neighborhood U of q 0 0 q0 0 in L2(S1; C), j ≥ 1 and C>0such that for q ∈ U ,j ≥j ,k ≥j/2 0 0 q0 0 1 (γ /2)2 (γ /2)2 1 C (4.8i) ξ(j) − − k − k ≤ |kγ |2 k 5 k 2 τk − τj 4 τk − λ0 j and, for j ≥ j0, k ≤ j/2 (γ /2)2 1 |kγ |2 (4.8ii) ξ(j) − − k ≤ C k . k 5 4 τk − λ0 k

Lemma 4.2. Let q ∈ H 1(S 1). Then there exists a neighborhood U of q 0 0 q0 0 1 1 in H0 (S ; C) so that Z !−1 ϕ (µ)dµ |jγ |2 (4.9i) p j = 2πψ (0) −1 + O j 2 j 5 Γj ∆(µ) − 4 j (4.9ii)   X 1 1 (γ /2)2 2ψ (0) −1 =(τ −λ )1/2 1− ξ(j) + k j j 0  τ − τ k 2 τ − τ k≤j/2 k j k j    2 !2 1 X ξ(j) 1 X ξ(j)  +  k  + k 2 τ − τ 2 τ − τ  k≤j/2 k j k≤j/2 k j  1 X 1 + O |kγ |2 + O . j4 k j5 k≥j/2 32 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

Proof. (i) Follows from (3.20). (ii) By Lemma 3.2,   X 1 1(γ /2)2 2ψ (0) −1 =(τ −λ )1/2 1− ξ(j) + k j j 0  τ −τ k 2τ − τ k6=j k j k j    2 !2 1 X ξ(j) 1 X ξ(j)  1 +  k  + k + O . 2 (τ − τ ) 2 (τ − τ )  j5 k6=j k j k6=j k j 

By (4.8i) one has

X 1 1 (γ /2)2 1 X ξ(j) + k ≤ O |γ k|2. τ − τ k 2 τ − τ j5 k k≥j/2 k j k j k≥j/2 Similarly, by (4.8i) and (4.8ii), X ξ(j) 1 X ξ(j) 1 k = O ; k = O . τ − τ j4 τ − τ j2 k≥j/2 k j k≤j/2 k j This leads to (4.9ii). Lemma 4.3. Let q ∈ H 1(S 1). Then there exists a neighborhood U of q 0 0 q0 0 in H1(S1; C) and C>0so that for any q in U and j ≥ 1 0 q0   X ∼ 1 (4.10) ω(2) − 8 (jπ)5 + jπ5τ ≤ C  |kγ |2 +  j j k j k≥j/2

√ ∼ where τ = jπ + τ j . j (jπ)3 Proof. Using (4.7), (4.6) and (4.5) one obtains   Z !−1 ( ϕ(µ)dµ λ X ω(2) =8π p j τ 2 + τ  0 − ξ(j) j ∆(µ)2 − 4 j j 2 k Γj k6=j ) 3 λ X 1 X 1 X X + λ2 − 0 ξ(j) + γ2 + ξ(j)ξ(j) − τ ξ(j) . 8 0 2 k 8 k 2 k l k k k6=j k6=j k,l6=j,k6=l k6=j

−1 R −1 2 √ϕj (µ)dµ |jγj| Therefore with = 2πψj(0) +O 5 and the asymp- Γj ∆(µ)2−4 j −1 totics of 2πψj(0) we obtain by a straight forward computation   1 X 1 ω(2) = τ 5/2 + O(1)  |kγ |2 + O . 8 j j k j k≥j/2 ON THE KORTEWEG-DE VRIES EQUATION 33

√ ∼ With τ = jπ + τ j we obtain j (jπ)3 1 τ 5/2 =(jπ)5 + 5∼τ jπ + O j j j which leads to (4.10).

Lemma 4.4. Let q ∈ H 1(S 1). Then there exists a neighborhood U of q 0 0 q0 0 in H1(S1; C) and C>0so that, uniformly for q in U , 0 q0 R 1 1q(x)2dx 1 (4.11i) τ (q)=j2π2 + 0 + O j 4 j2π2 j3 R q 1 1 q(x)2dx 1 (4.11ii) τ (q)=jπ + 0 + O . j 8 (jπ)3 j4

Proof. Notice that (4.11ii) follows immediately from (4.11i), so let us con- centrate on (4.11i). Recall that, for j ≥ j0 with j0 suﬃciently large, d2 2τ = tr − + q P j dx2 j where Pj is the Riesz projector on the subspace generated by the generalized eigenfunctions corresponding to the eigenvalues λ2j and λ2j−1.Itis convenient to introduce, for j ≥ j0, d2 L(t):=− + tq, 2 dxZ 1 −1 Pj(t):= λ − L(t) dλ, 2πi Γ j τj(t):=tr L(t)Pj(t)

2 2 where Γj is a circle with center j π and radius ρj so that the eigenvalues λ2j(tq) and λ2j−1(tq) are inside of Γj and all other eigenvalues are outside of Γ for any 0 ≤ t ≤ 1, q ∈ U . (The integer j has been chosen suﬃciently j q0 0 large and the neighborhood U suﬃciently small.) The functions τ (t) admit q0 j a Taylor expansion, 1 1 (4.12) τ (1) = τ (0) + τ 0(0) + τ 00(0) + τ 000(0) + E j j j j 2! j 3! j where Ej denotes the Taylor remainder term. Clearly

2 2 2 2 (4.13) 2τj(0) = n π trPj(0)=2j π . 34 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

Further 0 0 (4.14) 2τj(0)=tr Pj(0)L (0)Pj(0) =tr qPj(0) =2[q]=0.

To compute the second derivative it is convenient to introduce

Z 1 ∧ 1 −iπkx qk = e q(x)dx. 2 −1 Then Z X 1 dλ 2τ 00(0)=tr qP 0(0) = ∧q ∧q j j m−k k−m 2πi (λ − m2π2)(λ − k2π2) m,k Γj X ∧q ∧q 4 X ∧q ∧q =4 m−j j−m = n −n . π2(j2 −m2) π2 n(2j − n) m6=j k6=0,2j

Writing 1 1 1 1 1 = + n(2j − n) n 2j 2j − n 2j 1 1 1 n = + + n 2j (2j)2 (2j − n)(2j)2 and using that

X ∧q ∧q X ∧q ∧q ∧q ∧q ∧q ∧q (4.15) n −n = n −n − 2j −2j =0− 2j −2j, n n 2j 2j n6=0,2j n6=0 we conclude from (4.14)

(4.16) 4 X ∧q ∧q 4 X n∧q (−n)∧q 2τ 00(0) = n −n + n −n j π2 (2j)2 π2 (−n)(2j − n)(2j)2 n6=0 n6=0 n6=2j Z Z 1 1 1 1 1 1 1 = q(x)2dx +0 = q(x)2dx + O . 2 2 3 2 2 3 π j 2 −1 j j π 0 j

000 Similarly, we compute τj (0),

(4.17) 000 00 2τj (0)=tr qPj (0) ON THE KORTEWEG-DE VRIES EQUATION 35

Z 1 =tr q2 λ − L(0) −1q λ − L(0) −1q λ − L(0) −1dλ 2πi Γ j Z X 1 dλ =2 ∧q ∧q ∧q m−n n−k k−m 2πi (λ − n2π2)(λ − k2π2)(λ − m2π2) m,k,n Γj X ∧q ∧q ∧q =6·2 j−n n−k k−j (j2π2 −n2π2)(j2π2 − k2π2) n6=j k6=j 6 · 2 X ∧q ∧q ∧q = m `−m −` . π4 m(2j − m)`(2j − `) m,` m,`6=0,2j Again, writing 1 1 1 m ` 1 = + + m(2j − m) (2j − `)` m`(2j)2 m(2j − m)`(2j)2 m`(2j − `) (2j)2 m` 1 + `(2j − `)m(2j − m) (2j)2 and taking into account that q ∈ U ⊂ H1(S1; C) we conclude that q0 0 6 · 2 X ∧q ∧q ∧q 1 (4.18) 2τ 000(0) = m `−m −` + O j π4 m`(2j)2 j3 m,`6=0 1 =0+O . j3 Using the standard resolvent estimate one concludes that the Taylor remainder term Ej satisﬁes 1 (4.19) E = O . j j3 Substitute (4.13), (4.14), (4.16), (4.18) and (4.19) into the Taylor expansion (4.12) we conclude R 1 1 q(x)2dx 1 τ (1) = j2π2 + 4 0 + O . j j2π2 j3 This proves (4.11i).

Theorem 4.5. Let q ∈ H 1(S 1). Then there exists a neighborhood U of 0 0 q0 q in H1(S1; C) and C>0so that uniformly for q in U 0 0 q0

(2) 5 (4.20) ωj − 8[(jπ) − a1jπ] ≤ C 36 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

where a1 = a1(q) is given by

Z 1 5 2 (4.21) a1 = q(x) dx. 8 0

(2) ∼ Proof. Combine the estimate (4.10) for ωj with the estimate (4.11ii) for τ j

Z 1 ∼ 1 2 1 τ j = q(x) dx +0 8 0 j to obtain (4.20)-(4.21). Given real numbers c ,c we now construct weak solutions of (4.1)-(4.2) 1 3 in C R; H1 (S1) for initial data q in H1 (S1):={q∈H1(S1); [q]= c1,c3 c1,c3 c3;a1(q)=c1}.

(2) Theorem 4.6. Let c1,c3 ∈R. Then there exists a solution map S S(2) : H1 (S1) → C R,H1 (S1) c1,c3 c1,c3 with the following properties: (i) S(2)(q) is a weak solution of (4.1)-(4.2); 1 1 (ii) given q1,q2 in H (S ) with

[q1]=[q2]=c3; a1(q1)=a1(q2)=c1 there exists M>0so that for any t

(2) (2) S (q1)(·,t)−S (q2)(·,t) ≤M(1 + |t|)kq1 − q2kH1(S1). H1(S1) (iii) For any 0

Remark 1. According to [Bo2], Bourgain’s method cannot be applied to solve the initial value problem (4.1)-(4.2)inH1(S1). R Remark 2. Note that the KdV Hamiltonian H(q)= 1(∂ q)2 +q3 dx S1 2 xR 2 is a conserved quantity for (4.1), as well as the average [q] and S1 q(x) dx. Thus, given a real valued smooth solution v(x, t)of(4.1) one obtains an R 2 a priori bound for S1 |∂xv(x, t)| dx which leads to the existence of a weak solution of (4.1)-(4.2). This solution can be approximated by ﬁnite gap solutions of (4.1). Of course, the main point of the statement in Theorem 4.6 is that the solution map S(2) is real analytic. Proof of Theorem 4.6. The construction of the solution map S(2) and the proofs of its properties are similar to the ones of S(1) and we therefore omit it. ON THE KORTEWEG-DE VRIES EQUATION 37

Appendix: Proof of Theorem 2.1.

2 1 In the case where the potential q ∈ L0(S ; C) is real valued, the entire functions φj(λ, q)(j ≥ 1) are constructed in [MT2]. Actually, instead of φj, [MT2] construct entire functions 1j(λ, q) which coincide with φj(λ, q)upto a normalization factor. (In the sequel, we use the normalization introduced by McKean-Trubowitz.) By a straightforward perturbation argument one 1 shows that for given q0 ∈ L0(S ; R) there exist a (suﬃciently small) neighborhood U of q in L2(S1; C) so that for q ∈ U the functions 1 (λ, q)(j ≥ 1) q0 0 0 q0 j are uniquely deﬁned and analytic with respect to q. The main part of the proof of Theorem 2.1 consists in analyzing the zeroes of 1j(λ, q) for q in a neighborhood U ⊆ U of q in L2(S1; C) which does not depend on j and q0 0 0 proving the estimates (2.1). We proceed in two steps:

2 1 In Section A.2 we show, using Rouch´e’s theorem that, given q0 ∈ L0(S ; R), 0

2 1 In Section A.3, we show that there exist a neighborhood U of q0 in L0(S ; C) and N ≥ 1, depending on U only, so that the system of nonlinear equations (j) for the zeroes µk (q)of1j(λ, q) with k ≥ N +1,k 6=j, obtained from (2.2), can be solved by a contraction argument. At the same time we obtain the (j) estimates for µk (q) − τk(q) claimed in Theorem 2.1. Finally the analyticity (j) of the zeroes µk (q)of1j(λ) follows from Cauchy’s integral formula. We remark that by using the contraction argument of section A.3 we can obtain a new proof for the existence of 1j(λ, q) and their product representation for 2 real valued potentials q in L0. In Section A.4 we present the proof of Theorem 2.1, combining the results of Sections A.1, A.2 and A.3.

In order to avoid introducing cumbersome notation the same letter C will denote various constants and the same letter U will denote various neigh- borhoods.

A.1. Normalized Riesz basis of holomorphic diﬀerentials. Let us ﬁrst recall some notions, notations and results from [MT1] (cf. also [MT2]). Denote by I3/2 the complex Hilbert space of entire functions 38 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

φ of order ≤ 1/2 and type ≤ 1 with

Z ∞ (A.1) kφk2 := |φ(λ)|2λ3/2dλ < ∞. I3/2 0

The inner product in I3/2 is given by

Z ∞ hφ, ψi := φ(λ)ψ(λ)λ3/2dλ. 0

By the Paley-Wiener theorem

√ (A.2) |λφ(λ)|≤kφk e|Im λ|. I3/2 Introduce, for q ∈ L2(S1; R),K(q ):= 1 min λ (q ) − λ (q ) where 0 0 0 5 n≥0 2n+1 0 2n 0 λ (q ) is the anti/periodic spectrum of − d2 + q . Then K(q ) > 0 and n 0 n≥0 dx2 0 0 for 0

(A .3) sup |λj(q) − λj(q0)|≤K. j≥0

Denote by Γn =Λn(K, q0) the counterclockwise oriented circle in C with center τ = τ (q )= λ2n(q0)+λ2n−1(q0) and radius r := γn(q0) +2K where n n 0 2 n 2 γn(q0)=λ2n(q0)−λ2n−1(q0). Notice that, because of (A.3), for q in V , λ2n(q) and λ2n−1(q) are inside Γn and all other eigenvalues λk(q) are outside Γn. For q in V and n ≥ 1, denote by An(q):I3/2 →Cthe bounded linear functional Z φ(λ)dλ (A .4) A (q)(φ):= p . n 2 Γn ∆(λ, q) − 4

∗ Then An(q) is an analytic function of q ∈ V with values in the dual I3/2 of I3/2. Using (A.2) and (2.6) one veriﬁes (cf.[MT 1, 2]) that there exists 1 C ≥ 1 such that ≤knAn(q)kI∗ ≤ C for q in V . C 3/2 Recall (cf. [GK, p. 310]) that (un)n≥1 is said to be a Riesz basis of a Hilbert space H if there exist an orthonormal basis (ej)j≥1 of H and an invertible bounded linear operator B : H → H such that Bej = uj(j ≥ 1). In the case where the potential q is real valued McKean-Trubowitz (cf. [MT2]) showed that nA (q) is a Riesz basis of I∗ . To be able to n n≥1 3/2 consider complex valued potentials as well, we need the following auxilary result. We recall (use Marchenko’s asymptotics of the eigenvalues λj(q)) (cf ON THE KORTEWEG-DE VRIES EQUATION 39

[Ma, Theorem 1.5.2]) that the map q 7→ τ (q) − k2π2 is an analytic k k≥1 2 function on V with values in `1(N; C). Lemma A.1. There exist C ≥ 1 and N ≥ 1 such that for n ≥ N and q ∈ V (V deﬁned before (A.3)) 2 2 nkAn(q) − An(q0)kI∗ ≤ C |γn(q) − γn(q0) |  3/2  1/2 C  X + |λ (q) − λ (q )| +  k2|τ (q) − τ (q )|2 n2 0 0 0 k k 0 k≥1   1/2 X  2 2   + |γk(q) − γk(q0) |  . k≥1

Proof. For φ ∈ I 3/2 ,n≥1 and q ∈ V

(A.5)

An(q)(φ) − An(q0)(φ) Z 2 2 ∆(λ, q0) − 4 − ∆(λ, q) − 4 φ(λ)dλ = p p p p . 2 2 2 2 Γn ∆(λ, q) − 4 ∆(λ, q0) − 4 ∆(λ, q) − 4+ ∆(λ, q0) − 4

Recall that for λ ∈ Γn (cf. e.g. [Ka, p. 547]) 1 log(n +1) ∆(λ, q)2 − 4= λ (q)−λ λ − λ (q) 1+β (λ, q) n2π2 2n 2n−1 n n where βn(λ, q) satisﬁes |βn(λ, q)| 0. In view of the choice of K,Γn =Γn(K) and V , these asymptotics can be used to obtain an estimate of the denominator of the integrand in (A.5), for q in V ,

1 1 1 sup p · p · p p 2 2 2 2 λ∈Γn ∆(λ, q) − 4 ∆(λ, q0) − 4 ∆(λ, q) − 4+ ∆(λ, q0) − 4 ≤ Cn3 where C ≥ 1 can be chosen independently of q ∈ V and n ≥ 1. To estimate the nominator of the integrand in (A.5) we write ∆(λ, q)2 − 4 (A .6) ∆(λ, q )2−4 − ∆(λ, q)2−4 = ∆(λ, q )2−4 1 − . 0 0 2 ∆(λ, q0) − 4 40 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

Use the product representation Y∞ λ (q) − λ λ (q) − λ ∆(λ, q)2 − 4=4 λ (q)−λ 2k 2k−1 0 k4π4 k=1

2 2 and write λ (q) − λ λ (q) − λ = τ (q) − λ − γk(q) to obtain 2k 2k−1 k 2 (A.60) ∆(λ, q)2 − 4 λ (q )−λ (q) = 1+ 0 0 0 2 ∆(λ, q0) − 4 λ−λ0(q0) 2 2! τ (q)−τ (q ) τ (q)+τ (q )−2λ − γn(q) + γn(q0) n n 0 n n 0 2 2 1+ 2 2 ·En τ (q )−λ − γn(q0) n 0 2 where

2 2! Y τ (q)−τ (q ) τ (q)+τ (q )−2λ − γk(q) + γk(q0) k k 0 k k 0 2 2 En := 1+ 2 2 . τ (q )−λ − γk(q0) k6=n k 0 2 k≥1

One veriﬁes that there exist N0 ≥ 1 and C ≥ 1 such that for n ≥ N0,q ∈V and λ ∈ Γn

2 2 ! X |τ (q)−τ (q )||k2 − n2| | γk(q) − γk(q0) | |E − 1|≤C k k 0 + 2 2 . n |k2 − n2|2 |k2 − n2|2 k6=n k≥1 Notice that

2 2 X γk(q) − γk(q0) X 2 2 2 2 1 γk(q) γk(q0) ≤ − |k2 − n2|2 n2 2 2 k6=n k≥1 k≥1 and

X |τ (q) − τ (q )||k2 − n2| C X k k 0 ≤ |τ (q) − τ (q )|. |k2 − n2|2 n2 k k 0 n k≥1 |k−n|≥ 2 k≥1 Further

X 2 2 |τk(q) − τk(q0)||k − n | |k2 − n2|2 n |k−n|≤ 2 k6=n ON THE KORTEWEG-DE VRIES EQUATION 41

C X 1 ≤ k|τ (q) − τ (q )| n2 |k − n| k k 0 n |k−n|≤ 2 k6=n  1/2  1/2 C X 1 X ≤    k2|τ (q) − τ (q )|2 . n2 |k − n|2 k k 0 |k−n|≥1 k≥1 Combining the above estimates we conclude that there exist N ≥ 1 and

C ≥ 1 such that for n ≥ N,q ∈ V,λ ∈ Γn (A.600)

∆(λ, q)2 − 4 1 − ∆(λ, q )2 − 4 0 2 2 γn(q) γn(q0) C ≤ C|τ (q) − τ (q )| + C − + |λ (q) − λ (q )| n n 0 2 2 n2 0 0 0

 1/2  1/2 C X C X +  k2|τ (q) − τ (q )|2 +  |γ (q)2 − γ (q )2| . n2 k k 0 n2 k k 0 k≥1 k≥1

√ In view of (A.2) and sup e|Im λ| < ∞ one concludes that there exists λ∈∪Γn n≥1 C ≥ 1 such that for φ ∈ I3/2 and n ≥ 1 C (A.6000) sup |φ(λ)|≤ kφk . 2 I3/2 λ∈Γn n Combining the estimates (A.6)-(A.6000) we deduce from (A.5) that there exist N ≥ 1 and C ≥ 1 such that for n ≥ N,q ∈ V the claimed estimate for nkAn(q) − An(q0)kI∗ holds. 3/2

2 1 ∗ Since An(q) is continuous on V ⊆ L0(S ; C) with values in I3/2, Lemma A.1 leads to (use the asymptotics of the eigenvalues λj(q) to estimate the right hand side of the formula in Lemma A.3).

2 1 Corollary A.2. Let q0 ∈ L0(S ; C) be real valued. Then, for any >0, 2 1 there exists a neighborhhood U ≡ U ⊆ V of q0 in L0(S ; C) so that for q in U X 2 2 n kAn(q) − An(q0)kI∗ ≤ . 3/2 n≥1

2 1 Theorem A.3. Let q0 ∈ L0(S ; C) be a real valued potential. Then there exists a neighborhood U of q in L2(S1; C) with the following properties: 0 0 (i) ForanyqinU, nA (q) is a Riesz basis of I∗ ; n n≥1 3/2 42 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

(ii) there exists a uniquely determined sequence 1 1 (·,q) in I , each n n n≥1 3/2 of the elements 1 1 (·,q) depending analytically on q ∈ U, so that n n 1 nA (q) 1 (·,q) =δ ; n k k nk (iii) for any q in U, 1 1 (·,q) is a Riesz basis of I . n n n≥1 3/2 Proof. Let U be a neighborhood as in Corollary A.2. Since nA (q ) is n 0 n≥1 a Riesz basis of I∗ , we can define a linear bounded map T (q):I∗ →I∗ 3/2 3/2 3/2 by setting T (q) nAn(q0) := nAn(q). By Corollary A.2, Id − T(q)isa Hilbert-Schmidt operator and by choosing 0 <<1 sufficiently small we conclude from Corollary A.2 that kId − T(q)kHS < 1 for any q in U. (Given a linear operator A : H → H on a Hilbert space H with an orthonormal basis (ej)j≥1, kAkHS denotes the Hilbert-Schmidt norm P 2 1/2 kAkHS = j,k |hej,Aeki| .) In particular T (q) is an invertible, bounded ∗ linear operator and thus T (q) nAn(q0) is a Riesz basis of I3/2 as well. n≥1 Using the same symbols as McKean-Trubowitz [MT2], we denote by 1 1 (λ, q ) the Riesz basis of I uniquely determined by n n 0 n≥1 3/2 nA (q ) 1 1 (·,q ) =δ . n 0 k k 0 nk Let 1 1 (·,q):= T(q)−1 ∗ 11 (·,q ) where T (q)−1 ∗ denotes the adjoint n n n n 0 of T (q)−1. One verifies that nA (q) 1 1 (·,q) =δ . It remains to prove that n k k nk the 1n(·,q)’s depend analytically on q ∈ U. Denote by ∗ ∗ ∗ LHS(I3/2,I3/2) the Hilbert space of Hilbert-Schmidt operators on I3/2 with inner product X hR, SiHS := RijSji i,j where Rij = hRei,eji are the coefficients of R with respect to an orthonor- ∗ mal basis (ei)i≥1 of I3/2. From the analyticity of An(q) it follows that the coefficients of Id − T(q) are analytic on U. By Corollary A.2, Id − T(q)is ∗ ∗ bounded on U in LHS(I3/2,I3/2) and therefore, Id−T(q) is analytic (cf. e.g. −1 [PT, Appendix A]). As k Id − T(q) kHS < 1, X T (q)−1 =(Id − (Id − T(q)))−1 = (Id − T(q))k k≥0 where the series converges in L (I∗ ,I∗ ). Therefore T (q)−1 is analytic HS 3/2 3/2 which implies that 1 1 (·,q)= T(q)−1 ∗ 11(·,q ) is analytic on U with n n n n 0 values in I3/2 for any n ≥ 1. ON THE KORTEWEG-DE VRIES EQUATION 43

Remark. For q = 0, the Riesz basis nA (q) of I∗ and 1 1 (·,q) n n≥1 3/2 n n n≥1 of I3/2 are given by

n 2 2 nAn(0)=(−1) 2n π δ(n2π2) and √ 1 sin λ 1n(λ, 0) = √ . n λ(λ − n2π2)

A.2 Rouch´e’s theorem.

Denote by I−1/2 the complex Hilbert space of entire functions φ of order ≤ 1/2 and type ≤ 1 with Z ∞ dλ kφk2 := |φ(λ)|2 √ < ∞. I−1/2 0 λ

The inner product in I−1/2 is given by Z ∞ dλ hφ, ψi := φ(λ)ψ(λ)√ < ∞. 0 λ

By the Paley-Wiener theorem

√ (A.7) |φ(λ)|≤kφk e|Im λ|. I−1/2

∗ Denote by I−1/2 the Hilbert space dual to I−1/2. Notice that the identity embedding I3/2 ,→ I−1/2 is continuous. As in Section A.1, denote by V a 2 neighborhood of a real valued potential q0 ∈ L0 with the property (A.3). For q in V , the functionals An(q) introduced in (A.4) extend to linear bounded functionals on I−1/2, An(q):I−1/2 →C. Using (A.7) and the deﬁnition of 1 1 An(q) one concludes that there exists C>1 such that ≤k An(q)kI∗ ≤ C n −1/2 C for q in V . By the same arguments used in the proof of Lemma A.1 one obtains

Lemma A.10. There exist C ≥ 1 and N ≥ 1 such that for n ≥ N and q ∈ V

1 kAn(q) − An(q0)kI∗ n −1/2 2 ≤ C |γn(q) − γn(q0)| + |τn(q) − τn(q0)| 44 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

  1/2 C  X + |λ (q) − λ (q )| +  k2|τ (q) − τ (q )|2 n2 0 0 0 k k 0 k≥1   1/2 X  2 2   + |γk(q) − γk(q0) |  . k≥1

Next we need to estimate the I−1/2-norm of the functions n1n(λ, q0) for a real valued potential q0.

2 1 0 Lemma A.4. Let q0 ∈ L0(S ; R). Then there exists C > 0 such that kn1 (·,q )k ≤C0 (n≥1). n 0 I−1/2

Proof. It follows from the Paley-Wiener theorem that the Dirac measures √ √ ∗ πδ0, ( 2πδn2π2 )n≥1 are an orthonormal basis of I−1/2 (Kotelnikov theorem; cf. also Remark after Theorem A.3). Therefore X kj1 (·,q )k2 =π|j1 (0,q )|2 +2π |j1 (n2π2,q )|2. j 0 I−1/2 j 0 j 0 n≥1

Further recall that, as q0 is real valued, ([MT1])

Yµ(j) −λ (A .8) j1 (λ, q )=C k j 0 j k2π2 k6=j where λ (q ) ≤ µ(j) ≤ λ (q ) and 1 ≤ C ≤ C for all j ≥ 1 for some 2k−1 0 k 2k 0 C j constant C ≥ 1. For 1 ≤ n 6= j , write

Yµ(j) −n2π2 j1 (n2π2,q )=C k j 0 j k2π2 k6=j   Y µ(j) − n2π2 τ − n2π2 j2π2 µ(j) − n2π2 =C  k  j · n . j k2π2 j2π2 τ − n2π2 n2π2 k6=n,j j

Notice that there exists C ≥ 1 (cf. e.g. [PT, Appendix E, Lemma 3]) such that for j, n ≥ 1 with j 6= n  

Y µ(j) − n2π2 τ − n2π2  k  j ≤ C. k2π2 j2π2 k6=n,j ON THE KORTEWEG-DE VRIES EQUATION 45

This implies that there exists C ≥ 1 such that for j 6= n, j, n ≥ 1, j2 µ(j) −n2π2 |j1 (n2π2,q )|≤C n . j 0 |j2 −n2| n2π2 Similarly, there exists C ≥ 1 such that for j ≥ 1

2 2 |j1j(j π ,q0)|≤C and |j1j(0,q0)|≤C. Notice that for 1 ≤|n−j|≤j/2, j2 1 ≤ C and for |n − j| >j/2, |j2−n2| n2 |j2−n2| j2 1 ≤Cn2. Therefore we conclude that there exists C ≥ 1 such that for j2−n2 n2 any j ≥ 1, X X 2 2 2 (j) 2 2 2 |j1j(n π ,q0)| ≤2C+C |µn −n π | . n≥0 n6=j

(j) As λ2n−1(q0) ≤ µn ≤ λ2n(q0) the claimed result then follows from the asymptotics of the eigenvalues. In order to apply Rouch´e’s theorem we need to prove that | 1 1 (λ, q) − j j 1 1 (λ, q )| < | 1 1 (λ, q )| for λ ∈ Γ and q ∈ U. Notice that the diﬀerence j j 0 j j 0 n ψ (λ, q):= 11(λ, q) − 1 1 (λ, q )isinI and therefore, by Theorem A.3, j j j j j 0 3/2 can be written as X 1 ψ (λ, q)= nA (q ) ψ (·,q) 1 (λ, q ). j n 0 j n n 0 n≥1

∗ For any λ ∈ C, the Dirac measure δλ is in I and |δλ(ψ)|≤ √ −1/√2 |Im λ| | Im λ| kψkI e (cf. A.7). Introduce β := sup ∞ e . Then β<∞ −1/2 λ∈∪1 Γk and (A .9) sup |ψ (λ, q)|≤βkψ(·,q)k . j j I−1/2 ∞ λ∈∪1 Γk Further, by Lemma A.4, there exists C0 ≥ 1 such that for j ≥ 1, kψ (·,q)k j I−1/2

X∞ 1 ≤ |nA (q ) ψ (·,q) | 1 (·,q ) n 0 j n n 0 n≥1 I−1/2 X∞ 1 ≤C0 |nA (q ) ψ (·,q) |. n2 n 0 j n≥1 Rewrite nAn(q0) ψj(·,q) , using the deﬁnition of ψj, 1 nA (q ) ψ (·,q) =nA (q ) 1 (·,q) −δ n 0 j n 0 j j nj 46 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

1 = nA (q ) − nA (q) 1 (·,q) n 0 n j j 1 = nA (q ) − nA (q) 1 (·,q ) + nA (q ) − nA (q) ψ (·,q) n 0 n j j 0 n 0 n j to conclude that X 1 |nA (q ) ψ (·,q) | n2 n 0 j n≥1

X 1 1 ≤ knAn(q) − nAn(q0)kI∗ 1j(·,q0) n2 −1/2 j n≥1 I−1/2 X 1 + knAn(q) − nAn(q0)kI∗ kψj(·,q)kI . n2 −1/2 −1/2 n≥1

Apply Lemma A.10 in a way similar as Lemma A.1 to Corollary A.2 to conclude that for any 0 <≤ 1 there exists a neighborhood U ≡ U ⊂ V of 2 2 1 q0 in L0(S ; C), such that, for q ∈ U,

X 1 1 An(q) − An(q0) ≤ . n n I∗ n≥1 −1/2

Together, the two estimates above lead to

1 1 kψ (·,q)k ≤ 1 (·,q ) + kψ (·,q)k . j I−1/2 j j 0 2 j I−1/2 I−1/2

Together with Lemma A.4, this implies kψ (·,q)k ≤2k11 (·,q )k ≤ j I−1/2 j j 0 I−1/2 2C0 . Taking into account (A.9) and the deﬁnition of ψ this estimate leads j2 j to

1 1 (A .10) sup 1 (λ, q) − 1 (λ, q ) ≤ β2C0 , j j 0 2 ∞ j j j λ∈∪1 Γk for any q ∈ U and j ≥ 1. To be able to apply Rouch´e’s theorem we also need to estimate 1 1 (λ, q ) j j 0 from below for λ ∈ Γ .Asq is real valued, 1 1 (λ, q ) has product repre- n 0 j j 0 sentation (A.8) and in view of the deﬁnition of the circles Γn(n ≥ 1), we conclude that for any integer 1 ≤ N<∞there exists ρN > 0 such that for any j ≥ 1

1 1 (A .11) inf 1j(λ, q0) ≥ ρN . N 2 λ∈∪1 Γn j j ON THE KORTEWEG-DE VRIES EQUATION 47

Combining (A.10) and (A.11) we obtain

1 1 1 1 ρ − β2C0 1 (λ, q) ≥ 1 (λ, q ) − 1 (λ, q) − 1 (λ, q ) ≥ N j j j j 0 j j j j 0 j2

N for λ ∈ ∪Γn,q ∈U and j ≥ 1. 1

2 1 Theorem A.5. Let q0 ∈ L0(S ; C) be real valued, 1 ≤ N<∞an integer and 0 0 and a neighborhood U ≡ U,N 2 1 of q0 in L0(S ; C) suﬃciently small such that the statements of Theorem A.3 and (A.10) are valid and such that βC ≤ 1ρ where the product βC is as 2 N in (A.10) and ρN is given as in (A.11). Then

1 1 1 1 sup 1j(λ, q) − 1j(λ, q0) ≤ inf 1j(λ, q0) N N j j 2 λ∈∪ Γn j λ∈∪1 Γn 1 for any j ≥ 1 and q in U. Further notice that, due to the product representation (A.8), statement (ii) holds for q = q0. Therefore, by Rouch´e’s theorem, (ii) follows. Statement (iii) is a consequence of Cauchy’s integral formula

Z d 1 (λ, q) (j) 1 dλ j µk (q)= λ dλ 2πi Γk 1j(λ, q) together with the analyticity of 1j(λ, q) (cf. Theorem A.3).

A.3. Contraction mapping.

To localize the zeroes of the 1j(λ, q) outside the circles Γn(1 ≤ n ≤ N), we consider the following system (cf. notation of Section 2 or (A.16) below)

R1 √tdt C (t) − C (−t) (j) γk(q) 0 1−t2 jk jk (A .12) ξk (q)= R 1 (k ≥ N +1,k 6=j) 2 √ dt C (t)+C (−t) 0 1−t2 jk jk 48 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

(j) (j) (j) where Cjk(t)=Cjk t, q, ξ (q) and µk (q)=τk(q)+ξk (q). Let’s first outline how we obtain a solution of (A.12) by using a fixed point argument: 2 1 Let U be a neighborhood of q0 in L0(S ; C) with the properties of Theorem A.3. Let N ≥ 1 and chose arbitrarily q ∈ U and j ≥ 1. (We will choose N later sufficiently large, but N will be independent of q and j.) Let N\{j} =

A∪B where A≡ANj := {1 ≤ k ≤ N; k 6= j} and B≡BNj := (N\{j})\A. 2 Elements η =(ηk)k6=j in ` (N\{j}; C) are decomposed η =(ηA,ηB) with 2 2 ηA =(ηk)k∈A ∈ ` (A; C) and ηB =(ηk)k∈B ∈ ` (B; C). Given q ∈ U, denote by CA and CB the Hilbert cubes (with induced strong topology) CA ≡ C := {η ; |η |≤γ(q)+ 1K for k ∈A}and C ≡C := {η ; |η |≤ ANj A k k 0 k B BNj B k 1|γ(q)|+ min K, 1 |γ (q)| for k ∈B}. (Cf. (A.3) for the constant K.) 2 k 2 k (j) 2 Introduce the map T = Tq : CA ×CB →` (B;C) where the components of T (ηA,ηB) are given by the righthand side of (A.12)   R1 √tdt C (t) − C (−t) γ (q) 0 2 jk jk (A .13) T (η ,η )= k R 1−t  A B 1 dt 2 √ Cjk(t)+Cjk(−t) 0 1−t2 k∈B with Cjk(t)=Cjk(t, q, ηA,ηB). Using the same arguments as in Section 2 one shows that there exist a (suﬃciently small) neighborhood U of q0 in 2 1 L0(S ; C) and N ≥ 1 suﬃciently large (independent of q, j ≥ 1) so that T (ηA,ηB) ∈CB (cf. Proposition A.6 below). Moreover, for any ηA ∈CA, T(ηA,·):CB →CB is a contraction (cf. Proposition A.7 below). Therefore there exists a unique element in CB, denoted by κ(ηA), satisfying (A .14) κ(ηA)=T ηA,κ(ηA) .

In the remainder of this section we prove the statements claimed above.

Proposition A.6. There exist a (suﬃciently small) neighborhood U of 2 1 q0 in L0(S ; C) and N ≥ 1 so that for q in U and j ≥ 1 the map T = (j) 2 Tq : CA ×CB →` (B;C) satisﬁes the following estimate for k ∈Band any (ηA,ηB)∈CA ×CB |γ (q)|2 1 (A .15) |T (η ,η ) |≤C k ≤ |γ (q)|+ min K, |γ (q)| A B k k 2 k k where C>0is independent of q ∈ U, j ≥ 1 and k ∈B.

Proof. Recall that T (ηA,ηB) is given by   R1 √tdt C (t) − C (−t) γ (q) 0 2 jk jk T (η ,η )= k R 1−t  . A B 1 dt 2 √ Cjk(t)+Cjk(−t) 0 1−t2 k∈B ON THE KORTEWEG-DE VRIES EQUATION 49

As in Section 2, we write Cjk(t) ≡ Cjk(t, q, ηA,ηB) as a product of the form s s ∆0(0) ∆j(0) Ajk(t) Bjk(t) (A .16) Cjk(t)= . ∆0(t) ∆j(t) Ajk(0) Bjk(0) By the same proof as for Lemma 2.3-Lemma 2.5 one shows that there 2 1 exists a neighborhood U of q0 in L0(S ; C) and C>0 independent of j, k and q ∈ U so that for q ∈ U, k 6= j

A (t) X |γ (q)| X |γ (q)| sup jk − 1 ≤ C|γ (q)| k + C|γ (q)| α A (0) k α4 k α2 |t|≤1 jk k k |α−k|≥ 2 |α−k|≤ 2 |γ (q)| ≤ C k , k  

B (t) C|γ (q)| X 1 sup jk − 1 ≤ k  |γ |2 +  B (0) k3 α k3 |t|≤1 jk k |α−k|≤ 2 |γ (q)| ≤ C k k3 and

1/2 ∆0(0) |γk(q)| sup − 1 ≤ C 2 |t|≤1 ∆0(t) k

1/2 ∆j(0) |γk(q)| sup − 1 ≤ C . |t|<1 ∆j(t) k We proceed as in the proof of Theorem 2.7 and write ! 1/2 1/2 1/2 ∆j(0) ∆j(0) ∆0(0) Cjk(t)= + − 1 ∆j(t) ∆j(t) ∆0(t) ∆ (0) 1/2 ∆ (0) 1/2 B (t) + j 0 jk − 1 ∆j(t) ∆0(t) Bjk(0) ∆ (0) 1/2 ∆ (0) 1/2 B (t) A (t) + j 0 jk jk − 1 . ∆j(t) ∆0(t) Bjk(0) Ajk(0)

Then for |t|≤1, (ηA,ηB)∈CA ×CB

1/2 ∆j(0) |γk(q)| ≤ 1+C , ∆j(t) k ! 1/2 1/2 ∆j(0) ∆0(0) |γk(q)| |γk(q)| − 1 ≤ 1+C C 2 ∆j(t) ∆0(t) k k 50 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

1/2 1/2 ∆j(0) ∆0(0) Bjk(t) − 1 ∆j(t) ∆0(t) Bjk(0) |γ (q)| |γ (q)|2 |γ (q)| ≤ 1+C k 1+C k C k k k2 k3 and

1/2 1/2 ∆j(0) ∆0(0) Bjk(t) Ajk(t) − 1 ∆j(t) ∆0(t) Bjk(0) Ajk(0) |γ (q)| |γ (q)| |γ (q)| |γ (q)| ≤ 1+C k 1+C k 1+C k C k . k k2 k3 k We conclude, similar as in the proof of Theorem 2.7 (and using that

|ηk|≤|γk(q)|for k ∈B), that for q ∈ U, j ≥ 1, (ηA,ηB)∈CA ×CB |γ (q)|2 |T(j)(η ,η ) |≤C k . q A B k k P 2 2 Notice that |γk(q)| ≤ α≥1 |γα(q)| is bounded on U. Therefore there exists N ≥ 1 independent of q ∈ U, j ≥ 1 such that for k ≥ N,q ∈ U C|γ (q)| |γ (q)|2 |γ (q)| k ≤ 1; C k ≤ k + K. k k 2

(j) Notice that it follows from Proposition A.6 that the range of Tq : CA × 2 CB → ` (B; C) is contained in CB.

Proposition A.7. There exist a (suﬃciently small) neighborhood U of q0 2 1 (j) in L0(S ; C) and N ≥ 1 so that for q ∈ U and j ≥ 1, T = Tq : CA ×CB →CB satisﬁes, for (ηA,ηB)∈CA ×CB 1 kT(η0 ,η0 )−T(η ,η )k2 ≤ kη0 −η k2 +kη0 −η k2 . A B A B `2(B;C) 4 A A B B

In particular, T is continuous and, for any ηA ∈CA ﬁxed, the map T (ηA, ·): CB →CB is a contraction.

Ajk(t) Proof. Notice that in the decomposition (A.16)ofCjk(t) only the term Q Ajk(0) ηα depends on η =(ηA,ηB). Recall that Ajk(t)= 1+ . Thus, α∈N\{j,k} τα−λ(t) 0 0 with Djk(t) ≡ Djk(t, q, ηA,ηB,ηA,ηB), A (t, q, η0 ,η0 ) A (t, q, η ,η ) D (t):= jk A B − jk A B jk 0 0 Ajk(0,q,ηA,ηB) Ajk(0,q,ηA,ηB) ON THE KORTEWEG-DE VRIES EQUATION 51

  η0 X∞  Y 1+ α Y 1+ ηα  =  τα−λ(t) τα−λ(t)   η0 η  1+ α 1+ α β=1 α<β τα−τk α>β τα−τk β6=k,j α6=k,j α6=k,j   η0 1+ β 1+ ηβ · τβ−λ(t) − τβ−λ(t). η0 ηβ β 1+ 1+ τβ−τk τβ−τk Using the estimates in the proof of Proposition A.6 one shows that there 0 0 exists C>1 independent of q ∈ U, j ≥ 1 and (ηA,ηB),(ηA,ηB) ∈CA×CB such that X |η − η0 | |D (t)|≤C β β . jk |β2 − k2| β6=k,j This leads to the estimate (for some C>0 independent of q, j, η, η0)

0 0 2 kT(ηA,ηB)−T(ηA,ηB)k`2(B;C)

2 X γ (q) 2 X |η − η0 | X γ (q) 2 1 ≤C k β β ≤ C k kη − η0k2. 2 |β2 − k2| 2 k2 k∈B β6=k,j k∈B

By choosing N ≥ 1 suﬃciently large, but independent of j, q it follows that for any q in U

X γ (q) 2 1 1 C k ≤ . 2 k2 4 k∈B

Corollary A.8. There exist a (suﬃciently small) neighborhood U of q0 and (j) N ≥ 1 so that for q ∈ U and for all j ≥ 1, T = Tq : CA ×CB →CB has for (j) any ηA ∈CA a unique ﬁxed point κ(ηA)=κ (ηA,q)∈CB, T ηA,κ(ηA) =κ(ηA).

0 Moreover, for any ηA,ηA in CA, 1 kκ(η0 ) − κ(η )k2 ≤ kη0 − η k2 . A A `2(B;C) 3 A A `2(A,C)

In particular, the map κ : CA →CB is continuous.

Proof. Fix an arbitrary element ηA ∈CA. Then, by Proposition A.7, for any 0 ηB,ηB ∈CB

0 1 0 kT(η ,η )−T(η ,η )k 2 ≤ kη −η k 2 . A B A B ` (B;C) 4 B B ` (B;C) 52 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

Therefore T (η , ·) is a contraction map on C and there exists a unique ﬁxed A B point κ(ηA) ∈CB, i.e., κ(ηA)=T ηA,κB(ηA) . 0 For ηA ,ηA ∈CA one obtains, again by Proposition A.7, 0 2 0 0 2 kκ(ηA) − κ(ηA)k = kT ηA,κ(ηA) −T(ηA,κ(ηA))k 1 ≤ kη0 − η k2 + kκ(η0 ) − κ(η )k2 . 4 A A A A As a consequence, as claimed, 1 kκ(η0 ) − κ(η )k2 ≤ kη0 − η k2. A A 3 A A

Proof of Theorem 2.1. To prove Theorem 2.1 we need one more auxilary result. To formulate it, introduce for a real valued potential q ∈ L2(S1) circles Γ˜ (q )inCof radius 0 0 k 0 1 γ (q )+ 12K(q ) and center τ (q ) where K(q ):= 1min λ (q ) − 2 k 0 k 0 k 0 0 5 n≥0 2n+1 0 ˜ ˜ λ2n(q0) . Denote by Dk(q0) the closed disk with boundary Γk(q0).

2 1 Lemma A.9. Let q0 ∈ L0(S ; C) be real valued. Then there exists N(q0) ≥ 2 1 1 so that for any N ≥ N(q0) there exists a neighborhood U of q0 in L0(S ; C) with the following properties: For any q ∈ U, |λ (q) − λ (q )|≤ 1K(q) j j 0 k 0 (1 ≤ k ≤ N,j =2k, 2k − 1), and, given arbitrary complex numbers µ (1 ≤ k k ≤ N) inside Γ˜ (q ), the Dirac measures (−1)k2π2k2δ together k 0 µk 1≤k≤N with nA (q) form a Riesz basis of I∗ . n n≥N+1 3/2 Proof. The proof relies on the following three observations: ˜ (1) For any sequence (µk)k≥1 with µk in Dk(q0), the Dirac measures (−1)k2π2k2δ form a Riesz basis of I∗ . To see this we ﬁrst observe that µk 3/2 there exists N0 ≥ 1 such that X k 2 2 k 2 2 2 1 k(−1) 2π k δ − (−1) 2π k δ 2 2 k < µk π k I∗ 3/2 4 n≥N0+1 for any choice of µ in D˜ (q ). k k 0 k 2 2 Recall (cf. Remark after Theorem A.3) that (−1) 2π k δ 2 2 is an π k k≥1 orthonormal basis of I∗ .Thus (−1)k2π2k2δµ spans a subspace 3/2 k k≥N +1 of I∗ of codimension N . By Bari’s theorem, adding0 precisely N Dirac 3/2 0 0 k 2 2 measures (−1) 2π k δµk to this set will give rise to a Riesz basis as 1≤k≤N0 soon as (−1)k2π2k2δµ is ω-linearly independent. To prove the linear k k≥1 ON THE KORTEWEG-DE VRIES EQUATION 53

Q independence, introduce ψ (λ):= 1 µk−λ. These functions are ele- j j2 k6=j k2π2 ments of I and satisfy δ (ψ )=0fork6=jand δ (ψ ) 6= 0. This proves 3/2 µk j µj j that (−1)k2π2k2δµ is ω-linearly independent. k k≥1 (2) Introduce for any sequence µ =(µ ) with µ in D˜ (q ) the map T : k k≥ 1 k k 0 µ ∗ ∗ k 2 2 k 2 2 I → I given by T (−1) 2π k δ 2 2 =(−1) 2π k δ , corresponding 3/2 3/2 µ π k µk k 2 2 to the change of basis from the orthonormal basis (−1) 2π k δ 2 2 to π k k≥1 the Riesz basis (cf. (1)) (−1)k2π2k2δ . Then T is a bounded invertible µk k≥1 µ −1 operator, kTµkL(I∗ ) < ∞ and kTµ kL(I∗ ) < ∞. One veriﬁes that Tµ 3/2 Q3/2 depends continuously on µ =(µ ) ∈ D˜ (q )⊆`2(N;C). Due to the Q k k≥1 k≥1 k 0 ˜ 2 fact that the Hilbert cube k≥1 Dk(q0) is compact in ` (N; C) we conclude 2 −1 2 that there exists C(q0) > 0 such that kTµkL(I∗ ), kTµ kL(I∗ ) ≤ C(q0) for all 3/2 3/2 Q ˜ µ ∈ k≥1 Dk(q0).

(3) By standard arguments one shows that there exists N(q0) ≥ 1 such that (with C(q0) > 0 given as in (2)) X 1 1 (A .17) k(−1)k2π2k2δ − kA (q )k2 ≤ . τk(q0) k 0 I∗ 3/2 4 C(q0) k≥N(q0)+1

We now combine (1), (2) and (3) to prove the Lemma A.9: 2 1 For N ≥ N (q0 ) given, choose a neighborhood U of q0 in L0(S ; C) so that for q in U (cf. Corollary A.2) X 2 1 1 (A .18) knAn(q) − nAn(q0)kI∗ ≤ 3/2 4 C(q ) n≥1 0 and, in addition, for 1 ≤ k ≤ N and j ∈{2k, 2k − 1}, |λ (q) − λ (q )|≤ 1K j j 0 k as well as |λ0(q) − λ0(q0)|≤K. Combining these estimates with (A.18) and (A.17) one concludes that, for q ∈ U, X 1 1 knA (q) − (−1)n2n2π2δ k2 ≤ . n τn(q0) I∗ 3/2 2 C(q ) n≥N+1 0

In view of observation (2) and Bari’s theorem (cf. [GK, p. 310]) one sees that (δ ) 0 nA (q) is a Riesz basis for any q ∈ U and µk 1≤k≤N() n n≥N()+1 ˜ arbitrary µk ∈ Dk(q0), 1 ≤ k ≤ N().

2 1 Proof of Theorem 2.1. For a given real valued potential q0 ∈ L0(S ; C) let 2 1 K1 := K(q0) and choose a neighborhood U1 of q0 in L0(S : C) so that the statements of Theorem A.3 hold. Next choose N ≥ 1 and a neighborhood 2 1 U2 ⊂ U1 of q0 in L0(S ; C) so that the statements of Propositions A.6, A.7, 54 D. BATTIG,¨ T. KAPPELER AND B. MITYAGIN

Corollary A.8 and Lemma A.9 are valid. For this N, let K := 1 K(q ) and 2 N 0 2 1 choose a neighborhood U3 ⊂ U2 of q0 in L0(S ; C) so that the statements of Theorem A.5 hold. In particular, for any q in U3 and j ≥ 1, the function 1 (j) 1j(λ, q) has precisely one zero µ (q) inside the circle Γk(K2) for k in ANj := j k {1 ≤ k ≤ N,k 6= j}. Let U := µ(j)(q) (q ∈ U ,j ≥ 1) and denote ANj k 3 k∈ANj by η ≡ η the unique ﬁxed point κ(η ), provided by Corollary A.8. B BNj ANj Deﬁne the entire function ψj in I3/2 given by

1 Y µ(j)(q)−λ Y (η ) −λ ψ (λ, q):= k B k . j j2 k2π2 k2π2 k∈ANj k∈BNj By construction, 1 δ (j) ψ (·,q) =0=δ (j) 1(·,q) (k ∈A ) µ (q) j µ (q) j Nj k k j and 1 A (q) ψ (·,q) =0=A (q) 1(·,q) (k ∈B ). k j k j j Nj For 1 ≤ j ≤ N it follows from Theorem A.5 that δ 1 1 (·,q) 6=0as τj (q0) j j 11(·,q) has no zeroes inside Γ . By construction δ (ψ ) 6= 0. Therefore, j j j τj (q0) j there exists Cj ≡ Cj(q) 6= 0 such that 1 δ (C ψ )=δ 1(·,q) . τj (q0) j j τj(q0) j j k 2 2 j 2 2 By Lemma A.9, (−1) 2π k δ (j) ,(−1) 2π j δ , kA (q) is µ (q) τj (q0) k k k∈ANj k≥N+1 a Riesz basis of I∗ . This implies that C ψ (λ, q)= 11(λ, q) for 1 ≤ j ≤ N 3/2 j j j j and q ∈ U .Forj≥N+ 1, one shows, using standard asymptotic results, 3 that A (q) ψ (·,q) 6= 0. Recall that jA (q) 11 (·,q) = 1 and thus there j j j j j exists C ≡ C (q) 6= 0 with jA (q) C ψ (·,q) =1. j j j j j k 2 2 1 By Lemma A.9, (−1) 2π k δ (j) , A (q) is a Riesz basis µ (q) k k k 1≤k≤N k≥N+1 of I∗ . This implies that C ψ (λ, q)= 11(λ, q) for j ≥ N + 1 and q ∈ U . 3/2 j j j j 3 (j) (j) By deﬁnition of the Hilbert cube CB (cf. Section A.3) µk (q)−τk(q) =: ξk (k ∈B) satisfy the estimates stated in Theorem 2.1 and the analyticity of (j) the zeroes µk (q)(k∈B) follows again from Cauchy’s integral formula Z d 1 (λ, q) (j) 1 dλ j µk (q)= λ dλ. 2πi Γk 1j(λ, q) ON THE KORTEWEG-DE VRIES EQUATION 55

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Received February 26, 1996.

Universite´ Paris-Nord, France

Ohio State University Columbus, OH 43210