A degenerate KAM theorem for partial differential equations with periodic boundary conditions

In this paper, an infinite dimensional KAM theorem with double normal frequencies is established under qualitative non-degenerate conditions. This is an extension of the degenerate KAM theorem with simple normal frequencies in [ 3 ] by Bambusi, Berti and Magistrelli. As applications, for nonlinear wave equation and nonlinear Schr \begin{document}$ \ddot{\mbox{o}} $\end{document} dinger equation with periodic boundary conditions, quasi-periodic solutions of small amplitude and quasi-periodic solutions around plane wave are obtained respectively.

1. Introduction. In [3], the authors studied infinite dimensional Hamiltonian systems with their frequencies being simple and analytically depending on one parameter. A degenerate KAM theorem for lower dimensional elliptic tori was proved. It was novel that the non-degeneracy conditions are qualitative instead of quantitative. Then the abstract KAM theorem was applied to one dimensional nonlinear wave equations with Dirichlet boundary conditions.
It is a natural question whether if the above KAM theorem could be extended to infinite dimensional Hamiltonian systems with multiple normal frequencies, and applied to one dimensional partial differential equations with periodic boundary conditions or higher dimensional partial differential equations.
Therefore, the KAM theorem in [3] can be proved under the weaker qualitative non-degeneracy condition (i) (ii) . See the proof of Proposition 3 in [3] that, for 0 < |l| ≤ 2, integer combinations of frequencies are sufficient. This is also included in our proof of Proposition 1 in Section 3. For the condition (ii) with Z instead of Z + , generally (3) is still a counterexample. However, 'integer combinations of frequencies' provides a chance to avoid it by using some structure property of the equation. Now consider (1) with its nonlinearity not containing space variable x explicitly and periodic boundary conditions. The momentum is conserved. Passing to Fourier coefficients, the corresponding Hamiltonian consists of monomials q n1 · · · q nrqnr+1 · · ·q nr+s with n 1 + · · · + n r − n r+1 − · · · − n r+s = 0.
After introducing action-angle coordinates for tangential variables (see (53) (54)), the monomials of the Hamiltonian take the form e i(k1x1+···+knxn) y m1 1 · · · y mn n j∈Z\J z αj jzᾱ j j and (4) becomes with l j := α j −ᾱ j . We can prove the conclusion of condition (ii) for k, l additionally satisfying (5). In other words, such integer combinations as (3) are avoided by (5). Consequently, we can establish an infinite dimensional KAM theorem with double normal frequencies under qualitative non-degenerate conditions (i) and (ii) with (5). Historically, KAM theory for partial differential equations was originated by Kuksin [18] and Wayne [29], where one dimensional nonlinear wave and Schrödinger equations with Dirichlet boundary conditions were studied. Then infinite dimensional KAM theory was deeply developed with applications to more partial differential equations, including both simple and multiple normal frequencies. For the case with simple normal frequencies, also see [3,4,17,19,20,27,28,31] for example; for the case with multiple normal frequencies such as one dimensional nonlinear wave and Schrödinger equations with periodic boundary conditions and higher dimensional partial differential equations, see [6,7,8,9,10,11,12,13,15,16,23,26] for example; for partial differential equations with their nonlinearity containing spatial derivatives such as KdV, derivative nonlinear wave and Schrödinger equations, water wave equation, see [1,2,5,21,22,24] for example. (We can not list all papers in this field.) Especially, for the introduction of degenerate KAM theory, see [3] and the references therein.
We now lay out an outline of the present paper: In Subsection 2.1, we give a KAM iterative theorem (Theorem 2.1) and a measure estimate theorem (Theorem 2.2) which are copied from [3] with the following modifications: (1) the frequencies are double, and thus the index set is Z instead of Z + ; (2) the perturbation P is required to satisfy (9); (3) the assumption δ < 1 is added in assumption (A). The proof of Theorem 2.1 is completely parallel to that in [3], and the proof of Theorem 2.2 is given in Section 3.
In Subsection 2.2, as the first application to the nonlinear wave equation with periodic boundary conditions, we state the existence of Cantor families of quasiperiodic solutions with small amplitude, seeing Theorem 2.3.
In Subsection 2.3, as the second application, we consider the nonlinear Schrödinger equation with periodic boundary conditions where ι is an positive integer. For m ∈ Z, are plane wave solutions to equation (6), where ρ > 0 is the amplitude and µ m := m 2 + ρ 2ι is the frequency. Note that the amplitude ρ need not be small. Here, we state the existence of Cantor families of quasi-periodic solutions around these plane wave solutions, seeing Theorem 2.4. Actually in [14] [30], Sobolev stability of plane wave solutions has been proved even for higher dimensional nonlinear Schrödinger equations. We will study the existence of quasi-periodic solutions around plane wave solutions of higher dimensional nonlinear Schrödinger equations in our future work. In Section 3, Theorem 2.2 is proved. The key step is to use the analyticity of the frequencies to transform the qualitative non-degeneracy assumption (ND) into a quantitative non-resonance property, seeing Proposition 1. The authors in [3] discussed four cases, i.e., the Kolmogorov condition for l = 0, the first Melnikov condition for l = e i , the second Melnikov condition for l = e i + e j , and the second Melnikov condition for l = e i − e j . Then they proceeded by contradiction in every case separately. We will also prove Proposition 1 by contradiction. Due to the double multiplicity of normal frequencies, the essential difference compared with [3] is to handle the case l = e j − e −j additionally. Nevertheless, we will not discuss case by case as in [3]. Alternatively, we will give a summarized proof for all (k, l) ∈ X .
In Section 4, we give the proof of Theorem 2.3. We firstly write the equation (16) into the Hamiltonian form. Then we introduce standard angle-action coordinates and verify the conditions of Theorem 2.1 and Theorem 2.2. The verification of the first condition (I) of (ND) is completely the same as Lemma 6 in [3] for nonlinear wave equation with Dirichlet boundary conditions; in order to verify the second condition (II) of (ND), we prove that for every (k, l) satisfying momentum condition (5) and 0 < |l| ≤ 2, k, ω + l, Ω is a non-zero integer linear combination of j 2 + ξ, j ≥ 0, seeing Lemma 4.1.
In Section 5, we give the proof of Theorem 2.4. We firstly write the equation (18) in the Hamiltonian form. Then as in [14] [30], we eliminate the zero mode and normalize the quadratic Hamiltonian. Finally we introduce standard action-angle coordinates and verify the conditions of Theorem 2.1 and Theorem 2.2.
2. Main results. In this section, we give an abstract KAM theorem and its two applications. For convenience, we keep fidelity with the notation and terminology from [3].

2.
1. An abstract KAM theorem. Define the index set J = {j 1 < · · · < j n } ⊂ Z and denote J c = Z \ J. For a ≥ 0, p > 1 2 , define the Hilbert space a,p of all complex sequences z = (z j ) j∈J c such that where j := max{1, |j|} for j ∈ Z. Consider the direct product P a,p := T n × R n × a,p × a,p endowed with weighted norm v = (x, y, z,z) ∈ P a,p , ||v|| r,a,p = |x| + |y| where | · | denotes the sup-norm for finite dimensional vectors. In the whole of this paper the parameter a is fixed. Denote P a,p C as the complexification of P a,p , and define the toroidal domain We consider an infinite dimensional H = N + P , where P is a small perturbation to the parameter dependent normal form defined on the phase space P a,p with the symplectic structure The tangential frequencies ω = (ω 1 , · · · , ω n ) ∈ R n and the normal frequencies Ω = (Ω j ) j∈J c ∈ R J c are real analytic in ξ ∈ I, a compact set in R. The perturbation term P is real analytic in the space coordinates and the parameter.
For the frequencies ω, Ω and the perturbation P , we assume (A) and (R) respectively in the following.
(A) There exist d ≥ 1, 0 < η < 1 fixed, and functions v j : I → R such that where each v j (ξ) extends to an analytic function on the complex neighborhood of I Also the function ω : I → R n has an analytic extension on I η . Moreover there exist (R) The perturbation P is real analytic in ξ ∈ I η . There exist s > 0, r > 0 such that, for each ξ ∈ I, the Hamiltonian vector field X P := (∂ y P, −∂ x P, i∂zP, −i∂ z P ) defines in the neighborhood of T 0 := T n ×{y = 0}×{z = 0}×{z = 0} a real analytic map with p −p ≤ δ and ||X P || r,a,p,D(s,r)×Iη := sup D(s,r)×Iη ||X P || r,a,p < +∞.
Moreover, the perturbation is taken from a special class of real analytic functions, where k, x = n b=1 k b x b and k, α,ᾱ has the following relation for l ∈ Z J c , and denote the set For the normal frequencies Ω(ξ), define the norm |Ω| −δ := sup ξ∈I sup j∈J c | j −δ Ω j | and the C µ -norm, µ ∈ N as The | · | C µ -norm of ω : I → R n is defined analogously.
Then there is * > 0 such that, if the KAM-condition holds, then there exists a smooth family of real analytic torus embedding where I * is the Cantor set such that, for each ξ ∈ I * , the map Φ restricted T n × {ξ} is an embedding of a rotational torus with frequencies ω * (ξ) for the Hamiltonian system H, close to the trivial embedding T n × I → T 0 .
Theorem 2.1 is a modification of KAM theorem in [3]. The proof is completely parallel to that in [3], which follows by Theorem 5.1 and Remark 5.1 in [4]. Therefore, the difference between the single and double eigenvalues rely only in proving the measure estimates. (I) for any vector (c 1 , · · · , c n+1 ) ∈ R n+1 \{0}, the function c 1 ω 1 +· · ·+c n ω n +c n+1 is not identically zero on I, (II) for (k, l) ∈ X with 0 < |l| ≤ 2, the function k 1 ω 1 + · · · + k n ω n + j∈J c l j Ω j is not identically zero on I.
Remark 2. The condition (I) of (ND) is the non-degeneracy condition (i) in Section 1, while the condition (II) of (ND) is the non-degeneracy condition (ii) with (k, l) additionally satisfying momentum condition, i.e., (5).
where the mass ξ is a real parameter on an interval I := [ξ 1 , ξ 2 ], 0 < ξ 1 < ξ 2 , the nonlinearity f (u) is a real analytic function near u = 0 with We choose the index set J := {j 1 < · · · < j n } with such that the tangential frequencies will be different from each other. The linear equation u tt − u xx + ξu = 0 possesses quasi-periodic solutions where B b , θ b ∈ R, and λ j = j 2 + ξ, j ∈ Z denote the eigenvectors of A := √ −∂ xx + ξ with periodic boundary conditions. Theorem 2.3. Under the above assumptions, there exists R * > 0 such that, for any B = (B 1 , · · · , B n ) ∈ R n with |B| =: R ≤ R * , there is a Cantor set I * ⊂ I with asymptotically full measure as R → 0, such that, for any ξ ∈ I * , the nonlinear wave equation (16) has a quasi-periodic solution of the form where o(R) is small in some analytic norm andλ b → λ j b as R → 0.

Application to nonlinear Schrödinger equation.
The previous results also apply to the nonlinear Schrödinger equation with periodic boundary conditions where ι is an positive integer. For m ∈ Z, are plane wave solutions to equation (18), where ρ > 0 is the amplitude and µ m := m 2 + ρ 2ι is the frequency. We prove that (18) admits many quasi-periodic solutions near u m (x, t) corresponding to n-dimensional invariant tori of an associated infinite dimensional dynamical systems.
Theorem 2.4. Under the above assumptions, there exists R * > 0 such that, for any B = (B 1 , · · · , B n ) ∈ R n with |B| =: R ≤ R * , there is a Cantor set I * ⊂ I with asymptotically full measure as R → 0, such that, for all ρ ∈ I * , the nonlinear Schrödinger equation (18) has a quasi-periodic solution of the form where a b , a b are defined in (59), θ b ∈ R, o(R) is small in some analytic norm and 3. Proof of Theorem 2.2. The first step is to use the analyticity of the frequencies to transform the non-degeneracy assumption (ND) into a quantitative non-resonance property.
Note that for different λ in (23), l λ may be in different cases, i.e., l λ = 0, e i , e i + e j , e i − e j .
In view of µ = 0 in (23), we have In view of sup Iη |ω(ξ)| ≤ Γ in assumption (A), we have By (24) (25), we have In view of (8) and sup Iη sup j∈J c | j −δ v j (ξ)| ≤ Γ in assumption (A), we have where L * is a constant bigger than 1. By (26) (27), we obtain which means that, if k λ are bounded, then l λ d are bounded. We will discuss the case d > 1 and the case d = 1 separately in the following. In view of (28), for (k λ , l λ ) with l λ = e j − e −j , we have for some constant C 1 > 0. For (k λ , l λ ) with l λ = e j − e −j , we have 1≤b≤n k λ,b j b + 2j = 0 and thus By (29) (30), for all (k λ , l λ ), we have for some constant C 2 > 0. Note that the exponent min{1, d − 1} > 0, and thus (31) trivially holds true for [l λ ] = 0. By compactness there exist converging subsequences ξ λ h →ξ ∈ I. We will consider the case k λ h bounded and the case k λ h unbounded separately in the following. By analyticity, k , ω(ξ) + l , Ω(ξ) is identically zero on I. Ifl = 0, this contradicts the assumption (I) in (ND); otherwise, this contradicts the assumption (II) in (ND).
Note that for different λ in (32), l λ may be in different cases, i.e., l λ = 0, e i , e i + e j , and [l λ ] is bounded if k λ is bounded. Contrarily, for l λ = e i − e j , [l λ ] may be unbounded although k λ is bounded by (33). By compactness there exist converging subsequences ξ λ h →ξ ∈ I. We will discuss different cases according to the boundedness of k λ h and [l λ h ].
The proof is completely parallel to Subcase 1.1.
In view of (32) and (33), we obtain l λ h = e i λ h − e j λ h with |i λ h | − |j λ h | bounded. Then the quantity |i λ h | − |j λ h | converges, up to subsequence, tom ∈ Z. Since [l λ h ] are unbounded while |i λ h |−|j λ h | are bounded, we get both |i λ h | and |j λ h | are unbounded. In view of δ < 0 here, we obtain and for any µ ≥ 1, By the boundedness of k λ h , we extract constant subsequences k λ h ≡k. Therefore passing to the limit in (23), we get, for any µ ≥ 0, Then by analyticity, the function k , ω(ξ) +m is identically equal to zero on I. This contradicts the assumption (I) in (ND).
The proof is completely parallel to Subcase 1.2.
We now proceed with the proof of Theorem 2.2. By (14) we have Lemma 3.2. There exists a positive constant γ * depending on d, n, µ 0 , Γ, β, η, δ such that Proof. The proof is parallel to that of Lemma 4 and Lemma 5 in [3]. By Lemma 3.1, we only need to count the number of R kl . As usual, for any fixed k, we discuss the number of nonempty R kl with (k, l) ∈ X . For l = 0, e j , e i + e j and e i − e j with i = ± − j, the proof is parallel to that in [3]. The key difference lies in the case l = e j − e −j . In this case, we rewrite R kl as R kj(−j) . Then for every k, j, the measure of R kj(−j) can be estimated as usual. To count the number of R kj(−j) , we use the condition 1≤b≤n k b j b + 2j = 0, which implies that for any fixed k ∈ Z n , the number of R kj(−j) is at most 1. Also see the proof of Lemma 4.6 in [25] for details.
where g is a primitive of f . Letting then the system (36) is equivalent to the lattice Hamiltonian equationṡ where Next we introduce standard angle-action coordinates (x, y, z,z) ∈ T n × R n × a,p × a,p by letting Then the Hamiltonian in (39) is given by with the symplectic structure 1≤b≤n and P is just G expressed in terms of (x, y, z,z). More precisely, by (40), P is of the form k∈Z n ,l∈N n ,α,ᾱ∈N J c P klαᾱ e i k,x y l z αzᾱ i.e., P ∈ A. Now we verify the conditions of Theorem 2.1 and Theorem 2.2. The assumption (A) holds with d = 1, δ = −1 and η = ξ 1 /2. Also the assumption (R) holds with p = p + 1. Fix θ ∈ ( 2 3 , 1), γ := r σ , 0 < σ < 3θ − 2 µ 0 + 1 with r > 0 small enough. Then 9r 2 < γ < 1 and thus 2i+j1+j2=4 sup D(s,r)×Iη which verifies the condition (11). In view of the condition (12) and (15) hold true. It remains to verify the non-degeneracy condition (ND). By the choice of the index set J with (17), we have Then the verification of the first condition (I) of (ND) is completely the same as Lemma 6 in [3] for nonlinear wave equation with Dirichlet boundary conditions. In order to verify the second condition (II) of (ND), we give the following lemma.
Proof. In view of the definition of ω and Ω in (41) and the fact λ j (ξ) ≡ λ −j (ξ), it is equivalent to prove that the following combinations of k, ω + l, Ω do not exist: Calculating directly, the momentum n b=1 k b j b + j∈J c l j j for (42), (43) and (44) equal to ±2j b , ±2(j b + j b ) and ±2(j b − j b ) respectively, which are not zero. This contradicts (k, l) ∈ X .
Finally, the second condition (II) of (ND) follows from Lemma 6 in [3].

5.
Proof of Theorem 2.4. As in [21], the equation (18) can be rewritten in the Hamiltonian form Letting the system (45) is equivalent to the lattice Hamiltonian equationṡ Denote ρ := ||u(0)|| L 2 and assume that the initial datum is concentrated the zero mode. In order to eliminate the zero mode, as in [14] [30], define the symplectic reduction of u 0 : and then, up to a constant, and P is at least three order of v,v with its monomial v k1 · · · v krvl1 · · ·v ls satisfying k 1 + · · · + k r = l 1 + · · · + l s .
Denote J c =Z \ J. Next we introduce standard action-angle coordinates (x, y, z,z) ∈ T n × R n × a,p × a,p by letting Then the Hamiltonian in (49) is given by with the symplectic structure 1≤b≤n More precisely, by (52), P is of the form k∈Z n ,l∈N n ,α,ᾱ∈N J c P klαᾱ e i k,x y l z αzᾱ Now we verify the conditions of Theorem 2.1 and Theorem 2.2 withZ instead of Z, seeing Remark 3 in Section 2. The assumption (A) holds with d = 2, δ = 0 and η = 1 2 . Also the assumption (R) holds withp = p. Fix with r > 0 small enough. Then 9r 2 < γ < 1 and thus 2i+j1+j2=4 sup D(s,r)×Iη which verifies the condition (11). In view of := γ −1 ||X P || r,p,D(s,r)×Iη = O(γ −1 r 3θ−2 ), the condition (12) and (15) hold true. Note that the zero mode here is ruled off, i.e., the index set Z is substituted with Z. By Remark 3, in order to use Theorem 2.1 and Theorem 2.2, it only remains to verify the non-degeneracy condition (ND) withZ instead of Z. Obviously, Lemma 4.1 still holds true withZ instead of Z. Therefore, it is sufficient to to prove that, for any 0 < j 1 < · · · < j L , for any (c 0 , c 1 , · · · , c L ) ∈ R L+1 \ {0}, the function c 0 + c 1 λ j1 + · · · + c L λ j L is not identically zero on I, which is implied by the following lemma.
The last is a Vandermond determinant whose value is given by i<l≤L (χ i − χ l ). (58) Now one has By the conclusions of Theorem 2.1, we get quasi-periodic solutions in D(s, r) with frequencies ω * and Ω * . Returning to the coordinates q,q in (53) and (54), we have q j b = O(r θ ), 1 ≤ b ≤ n, q j = O(r), j ∈ Z * . Back to the coordinates v,v in (51), satisfy a 2 b − (a b ) 2 = 1; for j = ±j b , v j = O(r).
Finally, returning to the coordinates u in (48), we get (21) in Theorem 2.4.