Quasi-periodic solutions for perturbed generalized nonlinear vibrating string equation with singularities

{\bf Abstract}: The existence of 2-dimensional KAM tori is proved for the perturbed generalized nonlinear vibrating string equation with singularities $u_{tt}=((1-x^2)u_x)_x-mu-u^3$ subject to certain boundary conditions by means of infinite-dimensional KAM theory with the help of partial Birkhoff normal form, the characterization of the singular function space and the estimate of the integrals related to Legendre basis.


CHENGMING CAO AND XIAOPING YUAN
In the above papers, the potentials V are regular. In physics and mechanics the potentials sometimes contain some kind of singularity. As an example, let us consider the Legendre potential, Since lim the endpoints x = ± π 2 are actually singular. It is well-known that the singular differential expressioñ is in limit-circle case and is of deficiency index (2, 2). The expressionÃ is a selfadjoint operator in the domain Introducing the change of variable the operatorÃ with its domain can be written as In convention, we still write z(y) = u(x), y = x. The operator A has pure point spectrum σ(A ) = σ p (A ). And the property To ensure the singular differential operator's strict positive definiteness, we use the notation A = A + m (m > 0).
Let λ 2 j and φ j (j = 1, 2, . . .) be the eigenvalues and eigenfunctions of A,respectively. Here λ j > 0(j = 1, 2, . . .). Write Inserting (7) into the following equation we haveq j + λ 2 j q j + λ j u 3 , φ j = 0. (8) This is a hamiltonian system where the hamiltonian H is Denoting the invariant 2 × 2-dimensional linear space by E: where P 2 = {I ∈ R 2 : I j > 0 for j = 1, 2} is the positive quadrant in R 2 , T (I) = {(u, v) : q 2 j + p 2 j = I j for j = 1, 2}, then our main theorem is as follows. which is a higher order perturbation of the inclusion map Φ 0 : E → P restricted to T [C ], such that the restriction of Φ to each T (I) in the family is an embedding of a rotational invariant 2-torus for the nonlinear hamiltonian differential equation (12).
Here are some remarks. We compare our results with those of Pöschel [13]. By and large, the basic idea is the same in reducing the hamiltonian defined by the partial differential equations to a partial Birkhoff normal form such that the KAM theorem [14] (also see [7]) is applicable. However, there are several main differences because of the singularity of the differential operator A. In Pöschel [13] , the differential operatorÂ = − d 2 dx 2 + m with Dirichlet boundary conditions has eigenvaluesλ 2 j and eigenfunctionφ j : In contrast, the singular differential operator A has, respectively, the eigenvalues and eigenfunctions where P j (x) are Legendre polynomials. On the one hand, under the basis {φ j } the Hamiltonian of u 3 can be written aŝ Sinceφ j is a very simple triangle function 2 π sin x, it is easy to verify that and to fulfill the relationship where δ ij = 1 when i = j and δ ij = 0 when i = j. The relationship (15) leads immediately to that the HamiltonianĜ(q) is the convolution of q and q's, that is, from which the regularity of the vector field XĜ follows. At the same time, since the coefficientsĜ iijj can be explicitly calculated in (14), the resonant conditions in both Birkhoff normal form and the KAM theorem can be directly to verified. However, on the other hand, under the Legendre basis φ j 's, the Hamiltonian of u 3 can be written as Both the equation (14) and the relationship (15) do not hold true any more in this case. Actually, the calculation of the integral 1 −1 φ i φ j φ k φ l dx is not completely solved even in special function theory. Thus the fulfillment of the regularity of the vector filed X G and those resonant conditions in both Birkhoff normal form and the KAM theorem are not easy. Section 2 will be devoted to verify the regularity of X G . And the loss of (14) accounts for why we choose m ∈ (0, 1 4 ) ∪ ( 1 4 , 41 4 ) and consider only 2 dimensional KAM tori.

2.
Legendre polynomials and algebraic property . In the section, let us introduce some properties about Legendre polynomials P n (x) first. By using them, we can derive the estimate of G ijkl in next section.
For fixed n, the Legendre polynomial P n (x) is a n order polynomial. It has an usual expression as well as the Rodrigues's formula At the endpoint x = ±1, it satisfies and it has a uniform upper bound The recursion formula is important A routine computation from (20) gives rise to and (1 − x 2 )P n = n(P n−1 − xP n ).
From the Rodrigues's formula (17), we get A classical formula can express the product of two Legendre polynomials as a sum of such polynomials: where The result can also be expanded in a series using 3j symbol as: where k l m 0 0 0 Thus we could calculate the integral of three Legendre polynomials: We remark that the result we get in this paper is an extension of the research in special function and refer to [6] for details.
Next, let us verify the algebraic property of the function space given below. Employing the result, we can get the regularity of vectorfield in next section. Let where , and ·, · denotes the usual scalar product in L 2 [−1, 1]. The property is also necessary. If the norm of 2 s is defined by u s = ( j≥1 j 2s |u j | 2 ) 1 2 , then the following norms are equivalent A Our theorem is as follows Proof. First, we claim that Let f (x) = j(1 − x 2 ) − P 2j−2 (x) − xP 2j−1 (x) , according to (21), we have At the critical points x 0 such that P 2j−1 (x 0 ) = x 0 , and the endpoint x = ±1, we find Here we use the property (18) and (19). This derives the relationship Using (18) and (21) again, we obtain The same method gives Combining (33),(34),(35) and the property (22), we see that This leads to (32) because of φ j = 2j − 1 2 P 2j−1 . Due to the fact that u = j≥1 Then the sum j≥1 1]. It follows the estimate In view of (37) , (38),(39) and Poincaré inequality, we get 1874 CHENGMING CAO AND XIAOPING YUAN Using Theorem 7.2 (the Gagliardo-Nirenberg Inequality (130)) in Appendix A, for f (x) = u(x), we obtain Then we can conclude Noting (38) (39) (40) and using Poincaré inequality, we have the following inequality, Here In 1959, Nirenberg [12] observed a connection between L p -norms and the Hölder seminorms [·] α . Define By Theorem 7.1 (the General Nirenberg Inequality (128) ) in Appendix A, for f (x) = φ j (x), It follows that With the help of Theorem 7.1 (the General Nirenberg Inequality (128) Then, using (40) (41) and we can deduce that
3.1. the regularity of vectorfield. From introduction, we have already obtain the hamiltonian (10) with equations of motions(9) in some neighbourhood of the origin in the Hilbert space 2 s × 2 s with standard symplectic structure j dq j ∧ dp j . Then we have the following lemma , the gradient G q = ( ∂G ∂qj ) j≥1 is real analytic as a map from some neighbourhood of the origin in , with ).
Proof. From the notation we have Since G is independent of p, the associated hamiltonian vectorfield, is smoothing of order 1. By contrast, X Λ is unbounded of order 1.
3.2. The Legendre sequences. It is necessary to make clear the coefficient G ijkl (11) in hamiltonian H. In particular G iijj . Then we acquire the property of Legendre sequences denoted by P(m, n) = 1 −1 P m P m P n P n dx, m, n ∈ N below. Theorem 3.2. (Legendre sequences) The Legendre sequences P(m, n) satisfy the following recursion formula where with . (68) then we obtain the estimate of the following integral, In particular, there exist an absolute constant C > 0 such that The proof is left in section 6.
Using the property of the Legendre polynomials (23), we can obtain the property about G ijkl which we need in the next section.
Proof. From the definition of G ijkl (11) and φ i (13)(16), we know the product of 4. Partial Birkhoff normal form. Next we introduce complex coordinates Then we obtain a real analytic hamiltonian H = j λ j |z j | 2 + . . . on the complex Hilbert space 2 s with symplectic structure i j dz j ∧ dz j . In the following, A( 2 s , 2 s+1 ) denotes the class of all real analytic maps from some neighbourhood of the origin in 2 s into 2 s+1 . Thus we can also obtain the main proposition like Pöschel [13] but the handling of the small denominator is more complex.
with uniquely determined coefficient, and 3. Thus, the hamiltonian Λ + G is integrable with integrals |z j | 2 , j = 1, 2, while the not-normalized fourth order termĜ is not integrable, but independent of the first 2 modes.
Proof of property. Let us introduce another set of coordinates (. . . , w −2 , w −1 , w 1 , w 2 , . . .) in 2 s by setting z j = w j ,z j = w −j . The hamiltonian under consideration then reads The prime indicates that the subscripted indices run through all nonzero integers. The coefficients are defined for arbitrary integers by setting G ijkl = G |i||j||k||l| . Formally, the transformation Γ is obtained as the time-1-map of the flow of a hamiltonian vectorfield X F given by a hamiltonian Here, λ j =sgnj · λ |j| , L = (i, j, k, l) ∈ Z 4 : 0 = min(|i|, . . . , |l|) ≤ 2 , and N ⊂ L is the subset of all (i, j, k, l) ≡ (p, −p, q, −q). That is, they are of the form (p, −p, q, −q) or some permutation of it.
Next, we will estimate the denominator λ i + λ j + λ k + λ l to ensure the correction of the definition of (73), the proof of the lemma is left at the end of this section.
We continue the proof of the property. Expanding at t = 0 and using Taylor's formula we formally obtain where {H, F } denotes the Poisson bracket of H and F . The last line consists of terms of order six or more in w and constitutes the higher order term K. In the second to last line, Re-introducing the notations z j ,z j and counting multiplicities, we obtain that Thus, we have H • Γ = Λ +Ḡ +Ĝ + K as claimed formally.
To prove analyticity and regularity of the preceding transformation, we first show that ).
Assume a "threshold function"F where The natation σ comes from the estimate of denominator |λ i + λ j + λ k + λ l | σ.
It is easy to check that, by (71), the integral of Hence, the second inequality of (77) implies that ∂F ∂w l ∂F ∂w l , which means F w 9 2 F w 9 2 . On the other hand, In the end, we obtain F w 9 2 w 3 7 2 .
The analyticity of F w follows from the analyticity of each component function and its local boundedness.
In a sufficiently small neighbourhood of the origin in 2 7 2 , the time-1-map X t F | t=1 is well defined and gives rise to a real analytic symplectic change of coordinates Γ with the estimates where the operator norm · op r,s is defined by Aw r w s .

CHENGMING CAO AND XIAOPING YUAN
The same holds for the Lie bracket: the boundedness of DX F ).
These two facts show that X K ∈ A( 2 7 2 , 2 9 2 ). The analogue claims for XḠ and XĜ are obvious.
Proof of Lemma 4.2. In fact, we want to prove there exists the lower bound of λ i +λ j +λ k +λ l , it does not matter to use the renumbered notation λ 2 j = j(j +1)+m instead of λ 2 j = 2j(2j − 1) + m. This also makes it easier to use Lemma 4 in Pöschel [13] in our proof.
Proof. It is easy to get j − i ≥ l − k + 2, hence we have It is clear that If j − i = h ≥ 0, then the function with respect to h satisfies the following property, In fact, 1 4 < m < 41 4 comes from the last inequality of (79) if i = 1 .
In order to prove Case 2., we need to divide it into the following 9 subcases: , which convert to Subcase 2.4 below.
Using the idea of Lemma 4 in Pöschel [13] , one can obtain On the other hand, it is easy to obtain α ≤ i+j 2 . If α = i or α = i+j 2 , then it converts to Pöschel's case. So it suffices to consider the case i < α < i+j 2 , which means l − k < j − i. This can be solved by using Lemma 4.3. Subcase 2.3. i + j − k + l = 2α. Using the basic assumption, we can get l − k ≤ α ≤ i + j. The case α = l − k or α = i + j can be solved by using Lemma 4 in Pöschel [13] . If α > j, then i − j − k + l = 2(α − j), which converts to Subcase 2.5 below. So it suffices to consider the case l − k < α ≤ j, which means l − k ≤ j − i. Use Lemma 4 in Pöschel [13] when the equality holds, while use Lemma 4.3 when equality does not hold. Subcase 2.4. i − j + k + l = 2α. Using the basic assumption, we get α ≥ l − j. If α > k, then i − j − k + l = 2(α − k), which converts to Subcase 2.5 below. Otherwise j − i = k + l − 2α ≥ k + l − 2k = l − k, then use the same skill like Subcase 2.3.
It is easy to be solved when we observe that l − k = j − i + 2α by using the idea of Lemma 4 in Pöschel [13] . Subcase 2.6. i − j − k − l = 2β. If β + l ≥ 1, then it converts to Subcase 2.5. Next, using the basic assumption, we get −j − k ≤ β ≤ −l, which means i − j − k − l = 2β ≤ −2l. This concludes that l − k ≤ j − i, then we can use the same skill above.
Hence, we finish the proof of Lemma 4.2.
5. The Cantor Manifold Theorem. In this section, we will state Cantor manifold theorem in Pöschel's article [13] which is proven in [10] using the KAM-theorem for partial differential equations from [14]. The difficulty here is to check the nondegeneracy condition (86) for Cantor manifold theorem. In a neighbourhood of the origin in 2 s , we now consider more generally hamiltonian of the form H = Λ + Q + R, where Λ + Q is integrable and in normal form and R is a perturbation term. Letting z = (z,ẑ) withz = (z 1 , z 2 ) andẑ = (z 3 , z 4 , . . .), as well as we assume that Λ = α, I + β, Z , Q = AI, I + BI, Z , with constant vectors α, β and constant matrices A, B, In the Birkhoff normal form lemma, Λ +Ḡ is of that form. The equations of motion of the hamiltonian Λ + Q arė z = i(α + AI + B T z) jzj ,ż j = i(β + BI) jẑj .
Thus, the complex 2-dimensional manifold E =ẑ = 0 is invariant, and it is completely filled up to the origin by the invariant tori On T (I) the flow is given by the equationṡ and in its normal space bẏ for all (k, l) ∈ Z 2 × Z ∞ with 1 ≤ |l| ≤ 2.
B.Spectral asymptotics. There exists d ≥ 1 and δ < 1 such that where the dots stand for terms of order less than d in j. Note that the normalization of the coefficient of j d can always be achieved by a scaling of time. C.Regularity.
s > s for d = 1.
By the regularity assumption, the coefficients of B = (B ij ) 1≤j≤2<i satisfy the estimate B ij = O(i s−s ) uniformly in 1 ≤ j ≤ 2. Consequently, for d = 1 there exists a positive constant κ such that uniformly for bounded I. For d > 1, we set κ = ∞.
The following theorem is in Pöschel [13].
Then there exists a Cantor manifold E of real analytic, elliptic diophantine n− tori given by a Lipschitz continuous embedding Ψ : T [C ] → E , where C has full density at the origin, and Ψ is close to the inclusion map Ψ 0 : with some σ > 1. Consequently, E is tangent to E at the origin." We now verify the assumptions of the Cantor Manifold Theorem.
Proof. We already known that X Q , X R ∈ A( 2 ). On the other hand, we have So conditions B and C are satisfied with d = 1, δ = −1,s = 9 2 and s = 7 2 .
Here, we denote g j1 and g j2 according to (75), and By the basic computation as well as the definition of B jk , α i , β j and g j1 , g j2 in (80)-(83) and (94) (95). we can obtain the following result.
Then, by substituting (100) and (101) into (97), we obtain In (A 32 ), we have Since we have the following estimate by (88) Then for every fixed m ∈ (0, 1 and both monotonically increase with respect to x. Besides, In the end, by substituting (103) and (108) into (102), we get Thus the main theorem follows.
6. Proof of Theorem 3.2. Before our proof, we need the following four lemmas which will be proved in Appendix B.
The idea is using the recursion formula (20) to obtain lemma 6.2 (with the help of lemma 6.1 also ) and lemma 6.3. Then, we eliminate Q(m, n) and get the recursion (See lemma 6.4.) with respect to P(m, n) by some technical calculation.
The estimate of c(m)(= lim n→∞ nP(m, n)) in lemma 6.5 and the symmetry of P(m, n) give the final result. The idea using another sequence O(1) m+1 similar to c(m) to obtain a rough upper bound of c(m) and using the symmetry property to obtain a precise estimate.
Proposition 2. For every fixed m, P(m, n) is a rational fraction with respect to n, i.e.
An immediate consequence of the proposition is that, for every fixed m, which seems to be correct intuitively but need to be proved strictly. If we fix m, we can denote lim n→∞ nP(m, n) = c(m), and it is obvious that Observing the relationship , by letting n goes to infinity, we get where and α m−1 = 6m 3 − 2m 2 + 1 2(m + 1) 3 , α m = (2m + 1)(6m 3 + 2m 2 − 1) 2(2m − 1)(m + 1) 3 , Then from (111), letting n goes to infinity, we have the following unilateral sequence Lemma 6.5. The sequence c(m) satisfies the following properties: and The proof is given in Appendix B.
By Cauchy Inequality, we have If i ≤ j ≤ k ≤ l, then we already know 1 −1 P i P j P k P l dx = 0 (iff i + j + k + l = 2α and i + j + k ≥ l) and P i P j P k P l is an even function, from (25) we know is an easy deduction.
Then for any f ∈ C ∞ 0 [−1, 1], we have An immediate consequence of this theorem is the Gagliardo-Nirenberg Inequality.