Invariant Tori for a Two-Dimensional Completely Resonant Beam Equation with a Quintic Nonlinear Term

This paper focuses on a two-dimensional completely resonant beam equation with a quintic nonlinear term. This means studying u tt + Δ 2 u + ε f ð u Þ = 0, x ∈ T 2 , t ∈ ℝ , under periodic boundary conditions, where ε is a small positive parameter and f ð u Þ is a real analytic odd function of the form f ð u Þ = f 5 u 5 + ∑ ̂ i ≥ 3 f 2 ̂ i +1 u 2 ̂ i +1 , f 5 ≠ 0 : It is proved that the equation admits small-amplitude, Whitney smooth, linearly stable quasiperiodic solutions on the phase- ﬂ ow invariant subspace ℤ 2 † = f r = ð r 1 , r 2 Þ , r 1 ∈ 4 ℤ − 1, r 2 ∈ 4 ℤ g . Firstly, the corresponding Hamiltonian system of the equation is transformed into an angle-dependent block-diagonal normal form by using symplectic transformation, which can be achieved by selecting the appropriate tangential position. Finally, the existence of a class of invariant tori is proved, which implies the existence of quasiperiodic solutions for most values of frequency vector by an abstract KAM (Kolmogorov-Arnold-Moser) theorem for in ﬁ nite dimensional Hamiltonian systems.


Introduction
In this paper, a two-dimensional completely resonant beam equation with a quintic nonlinear term under periodic boundary conditions is considered, where ε is a small positive parameter and f ðuÞ is a real analytic odd function of the form It is proved that the equation admits the existence of a class of invariant tori, which implies the existence of quasiperiodic solutions for most values of frequency vector by an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional Hamiltonian systems.
In nature, periodic phenomenon is an ideal state, but in practical problems, such as data observation, extraction, and operation, it always has errors and even some interference. In fact, quasiperiodic functions are always needed to be introduced in a system when there are two disturbance factors with incommensurability of periods; thus, quasiperiodic phenomenon is more common than periodic phenomenon, such as the celestial mechanics, ecology system, and economic volatility in many practical problems which often can be classified as quasiperiodic problem of differential equations. Generally speaking, there is more than one variable that causes the change of a phenomenon, so it is of great practical value to study the quasiperiodic problem of partial differential equations (PDEs), and the quasiperiodic solution problem of nonlinear Hamiltonian system is an important branch of nonlinear scientific research. As a kind of important Hamiltonian system, beam equation has also received corresponding attention.
The classical KAM theory, proposed by Kolmogorov [1,2], Arnold [3], and Moser [4], is a theory about the longterm state of the solution of the integrable Hamiltonian system after it is perturbed, which is a significant progress of Newtonian mechanics in the 20th century and enables people to study the Hamiltonian system in a new way. In the late 1980s, in order to construct quasiperiodic solutions of one-dimensional Hamiltonian PDEs, the classical KAM theory was developed into infinite dimensional space by Wayne [5], Kuksin [6], and Pöschel [7]. Since then, KAM theory of Hamiltonian PDEs with one-dimensional spatial variables has been well developed and produced a lot of results, which we will not repeat here.
When the dimension of the spatial variable exceeds 1, the multiplicity of the normal frequency tends to infinity, which makes the small divisor problem and its measure estimation in KAM iteration more difficult to solve, resulting in fewer corresponding results and larger research space. The conclusion of the existence of quasiperiodic solutions of highdimensional Hamiltonian PDEs comes from Bourgain [8], but instead of using KAM theory, it uses multiscale analysis, so as to avoid a lot of tedious second Melnikov conditions. Since then, according to this idea, many important results have been obtained on high-dimensional Hamiltonian PDEs (refer to [9][10][11][12][13]). However, this method also has some disadvantages, such as it cannot give the normal form of the system, and thus, the linear stability and other related dynamic properties of small amplitude quasiperiodic solutions cannot be given. For these reasons, researchers have been trying to apply KAM theory to high-dimensional Hamiltonian PDEs. Yuan [14] and Geng and You [15,16] first applied KAM theory to the existence of quasiperiodic solutions of high-dimensional Hamiltonian PDEs. In [17], Eliasson and Kuksin studied high-dimensional Schrödinger equations with convolutionaltype potential and made a breakthrough in properly classifying normal frequencies by introducing the Toplitz-Lipschitz property, which perfectly solved the measure estimation problem brought by eigenvalue multiplicity. Eliasson et al. [18] considered a d-dimensional cubic beam equation that does not satisfy momentum conservation whereGðx, uÞ = u 4 + Oðu 5 Þ and d ≥ 2. The existence of quasiperiodic solutions of (1) was proved by the KAM theory. However, the above conclusions are dependent on external parameters and therefore cannot be applied to classical equations of complete resonance with physical background. Geng et al. [19] researched the KAM theory of twodimensional completely resonant Schrodinger equation under periodic boundary conditions. This equation has no external parameters and can only be supplied by amplitude. They proved the existence of quasiperiodic solution with special tangential frequencies of the equation by combining the Toplitz-Lipschitz idea in [17] and the idea of solving the homology equation of dependent angular variables proposed by Xu and You in [20]. By using the similar idea in [19], Geng and Zhou [21] researched the existence of quasiperiodic solutions of the two-dimensional completely resonant cubic beam equation on the phase flow invariant subspace ℤ 2 odd = fr = ðr 1 , r 2 Þ, r 1 ∈ 2ℤ − 1, r 2 ∈ 2ℤg of ℤ 2 . The reason why the existence of solution is only discussed in the invariant subspace of phase flow is that the nonlinear term of the Hamiltonian system corresponding to beam equation is relatively complex, and this idea was first proposed by [22].
The KAM theory is the compound of Newton iterative method and Birkhoff normal type. Through the normal type, parameters are introduced to adjust the frequency; that is, the spoke frequency modulation is realized through the parameters, so as to overcome the problem of small divisor related to homology equation in KAM iterative, which is an important link of the KAM theory. The nonlinear term of the Hamiltonian system directly affects its normal form. Therefore, once the nonlinearity changes, the corresponding normal form should be adjusted accordingly, so the KAM theory needs to be reconstructed. In 2021, the authors of the present paper Zhang and Si [23] applied the idea in [19] to the existence of quasiperiodic solutions of the two-dimensional completely resonant quintic Schrödinger equation Although only the nonlinear term has changed from juj 2 u to juj 4 u, its normal form is completely different, so its corresponding KAM theory has also undergone essential changes. In recent years, more attention has been paid to the existence of quasiperiodic solutions of quintic Hamiltonian PDEs in high-dimensional space. Relevant results can be referred to references [24][25][26]. However, using the KAM theory to prove the existence of quasiperiodic solutions for two-dimensional completed resonant beam equations with higher order nonlinear terms remains to be solved. This paper is focused on the study of ð1Þ + ð2Þ. The nonlinear term of the Hamiltonian system corresponding to (6) is p and that of (1) is p = 0, τ 1 = ±, τ 2 = ±, τ 3 = ±, τ 4 = ±,τ 5 = ±,τ 6 = ±, and p − = p. This difference leads to the essential difference of the normal form between the two, which leads to the fact that the KAM theory for (6) is not suitable for (1), and the phase flow invariant subspace where the quasiperiodic solution is located will also change. In this paper, we only discuss the existence of quasiperiodic solutions of (1) in the phase flow invariant subspace ℤ 2 † = fr = ðr 1 , r 2 Þ, r 1 ∈ 4ℤ − 1, r 2 ∈ 4ℤg. By selecting phase flow invariant subspace ℤ 2 † , it can be ensured that the eigenvalue λ r ≠ 0, and the normal form is simple and beautiful enough. However, due to the change of the phase flow invariant subspace where the quasiperiodic solution is located, we will have to reselect the tangential sites, that is, reconstruct the admissible set. Although this process does not require advanced mathematical knowledge and only involves elementary operations, it is cumbersome enough and requires strong skills. For the selected 2 Journal of Function Spaces admissible set, (1) is turned into normal form by symplectic coordinate transformation, whose integrable terms only come from p i p j p n p m p r p s / ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ i λ j λ n λ m λ r λ s q where i − j + n − m + r − s = 0; that is, the normal form of (1) is formally consistent with the normal form of (7). Of course, the two normal forms are only in the same form, and the coefficients of (1) are much more complex than (7). Therefore, we spend a lot of energy to prove that (1) meets the requirements of the KAM theory in [23] and then prove the existence of its quasiperiodic solution.
The following are the conditions that the tangential sites need to meet, which are proposed by Zhang and Si [23]. 5 1 , and the second one is defined as b 2 = 4c 5 1 − 1, c 2 = 4b 5 2 ; the others are defined inductively by The set K = fi * 1 = ðb 1 , c 1 Þ,⋯,i * d = ðb d , c d Þg ⊂ ℤ 2 † given above is admissible, and the proof is shown in the appendix.
The main result of this paper is as follows.
There exist a Cantor set Ξ * with positive measure, such that for arbitrary ðζ 1 ,⋯,ζ d Þ ∈ Ξ * , the beam equation ð1Þ + ð2Þ has a solution

The Hamiltonian Setting and Birkhoff Normal Form
Before turning equation (1) into a Hamiltonian system, we first introduce the following notations which will appear later.

Lemma 4.
For a given a ≥ 0 and ϖ > 1/2, the gradients Q 6 p , Q 6 p are real analytic as maps from some neighborhood of origin in l a,ϖ × l a,ϖ into l a,ϖ and kQ 6 p k a,ϖ = Oðkpk 5 a,ϖ Þ, kQ 6 p k a,ϖ = O ðkpk 5 a,ϖ Þ.

Partial Birkhoff Normal Form.
K is an admissible set with d points. Set ℤ 2,⋄ = ℤ 2 † \ K, and define four sets as follows: Journal of Function Spaces Obviously, the set is an empty set. It is proved by the reduction to absurdity that any six points The first polynomial is divisible by 8, but the second polynomial is not, which is a contradiction. Therefore, the set is an empty set. Similarly, any six points i, j, n, m, r, s on ℤ 2 † satisfy jij 2 + jjj 2 + jnj 2 + jmj 2 − jrj 2 − jsj 2 ≠ 0. Therefore, the set is an empty set. Let us introduce some partial Birkhoff form of order six.

Proposition 5.
K is an admissible set with d points; there exists a symplectic transformation X 1 R that converts the Hamiltonian (16) into with i · p i p j p n p m p r p s − p i p j p n p m p r p s and X 1 R be the time − 1 mapping of the Hamiltonian vector field of R. Set Then, where ∑ l∈K idp l ∧d p l + ∑ r∈ℤ 2,⋄ idv r ∧d v r is the corresponding symplectic structure of Poisson bracket f·, · g, ðr, sÞ is a resonant pair, and ði, jÞ and ði, j, n, mÞ are uniquely determined by ðr, sÞ. Introduce the action-angle variable Equation (37) converts the HamiltonianH intõ Scaling through time ζ ⟶ ε 3/2 ζ, the scaled Hamiltonian is H = ε −9H ðε 3/2 ζ, ε 5 I, θ, ε 5/2 v, ε 5/2 vÞ: The Hamiltonian H satisfies (23) and (24) where ζ ∈ ½ε 3/2 , 2ε 3/2 d .

An Infinite-Dimensional KAM Theorem for PDEs
We will use the KAM theorem in [23] to prove the main result (Theorem 3). For easy understanding, the KAM theorem in [23] is introduced below. Denote Journal of Function Spaces ðθ, IÞ are d-dimensional angle-action coordinates, ðv, vÞ are infinite-dimensional coordinates, and the corresponding symplectic structure is ∑ i∈K dθ i ∧dI i + i∑ r∈ℤ 2,⋄ dv r ∧d v r . Frequencies η = ðη i Þ i∈K and Ω = ðΩ r Þ r∈ℤ 2,⋄ depend on the parameter ζ ∈ Ξ ⊂ ℝ d , where Ξ is a closed bounded set with positive Lebesgue measure. In order to prove the existence of the invariant torus of small perturbation H = H ⋄ + W ⋄ of H ⋄ , we need the following assumptions.
whereΩ , r s are C 8 W functions of ζ with C 8 W -norm bounded by some small positive constant L.

Proof of the Main Theorem
Let us show that the Hamiltonian (23) satisfies the Assumptions (6)- (11).
Verifying Assumption (6): from (24), forn = 1, ⋯, d, where ζ ∈ Ξ; then, W is the submatrix of matrix f∂ 2 η/∂ ζ 2 g. According to (49) and (50), then Hence, Assumptions (6) is verified. Verifying Assumption (7): take ς = 4; the proof is obvious. Verifying Assumption (8): from (23), G r is represented as follows: Journal of Function Spaces where ði, jÞ and ði, j, n, mÞ are uniquely determined by ðr, sÞ. We only prove (A3) for det ½<k, ηðηÞ > I ± G r ⊗ I ± I ⊗ G r′ which is the most complicated. Let We will prove that jFðζÞj ≥ ðγ ′ /jkj τ Þ, ðk ≠ 0Þ: Let us only prove the case r, r′ ∈ I 4 , and everything else is similar. Let where with with Thus, where The eigenvalues of FðζÞ are Journal of Function Spaces It has been proved that none of the eigenvalues of FðζÞ are zero in [23]. Moreover, when r ∈ I 1 , r ′ ∈ I 1 or r ∈ I 1 , r ′ ∈ I 2 or r ∈ I 1 , r ′ ∈ I 3 and so on, the situations are similar, so omit the proofs. That is, none of the eigenvalues of FðζÞ are zero for k ≠ 0: From Lemma 3.1 in [19], then det ðFðζÞÞ is a polynomial function in the components of ζ with order at most eight and j∂ 8 ζ ðdetðW ðζÞÞÞj ≥ ð1/2Þjkj ≠ 0: By excluding some parameter set with measure Oð Þ, then jdet ðW ðζÞÞj ≥ ðγ ′ /jkj τ Þ, k ≠ 0:ðA3Þ is verified. Verifying Assumption (9): similar to [19], from Lemma 4, it is obvious that ðA4Þ holds, so omit its proof.
Verifying Assumption (11)  The above results are obvious, and we omit their proof.
(I) Let us prove the property (2) by reduction to absurdity. The proof for the property (1) is similar and simpler. Let us say i, j, n, m, r ∈ K satisfies <r − m, The set K given in Remark 2 is admissible.

ðA:15Þ
It is a contradiction.