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

Authors: Min Zhang; Zhe Hu; Yonggang Chen
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This paper focuses on a two-dimensional completely resonant beam equation with a quintic nonlinear term. This means studyingutt+Δ2u+εfu=0,x∈T2,t∈ℝ, under periodic boundary conditions, where ε is a small positive parameter and fu is a real analytic odd function of the form fu=f5u5+∑i^≥3f2i^+1u2i∧+1,f5≠0. It is proved that the equation admits small-amplitude, Whitney smooth, linearly stable quasiperiodic solutions on the phase-flow invariant subspace ℤ†2=r=r1,r2,r1∈4ℤ−1,r2∈4ℤ. 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 infinite dimensional Hamiltonian systems.


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## Body

## 1. Introduction

In this paper, a two-dimensional completely resonant beam equation with a quintic nonlinear term(1)utt+Δ2u+εfu=0,x∈T2,t∈ℝ,under periodic boundary conditions
(2)ut,x1,x2=ut,x1+2π,x2=ut,x1,x2+2πis considered, where ε is a small positive parameter and fu is a real analytic odd function of the form
(3)fu=f5u5+∑i^≥3f2i^+1u2i∧+1,f5≠0.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 long-term 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 high-dimensional 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 [9–13]). 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 convolutional-type 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
(4)utt+Δ2u+mu+∂uGx,u=0,x∈Td,t∈ℝ,whereGx,u=u4+Ou5 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 two-dimensional completely resonant Schrodinger equation
(5)iut−Δu+u2u=0,x∈T2,t∈ℝ,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
(6)utt+Δ2u+u3=0,x∈T2,t∈ℝ,on the phase flow invariant subspace ℤodd2=r=r1,r2,r1∈2ℤ−1,r2∈2ℤ of ℤ2. 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
(7)iut−Δu+u4u=0,x∈T2,t∈ℝ.Although only the nonlinear term has changed fromu2u to u4u, 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 [24–26]. 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 of1+2. The nonlinear term of the Hamiltonian system corresponding to (6) is piτ1pjτ2prτ3psτ4/λiλjλrλs and that of (1) is piτ1pjτ2pnτ3pmτ4prτ5psτ6/λiλjλnλmλrλs, where τ1i+τ2j+τ3n+τ4m+τ5r+τ6s=0,τ1=±,τ2=±,τ3=±,τ4=±,τ5=±,τ6=±, 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=r=r1,r2,r1∈4ℤ−1,r2∈4ℤ. By selecting phase flow invariant subspace ℤ†2, it can be ensured that the eigenvalue λr≠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 admissible set, (1) is turned into normal form by symplectic coordinate transformation, whose integrable terms only come from pip¯jpnp¯mprp¯s/λiλjλnλmλrλs 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].Definition 1.
A setK=i1∗=b1,c1,⋯,id∗=bd,cd⊂ℤ†2 is admissible, if
(1)
arbitrary three of them are not vertices of a rectangle(2)
for anyi,j,n,m,r,s⊂ℤ†2, if i−j+n−m+r−s=0 and i2−j2+n2−m2+r2−s2=0, then the intersection of i,j,n,m,r,s and K contains at most four elements, that is, #i,j,n,m,r,s⋂K≤4(3)
for anyr∈ℤ†2\K, there exists at most one triplet i,j,s, where i,j∈K,s∈ℤ†2\K such that i−j+r−s=0 and i2−j2+r2−s2=0. If such triplet exists, let us say that r,s are resonant in the first type and denote all such r by I1(4)
for anyr∈ℤ†2\K, there exists at most one triplet i,j,s, where i,j∈K,s∈ℤ†2\K such that i+j−r−s=0 and i2+j2−r2−s2=0. If such triplet exists, let us say that r,s are resonant in the second type and denote all such r by I2(5)
for anyr∈ℤ†2\K, there exists at most one quintuple i,j,n,m,s, where i,j,n,m∈K,s∈ℤ†2\K such that i−j+n−m+r−s=0 and i2−j2+n2−m2+r2−s2=0. If such quintuple exists, let us say that r,s are resonant in the third type and denote all such r by I3(6)
for anyr∈ℤ†2\K, there exists at most one quintuple i,j,n,m,s, where i,j,n,m∈K,s∈ℤ†2\K such that i−j+n+m−r−s=0 and i2−j2+n2+m2−r2−s2=0. If such quintuple exists, let us say that r,s are resonant in the fourth type and denote all such r by I4(7)
Anyr∈ℤ†2\K is not resonant of any two of the above four classes. Geometrically, any two of the above defined graphs cannot share vertex in ℤ†2\KRemark 2.
There are sets satisfying the above definition. For example, for any integerd≥4, the first point b1,c1∈ℤ†2 is defined as b1>d2,c1=4b15, and the second one is defined as b2=4c15−1,c2=4b25; the others are defined inductively by
(8)bj^+1=4cj^5∏2≤m^≤j^,1≤l^<m^cm∧−cl∧2−1,cj^+1=4bj^+15,2≤j^≤d−1.
The setK=i1∗=b1,c1,⋯,id∗=bd,cd⊂ℤ†2 given above is admissible, and the proof is shown in the appendix.The main result of this paper is as follows.Theorem 3 (main theorem).
LetK=i1∗=b1,c1,⋯,id∗=bd,cd⊂ℤ†2 be an admissible set with d≥4. There exist a Cantor set Ξ∗ with positive measure, such that for arbitrary ζ1,⋯,ζd∈Ξ∗, the beam equation 1+2 has a solution
(9)ut,x=∑l∈K12πζleiηlteil,x+e−iηlte−il,x+Oζ3/2,ηl=l2+Oζ2.


## 2. 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.LetK=i1∗,⋯,id∗⊂ℤ†2, ℤ2,⋄=ℤ†2\K, and la,ϖ be some Hilbert space of sequences v=⋯,vl,⋯l∈ℤ2,⋄, where the norm be va,ϖ=∑l∈ℤ2,⋄vleallϖ,a>0,ϖ>1/2. Set θ=θll∈K,I=Ill∈K,v=vll∈ℤ2,⋄ and ζ=ζll∈K, and introduce the phase space Pa,ϖ=T∧d×ℂd×la,ϖ×la,ϖ∋θ,I,v,v¯, where T∧d is complex neighborhood of Td. Let
(10)Da,ϖr∗,s∗≔θ,I,v,v¯∈Pa,ϖ:Imθ<r∗,I<s∗2,va,ϖ<s∗,v¯a,ϖ<s∗.For anyV∗=θ,I,v,v¯∈Pa,ϖ, define its norm be V∗a,ϖ=θ+1/s∗2I+1/s∗va,ϖ+1/s∗v¯a,ϖ. Let O be the parameter set. Denote α≡⋯,αl,⋯l∈ℤ2,⋄, β≡⋯,βl,⋯l∈ℤ2,⋄, and αl and βl∈ℕ have a finite number of positive integer components. Let Wθ,I,v,v¯=∑α,βWαβθ,Ivαv¯β, where vαv¯β is ∏lvlαlv¯lβl and Wαβ=∑k,lWklαβIlei<k,θ> are CW8 functions of the parameter ζ. Set the norm of W as WDa,ϖr∗,s∗,O≡supva,ϖ<s∗,v¯a,ϖ<s∗∑α,βWαβvαv¯β, where Wαβ≡∑k,lWklαβOs∗2lekr∗ and WklαβO≡supζ∈O∑0≤i^≤8∂ζi^Wklαβ. Denote XW as the Hamiltonian field W corresponding to the symplectic structure dθ∧dI+idv∧dv¯, which is XW=∂IW,−∂θW,i∇v¯W,−i∇vW. Set its norm as
(11)XWDa,ϖr∗,s∗,O≡WIDa,ϖr∗,s∗,O+1s∗2WθDa,ϖr∗,s∗,O+1s∗∑l∈ℤ2,⋄WvlDa,ϖr∗,s∗,Oelalϖ+∑l∈ℤ2,⋄Wv¯lDa,ϖr∗,s∗,Oelalϖ.The vector functionW˘:Da,ϖr∗,s∗×O⟶ℂm˘m˘<∞ is CW8 function of the parameter ζ, and its norm is W˘Da,ϖr∗,s∗,O=∑i^=1m˘W˘i^Da,ϖr∗,s∗,O. The vector function W^:Da,ϖr∗,s∗×O⟶la¯,ϖ is CW8 functions of the parameter ζ, and its norm is W^a¯,Da,ϖr∗,s∗,O=W^i^Da,ϖr∗,s∗,Oi^a¯,ϖ.


### 2.1. The Hamiltonian Setting

Without losing generality, we suppose thatfu=u5. Rewrite the beam equation (1) as
(12)utt+Δ2u+εu5=0,x∈T2,t∈ℝ.Introducing a variableut=v, then (12) can be turned into
(13)ut=v,vt=−Δ2u−εu5.Setp=1/2−Δ1/2u−i−Δ−1/2v; then, (13) is turned into ṗ=i∂H∗/∂p¯ whose corresponding Hamiltonian is
(14)H∗p=∫T2−Δpp¯+ε6−Δ−1/2p+p¯26dx.The eigenvalues and eigenvectors of operator−Δ with periodic boundary conditions are λl=l2 and ϕlx=1/2πei<l,x>, respectively. p has coordinates pll∈ℤ†2∈la,ϖ with respect to the bases ϕll∈ℤ†2. The corresponding symplectic structure is i∑d∈ℤ†2dpl∧dp¯l. In new coordinates, (13) becomes
(15)ṗl=i∂H∗∂p¯l,∀l∈ℤ†2.The corresponding Hamiltonian is(16)H∗=Λ+Q,where
(17)Λ=∑l∈ℤ†2λlpl2,Q=ε768π4∑i+j+n+m+r+s=0i,j,n,m,r,s∈ℤ†21λiλjλnλmλrλspipjpnpmprps+p¯ip¯jp¯np¯mp¯rp¯s+ε128π4∑i+j+n+m+r−s=0i,j,n,m,r,s∈ℤ†21λiλjλnλmλrλspipjpnpmprp¯s+p¯ip¯jp¯np¯mp¯rps+5ε256π4∑i+j+n+m−r−s=0i,j,n,m,r,s∈ℤ†21λiλjλnλmλrλspipjpnpmp¯rp¯s+p¯ip¯jp¯np¯mprps+5ε192π4∑i+j+n−m−r−s=0i,j,n,m,r,s∈ℤ†21λiλjλnλmλrλspipjpnp¯mp¯rp¯s.The regularity of the Hamiltonian system (15) is shown below, and its proof is similar to [21], which is omitted here.Lemma 4.
For a givena≥0 and ϖ>1/2, the gradients Qp6,Qp¯6 are real analytic as maps from some neighborhood of origin in la,ϖ×la,ϖ into la,ϖ and Qp6a,ϖ=Opa,ϖ5, Qp¯6a,ϖ=Opa,ϖ5.


### 2.2. Partial Birkhoff Normal Form

K is an admissible set with d points. Set ℤ2,⋄=ℤ†2\K, and define four sets as follows:
(18)K1=i,j,n,m,r,s∈ℤ†26:i−j+n−m+r−s=0i2−j2+n2−m2+r2−s2≠0#K∩i,j,n,m,r,s≥4,K2=i,j,n,m,r,s∈ℤ†26:i+j+n+m+r+s=0i2+j2+n2+m2+r2+s2≠0#K∩i,j,n,m,r,s≥4,K3=i,j,n,m,r,s∈ℤ†26:i+j+n+m+r−s=0i2+j2+n2+m2+r2−s2≠0#K∩i,j,n,m,r,s≥4,K4=i,j,n,m,r,s∈ℤ†26:i+j+n+m−r−s=0i2+j2+n2+m2−r2−s2≠0#K∩i,j,n,m,r,s≥4.Obviously, the set(19)i,j,n,m,r,s∈ℤ†26:i+j+n+m+r+s=0i2+j2+n2+m2+r2+s2=0is an empty set. It is proved by the reduction to absurdity that any six points i=4i^1−1,4i^2,j=4j^1−1,4j^2,n=4n^1−1,4n^2,m=4m^1−1,4m^2,r=4r^1−1,4r^2,s=4s^1−1,4s^2 on ℤ†2 satisfy i2+j2+n2+m2+r2−s2≠0. Suppose that i2+j2+n2+m2+r2−s2=0, then
(20)4i∧1−12+4j∧1−12+4n∧1−12+4m∧1−12+4r∧1−12−4s∧1−12=−16i∧22+j∧22+n∧22+m∧22+r∧22−s∧22.The first polynomial is divisible by 8, but the second polynomial is not, which is a contradiction. Therefore, the set(21)i,j,n,m,r,s∈ℤ†26:i+j+n+m+r−s=0i2+j2+n2+m2+r2−s2=0is an empty set. Similarly, any six points i,j,n,m,r,s on ℤ†2 satisfy i2+j2+n2+m2−r2−s2≠0. Therefore, the set
(22)i,j,n,m,r,s∈ℤ†26:i+j+n+m−r−s=0i2+j2+n2+m2−r2−s2=0is 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 XR1 that converts the Hamiltonian (16) into
(23)H=Λ+A1+B1+B1¯+A2+B2+B2¯+W,with
(24)Λ=∑i∈KηiζIi+∑r∈ℤ2,⋄Ωrζvr2,(25)ηiζ=ε−4λi+5ζi264π4λi3+∑j∈K,j≠i15ζj264π4λiλj2+15ζiζj32π4λi2λj+∑n∈K,n≠j,n≠i15ζjζn32π4λiλjλn,(26)Ωrζ=ε−4λr+∑i∈K15ζi264π4λi2λr+∑j∈K,j≠i15ζiζj32π4λiλjλr,(27)A1=∑r∈I1∑n∈K1516π4λiλjλn2λrλsζiζjζneiθi−θjvrv¯s,(28)B1=∑r∈I2∑n∈K1516π4λiλjλn2λrλsζiζjζnei−θi−θjvrvs,(29)B1¯=∑r∈I2∑n∈K1516π4λiλjλn2λrλsζiζjζneiθi+θjv¯rv¯s,(30)A2=∑r∈I31516π4λiλjλnλmλrλsζiζjζnζmeiθi−θj+θn−θmvrv¯s,(31)B2=∑r∈I41516π4λiλjλnλmλrλsζiζjζnζmei−θi+θj−θn−θmvrvs,(32)B2¯=∑r∈I41516π4λiλjλnλmλrλsζiζjζnζmeiθi−θj+θn+θmv¯rv¯s,(33)W=Oε2I2+ε2Iva,ϖ2+ε7/4ζ3/2I1/2va,ϖ2+ε7/4ζ3/2va,ϖ3+ε7/2ζva,ϖ4+ε9/4ζ9/2va,ϖ+ε4ζ4va,ϖ2+ε23/4ζ7/2va,ϖ3.Proof.
Let(34)R=5ε192π4∑i,j,n,m,r,s∈K1i·pip¯jpnp¯mprp¯sλiλjλnλmλrλs·λi−λj+λn−λm+λr−λs+ε768π4∑i,j,n,m,r,s∈K2i·pipjpdplprps−p¯ip¯jp¯np¯mp¯rp¯sλiλjλnλmλrλs·λi+λj+λn+λm+λr+λs+ε128π4∑i,j,n,m,r,s∈K3i·pipjpnpmprp¯s−p¯ip¯jp¯np¯mp¯rpsλiλjλnλmλrλs·λi+λj+λn+λm+λr−λs+5ε256π4∑i,j,n,m,r,s∈K4i·pipjpnpmp¯rp¯s−p¯ip¯jp¯dp¯lprpsλiλjλnλmλrλs·λi+λj+λn+λm−λr−λsand XR1 be the time−1 mapping of the Hamiltonian vector field of R. Set
(35)pl=pl,l∈K,vl,l∈ℤ2,⋄.
Then,XR1 converts H∗ into
(36)H~=H∗∘XR1=Λ+Q+Λ,R+Q,R+∫011−tH∗,R,R∘XRtdt=∑i∈Kλipi2+∑r∈ℤ2,⋄λrvr2+∑i∈K5ε192π4λi3pi6+∑i,j∈K,i≠j15ε64π4λi2λjpi4pj2+∑i,j,n∈K,i≠j,i≠n,j≠n15ε16π4λiλjλnpi2pj2pn2+∑i∈K,r∈ℤ2,⋄15ε64π4λi2λrpi4vr2+∑i,j∈K,r∈ℤ2,⋄,i≠j15ε16π4λiλjλrpi2pj2vr2+∑r∈I1∑n∈K15ε16π4λiλjλn2λrλspip¯jpn2vrv¯s+∑r∈I2∑n∈K15ε16π4λiλjλn2λrλsp¯ip¯jpn2vrvs+∑r∈I2∑n∈K15ε16π4λiλjλn2λrλspipjpn2v¯rv¯s+∑r∈I315ε16π4λiλjλnλmλrλspip¯jpnp¯mvrv¯s+∑r∈I415ε16π4λiλjλnλmλrλsp¯ipjp¯np¯mvrvs+∑r∈I415ε16π4λiλjλnλmλrλspip¯jpnpmv¯rv¯s+Oεp3va,ϖ3+εp2va,ϖ4+εpva,ϖ5+εva,ϖ6+ε2p10+ε2p9va,ϖ+ε2p8va,ϖ2+ε2p7va,ϖ3,where ∑l∈Kidpl∧dp¯l+∑r∈ℤ2,⋄idvr∧dv¯r is the corresponding symplectic structure of Poisson bracket ·,·, r,s is a resonant pair, and i,j and i,j,n,m are uniquely determined by r,s.
Introduce the action-angle variable(37)pl=Il+ζleiθl,p¯l=Il+ζle−iθl,l∈K.
Equation (37) converts the Hamiltonian H~ into
(38)H~=∑i∈Kλi+5εζi264π4λi3+∑j∈Kj≠i15εζj264π4λiλj2+15εζiζj32π4λi2λj+∑n∈Kn≠jn≠i15εζjζn32π4λiλjλnIi+∑r∈ℤ2,⋄λr+∑i∈K15ε64π4λi2λrζi2+∑j∈K,j≠i15ε32π4λiλjλrζiζjvr2+∑r∈I1∑n∈K15ε16π4λiλjλn2λrλsζiζjζneiθi−θjvrv¯s+∑r∈I2∑n∈K15ε16π4λiλjλn2λrλsζiζjζnei−θi−θjvrvs+∑r∈I2∑n∈K15ε16π4λiλjλn2λrλsζiζjζneiθi+θjv¯rv¯s+∑r∈I315ε16π4λiλjλnλmλrλsζiζjζnζmeiθi−θj+θd−θlvrv¯s+∑r∈I415ε16π4λiλjλnλmλrλsζiζjζnζmei−θi+θj−θn−θmvrvs+∑r∈I415ε16π4λiλjλnλmλrλsζiζjζnζmeiθi−θj+θn+θmv¯rv¯s+OεI2+εIva,ϖ2+εζ3/2I1/2va,ϖ2+εζ3/2va,ϖ3+εζva,ϖ4+ε2ζ9/2va,ϖ+ε2ζ4va,ϖ2+ε2ζ7/2va,ϖ3.
Scaling through time(39)ζ⟶ε3/2ζ,I⟶ε5I,θ⟶θ,v⟶ε5/2v,v¯⟶ε5/2v¯,the scaled Hamiltonian is H=ε−9H~ε3/2ζ,ε5I,θ,ε5/2v,ε5/2v¯. The Hamiltonian H satisfies (23) and (24) where ζ∈ε3/2,2ε3/2d.


## 2.1. The Hamiltonian Setting

Without losing generality, we suppose thatfu=u5. Rewrite the beam equation (1) as
(12)utt+Δ2u+εu5=0,x∈T2,t∈ℝ.Introducing a variableut=v, then (12) can be turned into
(13)ut=v,vt=−Δ2u−εu5.Setp=1/2−Δ1/2u−i−Δ−1/2v; then, (13) is turned into ṗ=i∂H∗/∂p¯ whose corresponding Hamiltonian is
(14)H∗p=∫T2−Δpp¯+ε6−Δ−1/2p+p¯26dx.The eigenvalues and eigenvectors of operator−Δ with periodic boundary conditions are λl=l2 and ϕlx=1/2πei<l,x>, respectively. p has coordinates pll∈ℤ†2∈la,ϖ with respect to the bases ϕll∈ℤ†2. The corresponding symplectic structure is i∑d∈ℤ†2dpl∧dp¯l. In new coordinates, (13) becomes
(15)ṗl=i∂H∗∂p¯l,∀l∈ℤ†2.The corresponding Hamiltonian is(16)H∗=Λ+Q,where
(17)Λ=∑l∈ℤ†2λlpl2,Q=ε768π4∑i+j+n+m+r+s=0i,j,n,m,r,s∈ℤ†21λiλjλnλmλrλspipjpnpmprps+p¯ip¯jp¯np¯mp¯rp¯s+ε128π4∑i+j+n+m+r−s=0i,j,n,m,r,s∈ℤ†21λiλjλnλmλrλspipjpnpmprp¯s+p¯ip¯jp¯np¯mp¯rps+5ε256π4∑i+j+n+m−r−s=0i,j,n,m,r,s∈ℤ†21λiλjλnλmλrλspipjpnpmp¯rp¯s+p¯ip¯jp¯np¯mprps+5ε192π4∑i+j+n−m−r−s=0i,j,n,m,r,s∈ℤ†21λiλjλnλmλrλspipjpnp¯mp¯rp¯s.The regularity of the Hamiltonian system (15) is shown below, and its proof is similar to [21], which is omitted here.Lemma 4.
For a givena≥0 and ϖ>1/2, the gradients Qp6,Qp¯6 are real analytic as maps from some neighborhood of origin in la,ϖ×la,ϖ into la,ϖ and Qp6a,ϖ=Opa,ϖ5, Qp¯6a,ϖ=Opa,ϖ5.


## 2.2. Partial Birkhoff Normal Form

K is an admissible set with d points. Set ℤ2,⋄=ℤ†2\K, and define four sets as follows:
(18)K1=i,j,n,m,r,s∈ℤ†26:i−j+n−m+r−s=0i2−j2+n2−m2+r2−s2≠0#K∩i,j,n,m,r,s≥4,K2=i,j,n,m,r,s∈ℤ†26:i+j+n+m+r+s=0i2+j2+n2+m2+r2+s2≠0#K∩i,j,n,m,r,s≥4,K3=i,j,n,m,r,s∈ℤ†26:i+j+n+m+r−s=0i2+j2+n2+m2+r2−s2≠0#K∩i,j,n,m,r,s≥4,K4=i,j,n,m,r,s∈ℤ†26:i+j+n+m−r−s=0i2+j2+n2+m2−r2−s2≠0#K∩i,j,n,m,r,s≥4.Obviously, the set(19)i,j,n,m,r,s∈ℤ†26:i+j+n+m+r+s=0i2+j2+n2+m2+r2+s2=0is an empty set. It is proved by the reduction to absurdity that any six points i=4i^1−1,4i^2,j=4j^1−1,4j^2,n=4n^1−1,4n^2,m=4m^1−1,4m^2,r=4r^1−1,4r^2,s=4s^1−1,4s^2 on ℤ†2 satisfy i2+j2+n2+m2+r2−s2≠0. Suppose that i2+j2+n2+m2+r2−s2=0, then
(20)4i∧1−12+4j∧1−12+4n∧1−12+4m∧1−12+4r∧1−12−4s∧1−12=−16i∧22+j∧22+n∧22+m∧22+r∧22−s∧22.The first polynomial is divisible by 8, but the second polynomial is not, which is a contradiction. Therefore, the set(21)i,j,n,m,r,s∈ℤ†26:i+j+n+m+r−s=0i2+j2+n2+m2+r2−s2=0is an empty set. Similarly, any six points i,j,n,m,r,s on ℤ†2 satisfy i2+j2+n2+m2−r2−s2≠0. Therefore, the set
(22)i,j,n,m,r,s∈ℤ†26:i+j+n+m−r−s=0i2+j2+n2+m2−r2−s2=0is 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 XR1 that converts the Hamiltonian (16) into
(23)H=Λ+A1+B1+B1¯+A2+B2+B2¯+W,with
(24)Λ=∑i∈KηiζIi+∑r∈ℤ2,⋄Ωrζvr2,(25)ηiζ=ε−4λi+5ζi264π4λi3+∑j∈K,j≠i15ζj264π4λiλj2+15ζiζj32π4λi2λj+∑n∈K,n≠j,n≠i15ζjζn32π4λiλjλn,(26)Ωrζ=ε−4λr+∑i∈K15ζi264π4λi2λr+∑j∈K,j≠i15ζiζj32π4λiλjλr,(27)A1=∑r∈I1∑n∈K1516π4λiλjλn2λrλsζiζjζneiθi−θjvrv¯s,(28)B1=∑r∈I2∑n∈K1516π4λiλjλn2λrλsζiζjζnei−θi−θjvrvs,(29)B1¯=∑r∈I2∑n∈K1516π4λiλjλn2λrλsζiζjζneiθi+θjv¯rv¯s,(30)A2=∑r∈I31516π4λiλjλnλmλrλsζiζjζnζmeiθi−θj+θn−θmvrv¯s,(31)B2=∑r∈I41516π4λiλjλnλmλrλsζiζjζnζmei−θi+θj−θn−θmvrvs,(32)B2¯=∑r∈I41516π4λiλjλnλmλrλsζiζjζnζmeiθi−θj+θn+θmv¯rv¯s,(33)W=Oε2I2+ε2Iva,ϖ2+ε7/4ζ3/2I1/2va,ϖ2+ε7/4ζ3/2va,ϖ3+ε7/2ζva,ϖ4+ε9/4ζ9/2va,ϖ+ε4ζ4va,ϖ2+ε23/4ζ7/2va,ϖ3.Proof.
Let(34)R=5ε192π4∑i,j,n,m,r,s∈K1i·pip¯jpnp¯mprp¯sλiλjλnλmλrλs·λi−λj+λn−λm+λr−λs+ε768π4∑i,j,n,m,r,s∈K2i·pipjpdplprps−p¯ip¯jp¯np¯mp¯rp¯sλiλjλnλmλrλs·λi+λj+λn+λm+λr+λs+ε128π4∑i,j,n,m,r,s∈K3i·pipjpnpmprp¯s−p¯ip¯jp¯np¯mp¯rpsλiλjλnλmλrλs·λi+λj+λn+λm+λr−λs+5ε256π4∑i,j,n,m,r,s∈K4i·pipjpnpmp¯rp¯s−p¯ip¯jp¯dp¯lprpsλiλjλnλmλrλs·λi+λj+λn+λm−λr−λsand XR1 be the time−1 mapping of the Hamiltonian vector field of R. Set
(35)pl=pl,l∈K,vl,l∈ℤ2,⋄.
Then,XR1 converts H∗ into
(36)H~=H∗∘XR1=Λ+Q+Λ,R+Q,R+∫011−tH∗,R,R∘XRtdt=∑i∈Kλipi2+∑r∈ℤ2,⋄λrvr2+∑i∈K5ε192π4λi3pi6+∑i,j∈K,i≠j15ε64π4λi2λjpi4pj2+∑i,j,n∈K,i≠j,i≠n,j≠n15ε16π4λiλjλnpi2pj2pn2+∑i∈K,r∈ℤ2,⋄15ε64π4λi2λrpi4vr2+∑i,j∈K,r∈ℤ2,⋄,i≠j15ε16π4λiλjλrpi2pj2vr2+∑r∈I1∑n∈K15ε16π4λiλjλn2λrλspip¯jpn2vrv¯s+∑r∈I2∑n∈K15ε16π4λiλjλn2λrλsp¯ip¯jpn2vrvs+∑r∈I2∑n∈K15ε16π4λiλjλn2λrλspipjpn2v¯rv¯s+∑r∈I315ε16π4λiλjλnλmλrλspip¯jpnp¯mvrv¯s+∑r∈I415ε16π4λiλjλnλmλrλsp¯ipjp¯np¯mvrvs+∑r∈I415ε16π4λiλjλnλmλrλspip¯jpnpmv¯rv¯s+Oεp3va,ϖ3+εp2va,ϖ4+εpva,ϖ5+εva,ϖ6+ε2p10+ε2p9va,ϖ+ε2p8va,ϖ2+ε2p7va,ϖ3,where ∑l∈Kidpl∧dp¯l+∑r∈ℤ2,⋄idvr∧dv¯r is the corresponding symplectic structure of Poisson bracket ·,·, r,s is a resonant pair, and i,j and i,j,n,m are uniquely determined by r,s.
Introduce the action-angle variable(37)pl=Il+ζleiθl,p¯l=Il+ζle−iθl,l∈K.
Equation (37) converts the Hamiltonian H~ into
(38)H~=∑i∈Kλi+5εζi264π4λi3+∑j∈Kj≠i15εζj264π4λiλj2+15εζiζj32π4λi2λj+∑n∈Kn≠jn≠i15εζjζn32π4λiλjλnIi+∑r∈ℤ2,⋄λr+∑i∈K15ε64π4λi2λrζi2+∑j∈K,j≠i15ε32π4λiλjλrζiζjvr2+∑r∈I1∑n∈K15ε16π4λiλjλn2λrλsζiζjζneiθi−θjvrv¯s+∑r∈I2∑n∈K15ε16π4λiλjλn2λrλsζiζjζnei−θi−θjvrvs+∑r∈I2∑n∈K15ε16π4λiλjλn2λrλsζiζjζneiθi+θjv¯rv¯s+∑r∈I315ε16π4λiλjλnλmλrλsζiζjζnζmeiθi−θj+θd−θlvrv¯s+∑r∈I415ε16π4λiλjλnλmλrλsζiζjζnζmei−θi+θj−θn−θmvrvs+∑r∈I415ε16π4λiλjλnλmλrλsζiζjζnζmeiθi−θj+θn+θmv¯rv¯s+OεI2+εIva,ϖ2+εζ3/2I1/2va,ϖ2+εζ3/2va,ϖ3+εζva,ϖ4+ε2ζ9/2va,ϖ+ε2ζ4va,ϖ2+ε2ζ7/2va,ϖ3.
Scaling through time(39)ζ⟶ε3/2ζ,I⟶ε5I,θ⟶θ,v⟶ε5/2v,v¯⟶ε5/2v¯,the scaled Hamiltonian is H=ε−9H~ε3/2ζ,ε5I,θ,ε5/2v,ε5/2v¯. The Hamiltonian H satisfies (23) and (24) where ζ∈ε3/2,2ε3/2d.


## 3. 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 H⋄=Λ⋄+A1⋄+B1⋄+B1⋄¯+A2⋄+B2⋄+B2⋄¯, where
(40)Λ⋄=∑j∈KηjζIj+∑r∈ℤ2,⋄Ωrζvrv¯r,A1⋄=∑r∈K1arζeiθi−θjvrv¯s,B1⋄=∑r∈K2arζe−iθi+θjvrvs,B1⋄¯=∑r∈K2a¯rζeiθi+θjv¯rv¯s,A2⋄=∑r∈K3arζeiθi−θj+θd−θlvrv¯s,B2⋄=∑r∈K4arζei−θi+θj−θd−θlvrvs,B2⋄¯=∑r∈K4a¯rζeiθi−θj+θd+θlv¯rv¯s.θ,I are d-dimensional angle-action coordinates, v,v¯ are infinite-dimensional coordinates, and the corresponding symplectic structure is ∑i∈Kdθi∧dIi+i∑r∈ℤ2,⋄dvr∧dv¯r. Frequencies η=ηii∈K and Ω=Ωrr∈ℤ2,⋄ depend on the parameter ζ∈Ξ⊂ℝd, 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.Assumption 6 (nondegeneracy).
Suppose that∀ζ∈Ξ,
(41)rank∂ηi1∗∂ζ,⋯,∂ηid∗∂ζ=σ,rank∂ιη∂ζι∀ι,1≤ι≤mind−σ+1,5=d,where σ is a given integer with 1≤σ≤d, ∂ηi1∗/∂ζ,⋯,∂ηid∗/∂ζ are vectors of all 1−order partial derivatives in ζ, and for a fixed ι, ∂ιη/∂ζι=∂ιηi1∗/∂ζι,⋯,∂ιηid∗/∂ζι. Moreover, ηζ belongs to CW8Ξ.Assumption 7 (asymptotics of normal frequencies).
(42)Ωr=ε−ςr2+Ω~r,ς≥0,where Ω~r,s are CW8 functions of ζ with CW8-norm bounded by some small positive constant L.Assumption 8 (Melnikov’s nondegeneracy).
LetGr=Ωr for r∈ℤ2,⋄\I1∪I2∪I3∪I4,(43)Gr=Ωr+ηiarasΩs+ηj,r∈I1,Gr=Ωr−ηi−ara¯s−Ωs+ηj,r∈I2,Gr=Ωr+ηi+ηnarasΩs+ηj+ηm,r∈I3,Gr=Ωr+ηj−ara¯s−Ωs+ηi+ηn+ηm,r∈I4,where r,s are resonant pairs and i,j and i,j,n,m are uniquely determined by r,s. There is γ′,τ>0 (here, I is identity matrix) such that
(44)<k,η>≥γ′kτ,k≠0,det<k,η>I±Gr≥γ′kτ,det<k,η>I±Gr⊗I±I⊗Gr′≥γ′kτ.Assumption 9 (regularity).
A1⋄+B1⋄+B1⋄¯+A2⋄+B2⋄+B2⋄¯+W⋄ is real analytic in θ,I,v,v¯ and CW8 in ζ and
(45)XA1⋄Da,ϖr∗,s∗,Ξ+XB1⋄Da,ϖr∗,s∗,Ξ+XA2⋄Da,ϖr∗,s∗,Ξ+XB2⋄Da,ϖr∗,s∗,Ξ<1,XW⋄Da,ϖr∗,s∗,Ξ<ε.Assumption 10 (special form).
A1⋄+B1⋄+B1⋄¯+A2⋄+B2⋄+B2⋄¯+W⋄ has the following special form:
(46)Y⋄=A1⋄+B1⋄+B1⋄¯+A2⋄+B2⋄+B2⋄¯+W⋄:A1⋄+B1⋄+B1⋄¯+A2⋄+B2⋄+B2⋄¯+W⋄=∑k∈ℤd,l∈ℕd,α,βA1⋄+B1⋄+B1⋄¯+A2⋄+B2⋄+B2⋄¯+W⋄klαβζIlei<k,θ>vαv¯β,where k,α,β satisfy ∑s^=1dks^is^∗+∑r∈ℤ2,⋄αr−βrr=0.Assumption 11 (Töplitz-Lipschitz property).
For givenr,s∈ℤ†2,c~∈ℤ†2, the limits
(47)limt~⟶∞∂2B1⋄+B2⋄+W2⋄∂vr+c~t~∂vs−c~t~,limt~⟶∞∂2∑l∈ℤ2,⋄Ωlvlv¯l+A1⋄+A2⋄+W⋄∂vr+c~t~∂v¯s+c~t~,limt~⟶∞∂2B1⋄¯+B2⋄¯+W⋄∂v¯r+c~t~∂v¯s−c~t~exist. Moreover, there is K∗>0, so that if t~>K∗, then Λ⋄+A1⋄+B1⋄+B1⋄¯+A2⋄+B2⋄+B2⋄¯+W⋄ satisfies
(48)∂2B1⋄+B2⋄+W⋄∂vr+c~t~∂vs−c~t~−limt~⟶∞∂2B1⋄+B2⋄+W⋄∂vr+c~t~∂vs−c~t~Da,ϖr∗,s∗,Ξ≤εt~e−r+sa,∂2∑l∈ℤ2,⋄Ωlvlv¯l+A1⋄+A2⋄+W⋄∂vr+c~t~∂v¯s+c~t~−limt~⟶∞∂2∑l∈ℤ2,⋄Ωlvlv¯l+A1⋄+A2⋄+W⋄∂vr+c~t~∂v¯s+c~t~Da,ϖr∗,s∗,Ξ≤εt~e−r−sa,∂2B1⋄¯+B2⋄¯+W⋄∂v¯r+c~t~∂v¯s−c~t~−limt~⟶∞∂2B1⋄¯+B2⋄¯+W⋄∂v¯r+c~t~∂v¯s−c~t~Da,ϖr∗,s∗,Ξ≤εt~e−r+sa.The following KAM theorem comes from Zhang and Si [23].Theorem 12 ([23] Theorem 2.1).
IfH∗=H⋄+W⋄=Λ⋄+A1⋄+B1⋄+B1⋄¯+A2⋄+B2⋄+B2⋄¯+W⋄ satisfies the Assumptions (6)–(11), and γ′>0 is small enough. Then, there exists a positive constant ε=εd,L,K∗,τ,γ′,r∗,s∗,a,ϖ, so that if XW⋄Da,ϖr∗,s∗,Ξ<ε, the following results true: there exists a Cantor subset Ξγ′⊂Ξ with measΞ\Ξγ′=Oγ′8 and two transformations (analytic in θ and CW8 in ζ) Ψ:Td×Ξγ′⟶Da,ϖr∗,s∗,η:Ξγ′⟶ℝd, where Ψ is ε/γ′8-close to the trivial embedding Ψ0:Td×Ξ⟶Td×0,0,0 and η~ is ε-close to the unperturbed frequency η, so that for ζ∈Ξγ′ and θ∈Td, the curve t⟶Ψθ+η~ζt,ζ is a quasiperiodic solution of the Hamiltonian equations corresponding to H∗.


## 4. Proof of the Main Theorem

Let us show that the Hamiltonian (23) satisfies the Assumptions (6)–(11).Verifying Assumption (6): from (24), for n^=1,⋯,d,
(49)∂2ηin^∗ζ∂ζin^∗2=532π4λin^∗3,(50)∂2ηin^∗ζ∂ζij^∗∂ζin^∗=∂2ηin^∗ζ∂ζin^∗∂ζij^∗=1532π4λin^∗2λij^∗,1≤j^≤d,j^≠n^.Let(51)W=∂2ηi1∗ζ∂ζi1∗2⋯∂2ηid∗ζ∂ζid∗∂ζi1∗⋯⋯⋯∂2ηi1∗ζ∂ζi1∗∂ζid∗⋯∂2ηid∗ζ∂ζid∗2,whereζ∈Ξ; then, W is the submatrix of matrix ∂2η/∂ζ2. According to (49) and (50), then
(52)limε⟶0W=532π41λi1∗33λi2∗2λi1∗⋯3λid∗2λi1∗3λi1∗2λi2∗1λi2∗3⋯3λid∗2λi2∗⋯⋯⋯⋯3λi1∗2λid∗3λi2∗2λid∗⋯1λid∗3d×d≔532π4W^.FromdetW^=1+3d−1−2d−1/∏j^=1dλij^∗3≠0, we have detW≠0 when 0<ε≪1, that is, rankW=d. Hence, Assumptions (6) is verified.Verifying Assumption (7): take ς=4; the proof is obvious.Verifying Assumption (8): from (23), Gr is represented as follows:
(53)Gr=Ωrr∈ℤ2,⋄\I1∪I2∪I3∪I4,Gr=Ωr+ηi∑n∗∈K15ζiζjζn∗16π4λiλjλn∗2λrλs∑n∗∈K15ζiζjζn∗16π4λiλjλn∗2λrλsΩs+ηj,r∈I1,Gr=Ωr−ηi−∑n∗∈K15ζiζjζn∗16π4λiλjλn∗2λrλs∑n∗∈K15ζiζjζn∗16π4λiλjλn∗2λrλs−Ωs+ηj,r∈I2,Gr=Ωr+ηi+ηn15ζiζjζnζm16π4λiλjλnλmλrλs15ζiζjζnζm16π4λiλjλnλmλrλsΩs+ηj+ηm,r∈I3,Gr=Ωr+ηj−15ζiζjζnζm16π4λiλjλnλmλrλs15ζiζjζnζm16π4λiλjλnλmλrλs−Ωs+ηi+ηn+ηm,r∈I4,wherei,j and i,j,n,m are uniquely determined by r,s. We only prove (A3) for det<k,ηη>I±Gr⊗I±I⊗Gr′ which is the most complicated. Let
(54)Fζ=<k,ηζ>I±Gr⊗I±I⊗Gr′.We will prove thatFζ≥γ′/kτ,k≠0. Let us only prove the case r,r′∈I4, and everything else is similar. Let
(55)Gr=ε−4Gr,1+Gr,2,∀r∈I4,where
(56)Gr,1=λr+λj00−λs+λi+λn+λm,Gr,2=Gr,211Gr,212Gr,221Gr,222,with
(57)Gr,211=516π4λj3ζj2−∑j∗∈K1532π4λj2λj∗ζjζj∗+1564π41λr+1λj<ζ,Aζ>,Gr,212=−15ζiζjζnζm16π4λiλjλnλmλrλs,Gr,221=15ζiζjζnζm16π4λiλjλnλmλrλs,Gr,222=516π4ζi2λi3+ζn2λn3+ζm2λm3−∑j∗∈K15ζj∗32π4λj∗ζiλi2+ζnλn2+ζmλm2+1564π41λi+1λn+1λm−1λs<ζ,Aζ>,with ζ=ζi1∗,ζi2∗,⋯,ζid∗ and
(58)A=1λi1∗22λi2∗λi1∗⋯2λid∗λi1∗2λi1∗λi2∗1λi2∗2⋯2λid∗λi2∗⋯⋯⋯⋯2λi1∗λid∗2λi2∗λid∗⋯1λid∗2d×d.Thus,(59)Fζ=ε−4<k,β∗>+<k,B>±r2+j2±r′2+j′2I±Gr,2⊗I±I⊗Gr′,2,where β∗=i1∗2,i2∗2,⋯,id∗2 and B=⋯,Bi∗,⋯i∗∈KT with
(60)Bi∗=5ζi∗264π4λi∗3+∑j∗∈Kj∗≠i∗15ζj∗264π4λi∗λj∗2+15ζi∗ζj∗32π4λi∗2λj∗+∑n∗∈Kn∗≠j∗n∗≠i∗15ζj∗ζn∗32π4λi∗λj∗λn∗.The eigenvalues ofFζ are
(61)ε−4<k,β∗>±r2+j2±r′2+j′2+<k,B>±5128π44ζi2λi3+ζj2λj3+ζn2λn3+ζm2λm3−6∑j∗∈Kζj∗λj∗ζiλi2+ζjλj2+ζnλn2+ζmλm2+31λi+1λj+1λn+1λm+1λr−1λs<ζ,Aζ>±hI4,I4ζi,ζj,ζn,ζm±5128π44ζi′2λi′3+ζj′2λj′3+ζn′2λn′3+ζm′2λm′3−6∑j∗∈Kζj∗λj∗ζi′λi′2+ζj′λj′2+ζn′λn′2+ζm′λm′2+31λi′+1λj′+1λn′+1λm′+1λr′−1λs′<ζ,Aζ>±hI4,I4ζi′,ζj′,ζn′,ζm′,where
(62)hI4,I4ζi,ζj,ζn,ζm=4ζi2λi3−ζj2λj3+ζn2λn3+ζm2λm3−6∑j∗∈Kζj∗λj∗ζiλi2−ζjλj2+ζnλn2+ζmλm2+31λi−1λj+1λn+1λm−1λr−1λs<ζ,Aζ>2−576ζiζjζnζmλiλjλnλmλrλs.It has been proved that none of the eigenvalues ofFζ are zero in [23]. Moreover, when r∈I1,r′∈I1 or r∈I1,r′∈I2 or r∈I1,r′∈I3 and so on, the situations are similar, so omit the proofs. That is, none of the eigenvalues of Fζ are zero for k≠0. From Lemma 3.1 in [19], then detFζ is a polynomial function in the components of ζ with order at most eight and ∂ζ8detWζ≥1/2k≠0. By excluding some parameter set with measure Oγ′8, then detWζ≥γ′/kτ,k≠0.A3 is verified.Verifying Assumption (9): similar to [19], from Lemma 4, it is obvious that A4 holds, so omit its proof.Verifying Assumption (10): similar to [19], the result is obvious.Verifying Assumption (11): similar to [23], the result is obvious.By Theorem12 ([23] Theorem 2.1), we have Theorem 3.



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*Source: 7106366-2022-10-05.xml*