# Quasiperiodic Solutions of Completely Resonant Wave Equations with Quasiperiodic Forced Terms

Authors: Yixian Gao; Weipeng Zhang; Jing Chang
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

This paper is concerned with the existence of quasiperiodic solutions with two frequencies of completely resonant, quasiperiodically forced nonlinear wave equations subject to periodic spatial boundary conditions. The solutions turn out to be, at the first order, the superposition of traveling waves, traveling in the opposite or the same directions. The proofs are based on the variational Lyapunov-Schmidt reduction and the linking theorem, while the bifurcation equations are solved by variational methods.


---

## Body

## 1. Introduction

This paper is devoted to the study of the existence of small-amplitude quasiperiodic solutions of completely resonant forced nonlinear wave equations like(1)utt-uxx+f(ω1t,ω2t,x,u)=0,
with periodic boundary conditions
(2)u(t,x)=u(t,x+2π),(t,x)∈R2,
while the nonlinear forced term f is
(3)f(ω1t,ω2t,x,u)∶=∑k≥3ak(ω1t+x,ω2t+x)uk,
when the traveling waves are in the same directions, and f is
(4)f(ω1t,ω2t,x,u)∶=∑k≥3ak(ω1t-x,ω2t+x)uk,
when the traveling waves are in the opposite directions. Moreover, the nonlinear forced terms are all analytic in a neighborhood of u=0.Periodic or quasiperiodic solutions in nonresonant PDEs have been obtained, for instance, in [1–12] by the Lyapunov-Schmidt reduction together with Nash-Moser theory and KAM theory, while the completely resonant autonomous PDEs have been originally studied by variational methods starting from Rabinowitz [13–19]. They obtained the existence of periodic solutions with period being a rational multiple of π, and such solutions correspond to a zero-measure set of values of the amplitudes. The case with period being irrational of π, which in principle could provide a large measure of values, has been mostly studied under strong Diophantine conditions; see [20–25] and the references therein. In [26, 27], using the Lindstedt series method, Gentile and Procesi obtained the existence of periodic solutions for a large measure set of frequencies for the nonlinear wave equations and nonlinear Schrödinger equations with periodic boundary conditions. In [28], Yuan obtained the existence of quasiperiodic solution for a large measure set of at least three dimensional rotation vectors by the KAM method. In [29], under the periodic boundary condition and with the periodic forced nonlinearities f(ω1t,u)=a(ω1t)u2d-1+O(u2d),d>2, Berti and Procesi got the existence of quasiperiodic solution of nonlinear wave equation in the form of v(t,x)=u(ω1t,ω2t+x). In [30], Procesi firstly obtained the quasiperiodic solutions with two frequencies in the form of v(t,x)=u(ω1t+x,ω2t-x) for the specific nonlinearities f=u3+O(u5), where the forced terms do not depend on the time and the bifurcation equations are solved by ODE methods. In [31], Baldi proved the existence of small-amplitude quasiperiodic solutions in the form of v(t,x)=u(ω1t+x,ω2t+x) with the general nonlinearities f=u3+O(u4), which also do not depend on the time. In [32], they considered the existence of quasiperiodic solution of the forced wave equation, in which the solutions are traveling in opposite directions. However, they did not give the regularity of the solutions, and the results are the special case of our results in Section 4. Moreover, we mention the work [33] of Bambusi, where a simple proof of an infinite-dimensional extension of the Lyapunov center theorem is given.In this paper, for the completely resonant wave equation (1) subjecting to the quasiperiodic forced terms, we will prove the existence and regularity of quasiperiodic solutions with two frequencies, ω1,ω2, in both of the following two cases.Case (A1). The first case considers the wave traveling in the same directions u(t,x)=v(ω1t+x,ω2t+x).Case (A2). The second case considers the wave traveling in opposite directions u(t,x)=v(ω1t-x,ω2t+x).


## 2. Main Results

We look for quasiperiodic solutionsu(t,x) of (1) of the following form:(in the same directions)(5)u(t,x)=v(ω1t+x,ω2t+x)=v(φ1,φ2),v(φ1,φ2)=v(φ1+2k1π,φ2+2k2π),∀k1,k2∈Z,with frequencies ω=(ω1,ω2)=(1+ϵ,1+aϵ2),a∈R-, or (in opposite directions)(6)u(t,x)=v(ω1t-x,ω2t+x)=v(φ1,φ2),v(φ1,φ2)=v(φ1+2k1π,φ2+2k2π),∀k1,k2∈Z,with frequencies ω=(ω1,ω2)=(1+ϵ,1+aϵ),a∈R-, imposing the frequencies ω=(ω1,ω2) to be close to linear frequency 1. Therefore, finding the quasiperiodic solutions of (1) with  frequencies, respectively, (ω1,ω2) is equivalent to finding 2π periodic solutions with respect to (φ1,φ2) for the following equations:(in the same directions)(7)(∂tt2-∂xx2)v=(∂t-∂x)∘(∂t+∂x)v=[ω1∂φ1+ω2∂φ2-∂φ1-∂φ2]∘[ω1∂φ1+ω2∂φ2+∂φ1+∂φ2]v=((ω12-1)∂φ12+(ω22-1)∂φ22k+2(ω1ω2-1)∂φ1∂φ2)v=-f(φ1,φ2,v),(in opposite directions)(8)(∂tt2-∂xx2)v=(∂t-∂x)∘(∂t+∂x)v=[ω1∂φ1+ω2∂φ2+∂φ1-∂φ2]∘[ω1∂φ1+ω2∂φ2-∂φ1+∂φ2]v=((ω12-1)∂φ12+(ω22-1)∂φ22l+2(ω1ω2+1)∂φ1∂φ2)v=-f(φ1,φ2,v).We assume that the quasiperiodic forced term f:T2×R→R,(9)f(φ1,φ2,v)=a3(φ1,φ2)v3+O(v4),
is analytic in v but has only finite regularity in φ1,φ2. More precisely,(H)
f(φ1,φ2,v)∶=∑k=3∞ak(φ1,φ2)vk, and the coefficients ak(φ1,φ2)∈H1(T2) verify, for some r>0, ∑k=3∞|ak|H1rk<∞. The function f(φ1,φ2,v) is not identically constant in (φ1,φ2).We look for solutionsv of (7)-(8) in the Banach space
(10)Hσ,s∶={v(φ)=∑(l1,l2)∈Z2v^(l1,l2)eil1φ1eil2φ2:v^(l1,l2)*=v^(-l1,-l2),kkkk|v|σ,s=∑(l1,l2)∈Z2|v^(l1,l2)|e|l2|σ(max⁡{0,|l1|})s<+∞},
where l=(l1,l2)∈Z2,v^(l1,l2)* denotes its complex conjugate, and σ>0,s≥0.The spaceHσ,s is a Banach algebra with respect to multiplications of functions, namely,
(11)v1,v2∈Hσ,s⟹v1v2∈Hσ,s,|v1v2|σ,s≤C|v1|σ,s|v2|σ,s.
We will prove the following theorems.Theorem 1.
Assume that the nonlinearityf satisfies (H) and a3(φ1,φ2)≠0,∀(φ1,φ2)∈T2. Let Bγ be the uncountable zero-measure Cantor set
(12)Bγ∶={(0,12)(a,ϵ)∈R-×R,1+ϵ≠0,1+aϵ2≠0,2+aϵ2≠0,kk(2+ϵ2+aϵ2,aϵ)∈Cγ1∩(1-ϵ0,1+ϵ0)kk×Cγ2∩(-ϵ0,ϵ0),1+ϵ1+aϵ2∉Q,forϵ0∈(0,12)},
where Cγi, i=1,2, are sets of badly approximate numbers defined as
(13)Cγ1∶={2+ϵ2+aϵ2:|l2+2+ϵ2+aϵ2l1|>γ|l1|},Cγ2∶={aϵ:|l1+aϵl2|>γ|l2|}
for ∀l1,l2∈Z∖{0}, and 0<γ<1/4. There exist constants σ->0,s->2, and ϵ-(R)>0, such that, for (a,ϵ)∈Bγ,|ϵ|/γ<ϵ-(R), there exists a solution v(ϵ,φ1,φ2)∈Hσ-,s- of (7), having the form
(14)v(ϵ,φ1,φ2)=|ϵ|(q^0+q^-(φ2)+q^+(φ1+φ2)+p^(ϵ,φ1,φ2)),
with
(15)|p^(ϵ,φ1,φ2)|σ-,s-≤C-(|ϵ|γ),
where C- is a constant. As a consequence, (1) possesses the quasiperiodic solutions, traveling in the same directions, u(ϵ,t,x)=v(ϵ,ω1t+x,ω2t+x), with two frequencies (ω1,ω2)=(1+ϵ,1+aϵ2).Theorem 2.
Assume thatf satisfies assumption (H) and a3(φ1,φ2)≠0,∀(φ1,φ2)∈T2. Let Dγ⊂(-ϵ0,ϵ0)×(-ϵ0,ϵ0) be the uncountable zero-measure Cantor set
(16)Dγ∶={(a,ϵ)∈R-×R,(aϵ2+ϵ,ϵ2+aϵ)∈Eγ,k1+ϵ1+aϵ∉Q,1+ϵ≠0,1+aϵ≠0,2+aϵ≠0},
where Eγ is a set of badly approximate numbers defined as
(17)Eγ∶={γ|l1|(aϵ2+ϵ,ϵ2+aϵ)∶=(ϵ1,ϵ2)∈(-ϵ0,ϵ0)×(-ϵ0,ϵ0):kkkkk|l1+ϵ1l2|>γ|l2|,|l2+ϵ2l1|>γ|l1|}
for ∀l1,l2∈Z∖{0}, and 0<γ<1/4,ϵ0∈(0,1/2). There exist positive numbers σ-,ϵ-,C-,s->2, such that, ∀(a,ϵ)∈Dγ, (8) admits solutions in the form of
(18)va,ϵ(t,x)=v(ϵ,φ1,φ2)=|ϵ|(p^(ϵ,φ1,φ2)q^0+q^-(φ2)+q^+(φ1)kkkk+p^(ϵ,φ1,φ2))∈Hσ-,s,
satisfying
(19)|p^(ϵ,φ1,φ2)|σ-,s-≤C-(|ϵ|γ).
As a consequence, (1) possesses the quasiperiodic solutions, ua,ϵ(t,x)=va,ϵ((1+ϵ)t+x,(1+aϵ)t-x), traveling in opposite directions.Remark 3.
The quasiperiodic solutions of traveling waves we obtained are different from the ones got by KAM methods since the quasiperiodic solutions we get depending onx and t are coupled and in the form of the traveling waves.Remark 4.
We can get the similar result with more general nonlinearity, such asf(ω1t,ω2t,x,u)=ad(φ1,φ2)ud+O(u(d+1)), for any d∈N,d≥3.This paper is organized as follows: we first prove the existence of quasiperiodic solutions, at the first order, to the superposition of traveling waves, traveling in the same directions. In Section4, we prove the existence of quasiperiodic solutions traveling in opposite directions.


## 3. Waves Traveling in the Same Directions

Substitutingω1=1+ϵ,ω2=1+aϵ2 into (7), we can obtain the equations
(20)Lv+f(φ1,φ2,v)=0,
where, see (7), we have
(21)L=((ω12-1)∂φ12+(ω22-1)∂φ22+2(ω1ω2-1)∂φ1∂φ2)v=ϵ(2∂φ12+2∂φ1∂φ2)+ϵ2(∂φ12+(2a+a2ϵ2)∂φ22+2a(1+ϵ)∂φ1∂φ2),
and f(φ1,φ2,v)=a3(φ1,φ2)v3+O(v4). To prove Theorem 1, instead of looking for solutions of (7) in a shrinking neighborhood of zero, it is convenient to perform the rescaling v(φ1,φ2)→ϵv(φ1,φ2), enhancing the relation between the amplitude and the frequencies. Without confusion, we define
(22)La,ϵ=(2∂φ12+2∂φ1∂φ2)+ϵ(∂φ12+(2a+a2ϵ2)∂φ22+2a(1+ϵ)∂φ1∂φ2)=L0+ϵL1,
so the problem becomes
(23)La,ϵ+ϵf(φ1,φ2,v,ϵ)=0.
To find the solutions of (23), we will apply the Lyapunov-Schmidt reduction method which leads to solving separately a “range equation” and a “bifurcation equation.” In order to solve the range equation (avoiding small divisor problems), we restrict ϵ to the uncountable zero-measure set Bγ for Theorem 1, and we apply the Contraction Mapping Theorem; similar nonresonance conditions have been employed, for example, in [21–23, 25, 29, 30].Equation (23) is the Euler-Lagrange equation of the action functional Ψϵ∈C1(Hσ,s,R) defined by
(24)Ψϵ(v)∶=∫T2(∂φ1v)2+(∂φ1v)(∂φ2v)kkl+ϵ(12(∂φ1v)2+2a+a2ϵ22(∂φ2v)2kkkkkkklk+a(1+ϵ)(∂φ1v)(∂φ2v)2a+a2ε22)kkl-ϵF(φ1,φ2,v,ϵ)=Ψ0(v)+ϵΨ1(v,ϵ),
where
(25)F(φ1,φ2,v,ϵ)∶=∫0vf(φ1,φ2,ξ,ϵ)dξ,Ψ0(v)=∫T2(∂φ1v)2+(∂φ1v)(∂φ2v),Ψ1(v,ϵ)∶=∫T2(12(∂φ1v)2+2a+a2ϵ22(∂φ2v)2kkkk+a(1+ϵ)(∂φ1v)(∂φ2v)2a+a2ε22)-ϵF(φ1,φ2,v,ϵ).
To find critical points of Ψϵ(v), we perform a variational Lyapunov-Schmidt reduction inspired by Berti and Bolle [22, 23, 29]; see also Ambrosetti and Badiale [34].


### 3.1. The Variational Lyapunov-Schmidt Reduction

The operatorLa,ϵ is diagonal defined on the Banach space Hσ,s under the Fourier basis el=eil1φ1eil2φ2 with eigenvalues
(26)-Dl=(2l12+2l1l2)+ϵ(l12+(2a+a2ϵ2)l22+2a(1+ϵ)l1l2)=(l1+aϵl2)((2+ϵ)l1+(2+aϵ2)l2).
So, we have
(27)La,ϵ[v]=∑(l1,l2)∈Z2Dlv^l1,l2eil1φ1eil2φ2,∀v∈Hσ,s.
The critical points of the unperturbed functional Ψ0:Hσ,s→R form an infinite-dimensional linear space Q, and they are the solutions of the equation
(28)L0q=(2∂φ12+2∂φ1∂φ2)q=0.
The space Q can be written as
(29)Q={q=∑(l1,l2)∈Z2q^l1,l2eil1φ1eil2φ2∈Hσ,s∣q^l1,l2=0,kkforl1(2l1+2l2)≠0∑(l1,l2)∈Z2q^l1,l2eil1φ1eil2φ2∈Hσ,s∣q^l1,l2=0,}.
In view of the variational argument that we will use to solve the bifurcation equation, we split Q as Q=Q++Q0+Q-, where
(30)Q+∶={q∈Q:q^l1,l2=0,for(l1,l2)∉Λ+}={q+∶=q+(φ)∈Hσ,s0},Q0∶={q:q0,0∈R},Q-∶={q∈Q:q^l1,l2=0,for(l1,l2)∉Λ-}={q-∶=q-(φ)∈Hσ,s0},
with
(31)Λ+∶={(l1,l2)∈Z2:l1=0,l2≠0},Λ-∶={(l1,l2)∈Z2:l1+l2=0,(l1,l2)≠(0,0)},Λ0∶={(l1,l2)∈Z2:(l1,l2)≡(0,0)}.
We will also use in Q the norm
(32)|q|H12=|q+|H1(T)2+q0,02+|q-|H1(T)2~∑(l1,l2)∈Λ+∪Λ0∪Λ+q^l1,l22(|l1|2+|l2|2+1).
So, we can decompose the space Hσ,s=Q+P, where
(33)P∶={p=∑l1,l2∈Zp^l1,l2eil1φ1eil2φ2∈Hσ,s∣p^l1,l2=0,kkkforl1(2l1+2l2)=0∑l1,l2∈Zp^l1,l2eil1φ1eil2φ2∈Hσ,s∣p^l1,l2=0,}.
Projecting (23) onto the closed subspaces Q and P, setting v=q+p∈Hσ,s with q∈Q and p∈P, we obtain
(34)(Q)L1[q]+ΠQf(φ1,φ2,q+p,ϵ)=0,(P)La,ϵ[p]+ϵΠPf(φ1,φ2,q+p,ϵ)=0,
where ΠQ:Hσ,s→Q,ΠP:Hσ,s→P are the projectors, respectively, onto Q and P.In order to prove analyticity of the solutions and to highlight the compactness of the problem, we perform a finite-dimensional Lyapunov-Schmidt reduction, introducing the decompositionQ=Q1+Q2, where
(35)Q1∶={q=∑|l1|+|l2|≤Nq^l1,l2eil1φ1eil2φ2∈Q},Q2∶={q=∑|l1|+|l2|≤Nq^l1,l2eil1φ1eil2φ2∈Q}.
Setting q=q1+q2 with q1∈Q1 and q2∈Q2, we finally get
(36)(Q1)L1[q1]+ΠQ1[f(φ1,φ2,q1+q2+p,ϵ)]=0⟺dΨϵ(v)[h]=0,∀h∈Q1;(37)(Q2)L1[q2]+ΠQ2[f(φ1,φ2,q1+q2+p,ϵ)]=0⟺dΨϵ(v)[h]=0,∀h∈Q2;(38)(P)La,ϵ[p]+ϵΠP[f(φ1,φ2,q1+q2+p,ϵ)]=0⟺dΨϵ(v)[h]=0,∀h∈P,
where ΠQi:Hσ,s→Qi are the projectors onto Qi  (i=1,2), and ΠP:Hσ,s→P is the projector onto P. We will solve first the (Q2)-(P)-equations for all |q1|H1≤2R, provided ϵ belongs to a suitable Cantor-like set, |ϵ|≤ϵ0(R) is sufficiently small, and N≥N0(R) is large enough (see Lemma 7). Next, we will solve the Q1-equation by means of a variational linking argument; see Section 3.4.


### 3.2. The(Q2)-(P)-Equations

We first prove thatLa,ϵ restricted to P has a bounded inverse when (a,ϵ) belongs to the uncountable zero-measure set
(39)Bγ∶={(2+ϵ2+aϵ2,aϵ)(a,ϵ)∈R-×R,1+ϵ≠0,1+aϵ2≠0,2+aϵ2≠0,kkkk(2+ϵ2+aϵ2,aϵ)∈Cγ1∩(1-ϵ0,1+ϵ0)kkkk×Cγ2∩(-ϵ0,ϵ0),1+ϵ1+aϵ2∉Q,forϵ0∈(0,12)(2+ϵ2+aϵ2,aϵ)},
where Cγi, i=1,2, is a set of badly approximate numbers defined by
(40)Cγ1∶={2+ϵ2+aϵ2:|l2+2+ϵ2+aϵ2l1|>γ|l1|},Cγ2∶={aϵ:|l1+aϵl2|>γ|l2|},
for ∀l1,l2∈Z∖{0}, and 0<γ<1/4. Cγi, i=1,2, accumulate at 1 and zero, respectively, from both the right and the left; see [21, 31, 35].The operatorLϵ is diagonal in the Fourier basis {eil1φ1eil2φ2,(l1,l2)∈Z2} with eigenvalues
(41)Dl1,l2=-(l1+aϵl2)((2+ϵ)l1+(2+aϵ2)l2)=-(2+aϵ2)(l1+aϵl1)(l2+2+ϵ2+aϵ2l1).Lemma 5.
For(a,ϵ)∈Bγ, the eigenvalues Dl1,l2 of La,ϵ restricted to P satisfy
(42)|Dl1,l2|=|(l1+aϵl2)((2+ϵ)l1+(2+aϵ2)l2)|>γ,kkkkkkkkkkkkkkkkkkkkkkkkklkkkk∀l1,l2≠0.
As a consequence, the operator La,ϵ:P→P has a bounded inverse La,ϵ-1 and satisfies
(43)|La,ϵ-1[h]|σ,s≤|h|σ,sγ,∀h∈P.Proof.
Denote by[x] the nearest integer close to x and by {x}=x-[x] its fractional part. If both l1≠-[aϵl2] and l2≠-[((2+ϵ)/(2+aϵ2))l1], we have
(44)|Dl1,l2|=|(l1+aϵl2)|·|(2+aϵ2)||(l2+2+ϵ2+aϵ2l1)|>1.
If l1=-[aϵl2], then |l1|<(1/2)|l2|, so that |l2+((2+ϵ)/(2+aϵ2))l1|>(1/2)|l2|. This implies that
(45)|Dl1,l2|=|l1+aϵl2|·|2+aϵ2||2+ϵ2+aϵ2l1+l2|≥γ|l2|·|(ϵ{aϵ2+ϵl2}+2l2+aϵ(1-ϵ2+ϵ)l2)(2+ϵ)kkkkkkkkk×(ϵ{aϵ2+ϵl2}+2l2+aϵ(1-ϵ2+ϵ)l2)|≥γ.
In the same way, if l2=-[((2+ϵ)/(2+aϵ2))l1], we have
(46)|Dl1,l2|≥γ|l1|·|(2+aϵ2)(ϵ{(2+ϵ)a2+aϵ2l1}+2-2aϵ2+aϵ2l1)|≥γ.
So, the operator La,ϵ restricted to P has a bounded inverse and satisfies
(47)|La,ϵ-1[h]|σ,s=∑(l1,l2)∈Z2|h^l1,l2|e|l2|σ(max⁡⁡{0,|l1|})sDl1,l2≤|h|σ,sγ,∀h∈P.Lemma 6.
The operatorL1:Q2→Q2 has a bounded inverse L1-1, satisfying
(48)|L1-1[h]|σ,s≤|h|σ,sN2.Proof.
L
1 is diagonal in the Fourier basis of Q:eil1φ1eil2φ2 with (l1,l2)∈Λ+∪Λ0∪Λ- with eigenvalues
(49)dl1,l2={(1+ϵ(-2a+a2ϵ))l22,ifl1+l2=0,a(2+aϵ2)l22,ifl1=0.
The eigenvalues of L1 restricted to Q2(N) verify |dl1,l2|≥N2/C, where the constant C depends on (ϵ,a), and (48) holds.Fixed points of the nonlinear operatorP:Q2⊕P→Q2⊕P defined by
(50)P(q2,p,q1)∶=(-L1-1ΠQ2f(φ1,φ2,q1+q2+p,ϵ),k-ϵLa,ϵ-1ΠPf(φ1,φ2,q1+q2+p,ϵ))
are solutions of the (Q2)-(P)-equations. Using the Contraction Mapping Theorem, we can prove the following lemma.Lemma 7.
For anyR>0, there exist an integer N0(R)∈N+ and positive constants ϵ0(R)>0,C0(R)>0 such that
(51)∀|q1|H1≤2R,∀ϵ∈Bγ,|ϵ|γ-1≤ϵ0(R),∀N≥N0(R):0≤Nσ≤1,
and there exists a unique solution (q2(q1),p(q1)):=(q2(ϵ,N,q1),p(ϵ,N,q1))∈Q2⊕P of the (Q2)-(P)-equations satisfying
(52)|q2(ϵ,N,q1)|σ,s≤C0(R)N2,|p(ϵ,N,q1)|σ,s≤C0(R)|ϵ|γ-1.
Moreover, the map q1→(q2(q1),p(q1)) is C1(B2R,Q2⊕P) and
(53)|q2′(q1)[h]|σ,s≤C0(R)N2|h|H1,|p′(q1)[h]|σ,s≤C0(R)|ϵ|γ-1|h|H1∀h∈Q1.Proof.
Let us consider the ball(54)B∶={(q2,p)∈Q2⊕P,|q2|σ,s≤ρ1,|p|σ,s≤ρ2}
with norm |(q2,p)|σ,s∶=|q2|σ,s+|p|σ,s. We can claim that, under assumptions (51), there exist ρ1,ρ2∈(0,1) such that the map (q2,p)→P(q2,p;q1) is a contraction mapping in B, that is, we have to prove (i)
(q2,p)∈B⇒P(q2,p;q1)∈B;(ii)
|P(q2,p;q1)-P(q^2,p^;q1)|σ,s≤η|(q2,p)-(q^2,p^)|σ,s,∀(q2,p),(q^2,p^)∈B,
where the constantη∈(0,1). In the following, κi  (i=1,2,…,5) denote different constants. By (48) and the Banach property of Hσ,s,
(55)|P1(q2,p;q1)|σ,s=|L1-1ΠQ2f(φ1,φ2,q1+q2+p,ϵ)|≤κ1N2(|q1|σ,s3+|q2|σ,s3+|p|σ,s3).
Similarly, for (a,ϵ)∈Bγ, by (43), we have
(56)|P2(q2,p;q1)|σ,s=|ϵLa,ϵ-1ΠPf(φ1,φ2,q1+q2+p,ϵ)|≤κ2|ϵ|γ(|q1|σ,s3+|q2|σ,s3+|p|σ,s3).
For all q1∈Q1(N), setting [x]=max⁡⁡{0,|x|}, we can get
(57)|q1|σ,s=∑|l2|≤N,l1=0|q^0,l2|e|l2|σ+∑|l2|≤N,l1+l2=0|q^-l2,l2|e|l2|σ[-l2]s≤eNσ(∑|l2|≤N,l1=0|q^l1,l2|([-l2-l22])skkkkk+∑|l2|≤N,l1+l2=0|q^-l2,l2|[-l2]s)≤κ3((∑|l2|≤N,l1=0|q^0,l2|2[l2]2)1/2(∑l2∈Z1[l2]2)1/2kkkk+(∑|l2|≤N,l1+l2=0|q^-l2,l2|2[l2]2)1/2kkkk×(∑l2∈Z1([l2])2(1-s))1/2(∑|l2|≤N,l1=0|q^0,l2|2[l2]2)1/2)≤κ4|q1|H1
whenever 0≤Nσ≤1,0≤s<1/2. Thus, ∀|q1|H1≤2R,|q2|σ,s≤ρ1,|p|σ,s≤ρ2, we get
(58)|P1(q2,p;q1)|σ,s≤κ5N2(R3+ρ13+ρ23),|P2(q2,p;q1)|σ,s≤κ5|ϵ|γ(R3+ρ13+ρ23).
Now, set C0(R)∶=κ5R3, and we define ρ1∶=2C0(R)/N3/2,ρ2∶=2C0(R)(|ϵ|/γ). By the inequality above, there exist N0(R)∈N+ and ϵ0(R)>0 such that ∀N≥N0(R) and ∀|ϵ|γ-1≤ϵ0(R),
(59)|P1(q2,p;q1)|σ,s≤ρ1,|P2(q2,p;q1)|σ,s≤ρ2.
So, we get the proof of (i). Item (ii) can be obtained with the same estimates.
By the Contraction Mapping Theorem, there exists a unique fixed point(q2(q1),p(q1))∶=(q2(ϵ,N,q1),p(ϵ,N,q1)) of P in B. The bounds of (52) follow by the definition of ρ1 and ρ2.
Since the mapP∈C1(Q2⊕P×Q1;Q2⊕P×Q1), the Implicit Function Theorem implies that the map q1→(q2(ϵ,N,q1),p(ϵ,N,q1)) is C1. Differentiating both sides of (q2(q1),p(q1))=P(q2(q1),p(q1),q1), we can get
(60)q2′(q1)[h]=-L1-1ΠQ2(∂vf)(φ1,φ2,q1+q2(q1)+p(q1),ϵ)×(h+q2′(q1)[h]+p′(q1)[h]),p′(q1)[h]=-ϵLϵ-1ΠP(∂vf)(φ1,φ2,q1+q2(q1)+p(q1),ϵ)×(h+q2′(q1)[h]+p′(q1)[h]).
Using (43)–(48) and the Banach property of Hσ,s, we get
(61)|q2′(q1)[h]|σ,s≤C(R)N2(|h|σ,s+|q2′(q1)[h]|σ,s+|p′(q1)[h]|σ,s),|p′(q1)[h]|σ,s≤C(R)|ϵ|γ(|h|σ,s+|q2′(q1)[h]|σ,s+|p′(q1)[h]|σ,s),
which implies the bounds (53), since when C(R)(|ϵ|/γ+1/N2) is small enough, we can get
(62)det⁡|1-C(R)N2-C(R)N2-C(R)|ϵ|γ1-C(R)|ϵ|γ|≥12.


### 3.3. The(Q1)-Equation

Once the(Q2)-(P)-equations have been solved by (q2(q1),p(q1))∈Q2⊕P, there remains the finite-dimensional Q1-equation
(63)L1[q1]+ΠQ1f(φ1,φ2,q1+q2(q1)+p(q1),ϵ)=0.
The geometric interpretation of the construction of (q2(q1),p(q1)) is that, on the finite- dimensional submanifold Z:={q1+q2(q1)+p(q1):|q1|<2R}, diffeomorphic to the ball
(64)B2R:={q∈Q1:|q1|H1<2R},
the partial derivatives of the action functional Ψϵ with respect to the variables (q2,p) vanish. We claim that, at a critical point of Ψϵ restricted to Z, also the partial derivative of Ψϵ with respect to the variable q1 vanishes, and therefore such a point is critical also for the nonrestricted functional Ψϵ:Hσ,s→R.Actually, the bifurcation equation (63) is the Euler-Lagrange equation of the reduced action functional
(65)Φϵ,N:B2R⊂Q1⟶R,Φϵ,N(q1)∶=Ψϵ(q1+q2(q1+p(q1))).Lemma 8.
Φ
ϵ
,
N
∈
C
1
(
B
2
R
,
R
) and a critical point q1∈B2R of Φϵ,N is a solution of the bifurcation (63). Moreover, Φϵ,N can be written as
(66)Φϵ,N(q1)=const+ϵ(Γ(q1)+Rϵ,N(q1)),
where
(67)Γ(q1)∶=∫T2‍12(∂φ1q1)2+(2a+a2ϵ2)2(∂φ2q1)2kkk+a(1+ϵ)(∂φ1q1)(∂φ2q1)-a3(φ1,φ2)q144,Rϵ,N(q1)∶=∫T2‍F(φ1,φ2,q1,ϵ=0)kkk-F(φ1,φ2,q1+q2+p,ϵ)kkk+12f(φ1,φ2,v,ϵ)(q2+p),
and for some positive constant C2(R)≥C1(R), we can get
(68)|Rϵ,N(q1)|≤C2(R)(ϵ+|ϵ|γ+1N2),|Rϵ,N′(q1)[h]|≤C2(R)(ϵ+|ϵ|γ+1N2)|h|H1,kkkkkkkkkkkkkkkkk∀h∈Q1.Proof.
By (37) and (38), we have that, at v∶=q1+q2(q1)+p(q1),
(69)dΨϵ(v)[h]=0,∀h∈Q2,dΨϵ(v)[h]=0,∀h∈P.
Since q2′(q1)[h]∈Q2 and p′(q1)[h]∈P,∀h∈Q1, we deduce that
(70)dΦϵ,N(q1)[h]=dΨϵ(v)[[h]+q2′(q1)[h]+p′(q1)[h]]=dΨϵ(v)[h],∀h∈Q1,
and therefore v:=q1+q2(q1)+p(q1) solves also the (Q1)-equation (36). Write Ψϵ(v):=Ψϵ(2)(v)-ϵ∫T2‍F(φ1,v,ϵ), where
(71)Ψϵ(2)(v)=∫T2‍(1+12ϵ)(∂φ1v)2+ϵ2a+a2ϵ22(∂φ2v)2kkk+(1+ϵa(1+ϵ))(∂φ1v)(∂φ2v)
is a homogeneous functional of degree two. By homogeneity,
(72)Ψϵ(v)=12dΨϵ(2)(v)[v]-ϵ∫T2‍F(φ1,φ2,v,ϵ),
and, according to (69), we have
(73)dΨϵ(2)(q1+q2(q1)+p(q1))[q2(q1)+p(q1)]=ϵ∫T2‍f(φ1,φ2,v,ϵ)(q2(q1)+p(q1)).
Substituting the above equality into (72), we obtain, at v∶=q1+q2(q1)+p(q1),
(74)Φϵ,N(q1)=Ψϵ(q1+q2(q1)+p(q1))=12dΨϵ(2)(v)[q1+q2(q1)+p(q1)]-ϵ∫T2F(φ1,φ2,v,ϵ)=12dΨϵ(2)(q1)[q1]+12ϵ∫T2f(φ1,φ2,v,ϵ)(q2(q1)+p(q1))-ϵ∫T2F(φ1,φ2,v,ϵ)=∫T2(∂φ1q1)2+(∂φ1q1)(∂φ2q1)+ϵ(∫T212(∂φ1q1)2+(2a+a2ϵ2)2(∂φ2q1)2kkkkkik+a(1+ϵ)(∂φ1q1)(∂φ2q1)kkkkkik+12f(φ1,φ2,v,ϵ)(q2(q1)+p(q1))kkkkkik-F(φ1,φ2,v,ϵ)∫T212(∂φ1q1)2)=Ψ0(q1)+ϵ(∫T212(∂φ1q1)2+(2a+a2ϵ2)2(∂φ2q1)2kkkkkik+a(1+ϵ)(∂φ1q1)(∂φ2q1)kkkkkik+12f(φ1,φ2,v,ϵ)(q2(q1)+p(q1))kkkkkik-F(φ1,φ2,v,ϵ)∫T212(∂φ1q1)2+(2a+a2ϵ2)2).
Because Ψ0(q1)≡const, by (52), we can get the bounds of (68).The problem of finding nontrivial solutions of theQ1-equation is reduced to finding nontrivial critical points of the reduced action functional Φϵ,N in B2R. By (66), this is equivalent to finding critical points of the rescaled functional
(75)Φ^ϵ,N=Γ(q1)+Rϵ,N(q1)=(Q(q1)-∫T2a3(φ1,φ2)q144)+Rϵ,N(q1),
where the quadratic form
(76)Q(q)∶=∫T212(∂φ1q1)2+(2a+a2ϵ2)2(∂φ2q1)2kkk+a(1+ϵ)(∂φ1q1)(∂φ2q1)
is positive definite on Q+, negative definite on Q-, and zero definite on Q0. For q1=q++q-+q0∈Q1 and a<0, we have
(77)Q(q+)=∫T212(∂φ1q+)2+(2a+a2ϵ2)2(∂φ2q+)2kki+a(1+ϵ)(∂φ1q+)(∂φ2q+)=∫T2∑(l1,l2)∈Z2l22(-a-a2ϵ22)q^l1,l2eil1φ1eil2φ2=α+(a)2|q+|H12,Q(q0)=0,Q(q-)=∫T2∑(l1,l2)∈Z2l22(1+aϵ)22q^l1,l2eil1φ1eil2φ2=-α-(a)2|q-|H12,
where the positive constants α+(a),α-(a) are bounded away from zero and independent of ϵ. We will prove the existence of critical point of Φ^ϵ,N in B2R of linking type.


### 3.4. Linking Critical Points of the Reduced Action FunctionalΦ^ϵ,N

We cannot directly apply the linking theorem becauseΦ^ϵ,N is defined only in the ball B2R. Therefore, we first extend Φ^ϵ,N to the whole space Q1. We define the extended action function Φ~ϵ,N∈C1(Q1,R) as
(78)Φ~ϵ,N(q1)=Γ(q1)+R~ϵ,N(q1),
where R~ϵ,N(q1):Q1→R is
(79)R~ϵ,N(q1):=λ(|q1|H12R2)Rϵ,N(q1),
and λ:[0,+∞)→[0,1] is a smooth, nonincreasing, cut-off function such that
(80)λ(x)=1,if|x|≤1;λ(x)=0,if|x|≥4,|λ′(x)|<1.
By definition, Φ~ϵ,N(q1)≡Φ^ϵ,N(q1)on BR:={q1∈Q1:|q1|H1≤R} and Φ~ϵ,N(q1)=Γ(q1) outside B2R. Moreover, by (68), there is a constant C3(R)≥C2(R)>0 such that ∀|q|H1≤2R, and
(81)|R~ϵ,N(q1)|≤C3(R)(ϵ+|ϵ|γ-1+1N2),(82)|R~ϵ,N′(q1)[h]|≤C3(R)(ϵ+|ϵ|γ-1+1N2)|h|H1,kkkkkkkkkkkkkkkkkk∀h∈Q1.
Second, we will verify that Φ~ϵ,N(q1) satisfies the geometrical hypotheses of the linking theorem.Lemma 9.
There exist positive constantsρ,β,r1,r2>ρ, and 0<ϵ1(R)≤ϵ0(R),N1(R)≥N0(R), which are independent of (ϵ,N,γ), such that, ∀(|ϵ|/γ)≤ϵ1 and ∀N≥N1(R),(i)
inf⁡q1∈S+⁡Φ~ϵ,N(q1)≥β>0,∀q1∈S+∶={q1∈Q1∩Q+:|q1|H1=ρ},(ii)
∑q1∈∂W-Φ~ϵ,N(q1)≤β/2,∀q1∈∂W-,
whereW- is the rectangle in Q-⊕Q0,
(83)W-∶={q1=q0+q++re+,|q0+q-|≤r1,q-∈Q1∩Q-,kq0∈R,r∈[0,r2]},
and e+ is the unit vector in Q1⊕Q0.Proof.
(i)∀q+∈Q1∩Q+ with |q+|H1=ρ<R, we have
(84)Φ~ϵ,N(q+)=Φ^ϵ,N(q+)=Q(q+)-∫T2a3(φ1,φ2)q+44+Rϵ,N(q+)≥α+(a)2ρ2-κ1ρ4-C3(R)(ϵ+ϵγ+1N2).
Now, we fix ρ>0 small such that (α+(a)/2)ρ2-κ1ρ4≥(α+(a)/4)ρ2. Since C3(R)(ϵ+ϵ/γ+1/N2)≤(α+(a)/8)ρ2, by (84), we can get
(85)Φ~ϵ,N(q+)≥α+(a)8ρ2∶=β>0,∀q+∈Q1∩Q+,kkkkkwith|q+|H1=ρ.
(ii) Let(86)B1∶={q1=q0+q-+r2e+with|q0+q-|H1≤r1,kq-∈Q1∩Q-}⊂∂W-,(87)B2∶={q1=q0+q-+re+with|q0+q-|H1=r1,kq-∈Q1∩Q-,r∈[0,r2]}⊂∂W-,
with r1,r2>2R. For q1=q0+q-+re+∈B1∪B2,
(88)Φ~ϵ,N(q1)=γ(q1)=Q(q1)-∫T2a3(φ1,φ2)(q0+q-+re+)44≤-α-(a)2|q-|H12+r2Q(e+)-∫T2a3(φ1,φ2)(q0+q-+re+)44≤-α-(a)2|q-|H12+r2Q(e+)-α∫T2(q0+q-+re+)4,
because a3(φ1,φ2)/4≥α>0. Now, by Hölder inequality and orthogonality,
(89)∫T2(q0+q-+re+)4≥κ2(∫T2(q0+q-+re+)2)2=κ2(q02+q-2+r2e+2)2≥κ3(q02+r2)2≥κ3(q04+r4),
and by (88) we deduce that
(90)Φ~ϵ,N(q0+q-+re+)≤(κ4r2-κ3r4)-(α-(a)2|q-|H12+κ3q04).
Now, we fix r2 large such that κ4r22-κ3r24≤0, and therefore
(91)Φ~ϵ,N(q1)≤κ4r22-κ3r24≤0,∀q1∈B1.
Next, setting M∶=max⁡[0,r2]⁡κ4r2-κ3r4, we fix r1 large such that
(92)α-(a)2|q-|H12+κ3q04≥M,∀|q-+q0|=r1,
and therefore
(93)Φ~ϵ,N(q1)≤M-(α-(a)2|q-|H12+κ3q04)≤0,∀q1∈B2.
Finally, if q1=q-+q0,
(94)Φ~ϵ,N(q1)=Q(q-)-∫T2a3(φ1,φ2)q144+R~ϵ,N(q1)≤|R~ϵ,N(q1)|≤C3(R)(ϵ+|ϵ|γ-1+1N2).
So, if C3(R)(ϵ+|ϵ|γ-1+(1/N2))≤β/2, we can get Φ~ϵ,N(q1)≤β/2,∀q1∈∂W-.We introduce the minimax classS∶={ψ∈C(W--,Q)∣ψ=Idon⁡∂W-}. According to Proposition 5.9 of [19], the maps of S have an important intersection property as follows.Proposition 10.
(S+ and W- link with respect to ψ). Consider
(95)ψ∈S⟹ψ(W-)∩S+≠∅.One defines the minimax linking level as follows:(96)c~ϵ∶=inf⁡ψ∈S⁡max⁡q1∈W-⁡Φ~ϵ,N(ψ(q1)).
Obviously, by Proposition 10 and Lemma 9,
(97)max⁡q1∈W-⁡Φ~ϵ,N(ψ(q1))≥min⁡q1∈S+⁡Φ~ϵ,N(q1)≥β>0,∀ψ∈S,
and, therefore, c~ϵ>β>0.SinceId∈S, so
(98)c~ϵ≤max⁡q∈W-⁡Φ~ϵ,N((q1))≤max⁡q∈W-⁡(Γ(q1)+R~ϵ,N(q1))≤max⁡q∈W-⁡(α+(a)2|q+|H12+α-(a)2|q-|H12+∫T2κq14)+1≤c~∞≤+∞,
where c~∞ is independent of N,ϵ,γ. By the linking theorem, we deduce the existence of a Palais-Smale sequence (qn)∈Q1 at the level c~ϵ, namely,
(99)Φ~ϵ,N(qn)⟶c~ϵ,Φ~ϵ,N′(qn)⟶0whenn⟶∞.
Third, we will prove that the Palais-Smale sequence (qn) (up to subsequence) converges to some nontrivial critical point q^1≠0 in some open ball of Q1, where Φ~ϵ,N and Φ^ϵ,N coincide. Because the space Q1 is finite dimensional, we only need to prove that the sequence (qn) is bounded.Lemma 11.
There is a constantM>0 independent of R,ϵ,N,γ, such that, for all |ϵ|/γ small enough and N large enough, the Palais-Smale sequence (qn) is bounded; that is, |qn|H1<M. So, there exists a subsequence of the P-S sequence that converges to some critical point q^1≠0, and the functional Φ~ϵ,N possesses a nontrivial critical point q^1∈Q1 with critical value Φ~ϵ,N(q^1)=c~ϵ.Proof.
In the sequel, we will always assume that(ϵ+|ϵ|γ-1+1/N2)<1. Writing Φ~ϵ,N(q)=Γ(q)+R~ϵ,N(q), by (81)-(82), we can get
(100)Φ~ϵ,N(qn)-12Φ~ϵ,N′(qn)[qn]=Γ(qn)-12Γ′(qn)[qn]+(R~ϵ,N(qn)-12R~ϵ,N′(qn)[qn])=(12-14)∫T2a3(φ1)qj4+(R~ϵ,N(qn)-12R~ϵ,N′(qn)[qn])≥α(12-14)∫T2qj4-(ϵ+|ϵ|γ-1+1N2).
Then, by c~ϵ<c~∞<+∞ and Φ~ϵ,N′(qn)→0(n→∞),
(101)c~∞+1+|qn|H1≥κ6∫T2qn4:=κ6|qn|L44.
By Hölder inequality and orthogonality,
(102)c~∞+1+|qn|H1≥κ7(∫T2(q0,n+q-,n+q+,n)2)2=κ7(∫T2q0,n2+q-,n2+q+,n2)2≥κ8(q0,n)4,
and therefore |q0,n|≤(1+|qn|H1)1/4. In the same way, by Hölder inequality and (82),
(103)Φ~ϵ,N′(qn)[q+,n]=α+(a)|q+,n|H12-∫T2a3(φ1)qn3q+,n+R~ϵ,N′(qn)[q+,n]≥α+(a)|q+,n|H12-κ9|q+,n|H1∫T2|qn|3-C3(R)(ϵ+|ϵ|γ-1+1N3/2)|q+,n|H1≥κ10|q+,n|H1(|q+,n|H1-|qn|L43-1).
By (101) and the above inequalities, using Φ~ϵ,N′(qn)→0(n→∞), we conclude that |q+,n|H1≤κ11(1+|qn|H13/4). Estimating analogously, we derive |q-,n|H1≤κ12(1+|qn|H13/4). Finally, we deduce that
(104)|qn|H1=|q0,n|+|q+,n|H1+|q-,n|H1≤κ13(1+|qn|H13/4+|qn|H13/4).
So, we can conclude that |qn|H1≤M for a suitable constant M>0, which is independent of R,ϵ,N,γ. Since Q1 is finite dimensional, {qn} converges, up to subsequence, to some critical point q^ of Φ~ϵ,N with |q^|H1<M. Since Φ~ϵ,N(q^)=c~ϵ≥β>0, we conclude that q^≠0.Proof of Theorem1.
Let us fixR-∶=M+1 and take |ϵ|γ-1≤ϵ2(R-):=ϵ-. According to Lemma 7, we can get, for 0<σN(R-)≤1,0≤s<1/2, a solution (q2(q1),p(q1))∈(Q2(N-)⊕P)∩Hσ,s of the (Q2)-(P)-equations with ∀|q1|H1≤2R. By Lemma 11, the extended functional Φ~ϵ,N(q1) possesses a nontrivial critical point q^ with |q1|H1≤M<R-. Since Φ~ϵ,N coincides with Φϵ,N on the ball BR-, by Lemma 8, there exists a nontrivial weak solution q^1+q2(q^1)+p(q^1)∈Hσ,s of (23). Finally, u=ϵ[q^1+q2(q^1)+p(q^1)]=ϵ[q^ϵ+p(q^1)] solves (7).
According to Lemma6, by the regularizing property of the operator L1, the solution q^ϵ∶=q^1+q2(q^1) of the (Q)-equation belongs to Hσ,s+2∩Q. By the P-equation
(105)((2+ϵ)∂φ12+2(1+a(ϵ+ϵ2))∂φ1∂φ2)p^=-ϵ((2a+a2ϵ2)∂φ22+ΠPf(φ1,φ2,q^ϵ+p,ϵ))p^,
where p^=p(q^1), we can get that
(106)-ϵ((2a+a2ϵ2)∂φ22+ΠPf(φ1,φ2,q^ϵ+p,ϵ))p^∈Hσ′,s,
for 0<σ′<σ, satisfying |ϵ(2a+a2ϵ2)l22|e|l2|σ′<e|l2|σ. For (a,ϵ)∈Bγ, the eigenvalues of operator (2+ϵ)∂φ12+2(1+a(ϵ+ϵ2))∂φ1∂φ2 restricted to P satisfy |(2+ϵ)l12+2(1+aϵ+aϵ2)l1l2|≥|2+ϵ|(γ|l1|/|l2|), ∀l1≠0,l1+l2≠0, and, thus, we deduce that p^∈Hσ′′,s+1, for all 0<σ′′<σ′ (satisfying C|l2|e|l2|σ′′<e|l2|σ′), and |∂φ1p^|σ′′,s=O(|ϵ|/γ). By (105),
(107)(2+ϵ)∂φ12p^=-2(1+a(ϵ+ϵ2))∂φ1∂φ2p^-ϵ((2a+a2ϵ2)∂φ22+ΠPf(φ1,φ2,q^ϵ+p,ϵ))p^,
we get p^∈Hσ-,s+2, and |p^|σ-,s+2=O(|ϵ|γ-1), with 0<σ-<σ′′. Thus, (15) follows with s-=s+2, 0<σ-<σ′′. By (5), u(t,x)=ϵv((1+ϵ)t+x,(1+aϵ2)t+x) is the solution of (1), for all (a,ϵ)∈Bγ. To show that u(t,x) is quasiperiodic, it remains to prove that v depends on both variables (φ1,φ2) independently. According to Lemma 9, ∀|q^1|H1≤R-,Φ~ϵ,N(q^1)≥β. On the other hand, Φ~ϵ,N(q^-+q^0)≤β/2, ∀|q^-+q^0|H1≤R-, so that q^1∉Q-⊕Q0, and, therefore, v depends on φ2. In fact, any solution v of (23) depending only on φ2, that is, the solutions of
(108)(2a+a2ϵ2)d2v(φ2)dφ22+f(φ1,φ2,v(φ2),ϵ)=0,
is v(φ2)≡0. Indeed, by the homogeneity of f(φ1,φ2,v,ϵ), we have
(109)ϵ3f(φ1,φ2,v,ϵ)=f(φ1,φ2,ϵv)=∑k=3∞ak(φ1,φ2)(ϵv)k.
Now consider a smooth function h(φ1) with zero mean, ant it satisfies ∫T‍ak(φ1,φ2)h(φ1)dφ1≠0 for some k. Multiplying (108) by h(φ1) and integrating over [0,π], we have(110)∫02π‍(2a+a2ϵ2)d2v(φ2)dφ22h(φ1)dφ1+∫02π‍f(φ1,φ2,v(φ2),ϵ)h(φ1)dφ1=0.
According to h(φ1) that has zero mean and multiplying above equation by ϵ3, we get
(111)∫02πf(φ1,φ2,ϵv(φ2))h(φ1)dφ1=∑k=3∞(ϵv(φ2))k∫02πak(φ1,φ2)h(φ1)dφ1=0.
The function G(z(φ2))=∑k=3∞bk(φ2)(z(φ2))k, with bk=∫02πak(φ1,φ2)h(φ1)dφ1, is a nontrivial analytic function. Thus, the equation G(ϵv(φ2))=0 cannot have a sequence of zeros accumulating to zero. So, for ϵ small enough, v(φ2)≡0.


## 3.1. The Variational Lyapunov-Schmidt Reduction

The operatorLa,ϵ is diagonal defined on the Banach space Hσ,s under the Fourier basis el=eil1φ1eil2φ2 with eigenvalues
(26)-Dl=(2l12+2l1l2)+ϵ(l12+(2a+a2ϵ2)l22+2a(1+ϵ)l1l2)=(l1+aϵl2)((2+ϵ)l1+(2+aϵ2)l2).
So, we have
(27)La,ϵ[v]=∑(l1,l2)∈Z2Dlv^l1,l2eil1φ1eil2φ2,∀v∈Hσ,s.
The critical points of the unperturbed functional Ψ0:Hσ,s→R form an infinite-dimensional linear space Q, and they are the solutions of the equation
(28)L0q=(2∂φ12+2∂φ1∂φ2)q=0.
The space Q can be written as
(29)Q={q=∑(l1,l2)∈Z2q^l1,l2eil1φ1eil2φ2∈Hσ,s∣q^l1,l2=0,kkforl1(2l1+2l2)≠0∑(l1,l2)∈Z2q^l1,l2eil1φ1eil2φ2∈Hσ,s∣q^l1,l2=0,}.
In view of the variational argument that we will use to solve the bifurcation equation, we split Q as Q=Q++Q0+Q-, where
(30)Q+∶={q∈Q:q^l1,l2=0,for(l1,l2)∉Λ+}={q+∶=q+(φ)∈Hσ,s0},Q0∶={q:q0,0∈R},Q-∶={q∈Q:q^l1,l2=0,for(l1,l2)∉Λ-}={q-∶=q-(φ)∈Hσ,s0},
with
(31)Λ+∶={(l1,l2)∈Z2:l1=0,l2≠0},Λ-∶={(l1,l2)∈Z2:l1+l2=0,(l1,l2)≠(0,0)},Λ0∶={(l1,l2)∈Z2:(l1,l2)≡(0,0)}.
We will also use in Q the norm
(32)|q|H12=|q+|H1(T)2+q0,02+|q-|H1(T)2~∑(l1,l2)∈Λ+∪Λ0∪Λ+q^l1,l22(|l1|2+|l2|2+1).
So, we can decompose the space Hσ,s=Q+P, where
(33)P∶={p=∑l1,l2∈Zp^l1,l2eil1φ1eil2φ2∈Hσ,s∣p^l1,l2=0,kkkforl1(2l1+2l2)=0∑l1,l2∈Zp^l1,l2eil1φ1eil2φ2∈Hσ,s∣p^l1,l2=0,}.
Projecting (23) onto the closed subspaces Q and P, setting v=q+p∈Hσ,s with q∈Q and p∈P, we obtain
(34)(Q)L1[q]+ΠQf(φ1,φ2,q+p,ϵ)=0,(P)La,ϵ[p]+ϵΠPf(φ1,φ2,q+p,ϵ)=0,
where ΠQ:Hσ,s→Q,ΠP:Hσ,s→P are the projectors, respectively, onto Q and P.In order to prove analyticity of the solutions and to highlight the compactness of the problem, we perform a finite-dimensional Lyapunov-Schmidt reduction, introducing the decompositionQ=Q1+Q2, where
(35)Q1∶={q=∑|l1|+|l2|≤Nq^l1,l2eil1φ1eil2φ2∈Q},Q2∶={q=∑|l1|+|l2|≤Nq^l1,l2eil1φ1eil2φ2∈Q}.
Setting q=q1+q2 with q1∈Q1 and q2∈Q2, we finally get
(36)(Q1)L1[q1]+ΠQ1[f(φ1,φ2,q1+q2+p,ϵ)]=0⟺dΨϵ(v)[h]=0,∀h∈Q1;(37)(Q2)L1[q2]+ΠQ2[f(φ1,φ2,q1+q2+p,ϵ)]=0⟺dΨϵ(v)[h]=0,∀h∈Q2;(38)(P)La,ϵ[p]+ϵΠP[f(φ1,φ2,q1+q2+p,ϵ)]=0⟺dΨϵ(v)[h]=0,∀h∈P,
where ΠQi:Hσ,s→Qi are the projectors onto Qi  (i=1,2), and ΠP:Hσ,s→P is the projector onto P. We will solve first the (Q2)-(P)-equations for all |q1|H1≤2R, provided ϵ belongs to a suitable Cantor-like set, |ϵ|≤ϵ0(R) is sufficiently small, and N≥N0(R) is large enough (see Lemma 7). Next, we will solve the Q1-equation by means of a variational linking argument; see Section 3.4.


## 3.2. The(Q2)-(P)-Equations

We first prove thatLa,ϵ restricted to P has a bounded inverse when (a,ϵ) belongs to the uncountable zero-measure set
(39)Bγ∶={(2+ϵ2+aϵ2,aϵ)(a,ϵ)∈R-×R,1+ϵ≠0,1+aϵ2≠0,2+aϵ2≠0,kkkk(2+ϵ2+aϵ2,aϵ)∈Cγ1∩(1-ϵ0,1+ϵ0)kkkk×Cγ2∩(-ϵ0,ϵ0),1+ϵ1+aϵ2∉Q,forϵ0∈(0,12)(2+ϵ2+aϵ2,aϵ)},
where Cγi, i=1,2, is a set of badly approximate numbers defined by
(40)Cγ1∶={2+ϵ2+aϵ2:|l2+2+ϵ2+aϵ2l1|>γ|l1|},Cγ2∶={aϵ:|l1+aϵl2|>γ|l2|},
for ∀l1,l2∈Z∖{0}, and 0<γ<1/4. Cγi, i=1,2, accumulate at 1 and zero, respectively, from both the right and the left; see [21, 31, 35].The operatorLϵ is diagonal in the Fourier basis {eil1φ1eil2φ2,(l1,l2)∈Z2} with eigenvalues
(41)Dl1,l2=-(l1+aϵl2)((2+ϵ)l1+(2+aϵ2)l2)=-(2+aϵ2)(l1+aϵl1)(l2+2+ϵ2+aϵ2l1).Lemma 5.
For(a,ϵ)∈Bγ, the eigenvalues Dl1,l2 of La,ϵ restricted to P satisfy
(42)|Dl1,l2|=|(l1+aϵl2)((2+ϵ)l1+(2+aϵ2)l2)|>γ,kkkkkkkkkkkkkkkkkkkkkkkkklkkkk∀l1,l2≠0.
As a consequence, the operator La,ϵ:P→P has a bounded inverse La,ϵ-1 and satisfies
(43)|La,ϵ-1[h]|σ,s≤|h|σ,sγ,∀h∈P.Proof.
Denote by[x] the nearest integer close to x and by {x}=x-[x] its fractional part. If both l1≠-[aϵl2] and l2≠-[((2+ϵ)/(2+aϵ2))l1], we have
(44)|Dl1,l2|=|(l1+aϵl2)|·|(2+aϵ2)||(l2+2+ϵ2+aϵ2l1)|>1.
If l1=-[aϵl2], then |l1|<(1/2)|l2|, so that |l2+((2+ϵ)/(2+aϵ2))l1|>(1/2)|l2|. This implies that
(45)|Dl1,l2|=|l1+aϵl2|·|2+aϵ2||2+ϵ2+aϵ2l1+l2|≥γ|l2|·|(ϵ{aϵ2+ϵl2}+2l2+aϵ(1-ϵ2+ϵ)l2)(2+ϵ)kkkkkkkkk×(ϵ{aϵ2+ϵl2}+2l2+aϵ(1-ϵ2+ϵ)l2)|≥γ.
In the same way, if l2=-[((2+ϵ)/(2+aϵ2))l1], we have
(46)|Dl1,l2|≥γ|l1|·|(2+aϵ2)(ϵ{(2+ϵ)a2+aϵ2l1}+2-2aϵ2+aϵ2l1)|≥γ.
So, the operator La,ϵ restricted to P has a bounded inverse and satisfies
(47)|La,ϵ-1[h]|σ,s=∑(l1,l2)∈Z2|h^l1,l2|e|l2|σ(max⁡⁡{0,|l1|})sDl1,l2≤|h|σ,sγ,∀h∈P.Lemma 6.
The operatorL1:Q2→Q2 has a bounded inverse L1-1, satisfying
(48)|L1-1[h]|σ,s≤|h|σ,sN2.Proof.
L
1 is diagonal in the Fourier basis of Q:eil1φ1eil2φ2 with (l1,l2)∈Λ+∪Λ0∪Λ- with eigenvalues
(49)dl1,l2={(1+ϵ(-2a+a2ϵ))l22,ifl1+l2=0,a(2+aϵ2)l22,ifl1=0.
The eigenvalues of L1 restricted to Q2(N) verify |dl1,l2|≥N2/C, where the constant C depends on (ϵ,a), and (48) holds.Fixed points of the nonlinear operatorP:Q2⊕P→Q2⊕P defined by
(50)P(q2,p,q1)∶=(-L1-1ΠQ2f(φ1,φ2,q1+q2+p,ϵ),k-ϵLa,ϵ-1ΠPf(φ1,φ2,q1+q2+p,ϵ))
are solutions of the (Q2)-(P)-equations. Using the Contraction Mapping Theorem, we can prove the following lemma.Lemma 7.
For anyR>0, there exist an integer N0(R)∈N+ and positive constants ϵ0(R)>0,C0(R)>0 such that
(51)∀|q1|H1≤2R,∀ϵ∈Bγ,|ϵ|γ-1≤ϵ0(R),∀N≥N0(R):0≤Nσ≤1,
and there exists a unique solution (q2(q1),p(q1)):=(q2(ϵ,N,q1),p(ϵ,N,q1))∈Q2⊕P of the (Q2)-(P)-equations satisfying
(52)|q2(ϵ,N,q1)|σ,s≤C0(R)N2,|p(ϵ,N,q1)|σ,s≤C0(R)|ϵ|γ-1.
Moreover, the map q1→(q2(q1),p(q1)) is C1(B2R,Q2⊕P) and
(53)|q2′(q1)[h]|σ,s≤C0(R)N2|h|H1,|p′(q1)[h]|σ,s≤C0(R)|ϵ|γ-1|h|H1∀h∈Q1.Proof.
Let us consider the ball(54)B∶={(q2,p)∈Q2⊕P,|q2|σ,s≤ρ1,|p|σ,s≤ρ2}
with norm |(q2,p)|σ,s∶=|q2|σ,s+|p|σ,s. We can claim that, under assumptions (51), there exist ρ1,ρ2∈(0,1) such that the map (q2,p)→P(q2,p;q1) is a contraction mapping in B, that is, we have to prove (i)
(q2,p)∈B⇒P(q2,p;q1)∈B;(ii)
|P(q2,p;q1)-P(q^2,p^;q1)|σ,s≤η|(q2,p)-(q^2,p^)|σ,s,∀(q2,p),(q^2,p^)∈B,
where the constantη∈(0,1). In the following, κi  (i=1,2,…,5) denote different constants. By (48) and the Banach property of Hσ,s,
(55)|P1(q2,p;q1)|σ,s=|L1-1ΠQ2f(φ1,φ2,q1+q2+p,ϵ)|≤κ1N2(|q1|σ,s3+|q2|σ,s3+|p|σ,s3).
Similarly, for (a,ϵ)∈Bγ, by (43), we have
(56)|P2(q2,p;q1)|σ,s=|ϵLa,ϵ-1ΠPf(φ1,φ2,q1+q2+p,ϵ)|≤κ2|ϵ|γ(|q1|σ,s3+|q2|σ,s3+|p|σ,s3).
For all q1∈Q1(N), setting [x]=max⁡⁡{0,|x|}, we can get
(57)|q1|σ,s=∑|l2|≤N,l1=0|q^0,l2|e|l2|σ+∑|l2|≤N,l1+l2=0|q^-l2,l2|e|l2|σ[-l2]s≤eNσ(∑|l2|≤N,l1=0|q^l1,l2|([-l2-l22])skkkkk+∑|l2|≤N,l1+l2=0|q^-l2,l2|[-l2]s)≤κ3((∑|l2|≤N,l1=0|q^0,l2|2[l2]2)1/2(∑l2∈Z1[l2]2)1/2kkkk+(∑|l2|≤N,l1+l2=0|q^-l2,l2|2[l2]2)1/2kkkk×(∑l2∈Z1([l2])2(1-s))1/2(∑|l2|≤N,l1=0|q^0,l2|2[l2]2)1/2)≤κ4|q1|H1
whenever 0≤Nσ≤1,0≤s<1/2. Thus, ∀|q1|H1≤2R,|q2|σ,s≤ρ1,|p|σ,s≤ρ2, we get
(58)|P1(q2,p;q1)|σ,s≤κ5N2(R3+ρ13+ρ23),|P2(q2,p;q1)|σ,s≤κ5|ϵ|γ(R3+ρ13+ρ23).
Now, set C0(R)∶=κ5R3, and we define ρ1∶=2C0(R)/N3/2,ρ2∶=2C0(R)(|ϵ|/γ). By the inequality above, there exist N0(R)∈N+ and ϵ0(R)>0 such that ∀N≥N0(R) and ∀|ϵ|γ-1≤ϵ0(R),
(59)|P1(q2,p;q1)|σ,s≤ρ1,|P2(q2,p;q1)|σ,s≤ρ2.
So, we get the proof of (i). Item (ii) can be obtained with the same estimates.
By the Contraction Mapping Theorem, there exists a unique fixed point(q2(q1),p(q1))∶=(q2(ϵ,N,q1),p(ϵ,N,q1)) of P in B. The bounds of (52) follow by the definition of ρ1 and ρ2.
Since the mapP∈C1(Q2⊕P×Q1;Q2⊕P×Q1), the Implicit Function Theorem implies that the map q1→(q2(ϵ,N,q1),p(ϵ,N,q1)) is C1. Differentiating both sides of (q2(q1),p(q1))=P(q2(q1),p(q1),q1), we can get
(60)q2′(q1)[h]=-L1-1ΠQ2(∂vf)(φ1,φ2,q1+q2(q1)+p(q1),ϵ)×(h+q2′(q1)[h]+p′(q1)[h]),p′(q1)[h]=-ϵLϵ-1ΠP(∂vf)(φ1,φ2,q1+q2(q1)+p(q1),ϵ)×(h+q2′(q1)[h]+p′(q1)[h]).
Using (43)–(48) and the Banach property of Hσ,s, we get
(61)|q2′(q1)[h]|σ,s≤C(R)N2(|h|σ,s+|q2′(q1)[h]|σ,s+|p′(q1)[h]|σ,s),|p′(q1)[h]|σ,s≤C(R)|ϵ|γ(|h|σ,s+|q2′(q1)[h]|σ,s+|p′(q1)[h]|σ,s),
which implies the bounds (53), since when C(R)(|ϵ|/γ+1/N2) is small enough, we can get
(62)det⁡|1-C(R)N2-C(R)N2-C(R)|ϵ|γ1-C(R)|ϵ|γ|≥12.


## 3.3. The(Q1)-Equation

Once the(Q2)-(P)-equations have been solved by (q2(q1),p(q1))∈Q2⊕P, there remains the finite-dimensional Q1-equation
(63)L1[q1]+ΠQ1f(φ1,φ2,q1+q2(q1)+p(q1),ϵ)=0.
The geometric interpretation of the construction of (q2(q1),p(q1)) is that, on the finite- dimensional submanifold Z:={q1+q2(q1)+p(q1):|q1|<2R}, diffeomorphic to the ball
(64)B2R:={q∈Q1:|q1|H1<2R},
the partial derivatives of the action functional Ψϵ with respect to the variables (q2,p) vanish. We claim that, at a critical point of Ψϵ restricted to Z, also the partial derivative of Ψϵ with respect to the variable q1 vanishes, and therefore such a point is critical also for the nonrestricted functional Ψϵ:Hσ,s→R.Actually, the bifurcation equation (63) is the Euler-Lagrange equation of the reduced action functional
(65)Φϵ,N:B2R⊂Q1⟶R,Φϵ,N(q1)∶=Ψϵ(q1+q2(q1+p(q1))).Lemma 8.
Φ
ϵ
,
N
∈
C
1
(
B
2
R
,
R
) and a critical point q1∈B2R of Φϵ,N is a solution of the bifurcation (63). Moreover, Φϵ,N can be written as
(66)Φϵ,N(q1)=const+ϵ(Γ(q1)+Rϵ,N(q1)),
where
(67)Γ(q1)∶=∫T2‍12(∂φ1q1)2+(2a+a2ϵ2)2(∂φ2q1)2kkk+a(1+ϵ)(∂φ1q1)(∂φ2q1)-a3(φ1,φ2)q144,Rϵ,N(q1)∶=∫T2‍F(φ1,φ2,q1,ϵ=0)kkk-F(φ1,φ2,q1+q2+p,ϵ)kkk+12f(φ1,φ2,v,ϵ)(q2+p),
and for some positive constant C2(R)≥C1(R), we can get
(68)|Rϵ,N(q1)|≤C2(R)(ϵ+|ϵ|γ+1N2),|Rϵ,N′(q1)[h]|≤C2(R)(ϵ+|ϵ|γ+1N2)|h|H1,kkkkkkkkkkkkkkkkk∀h∈Q1.Proof.
By (37) and (38), we have that, at v∶=q1+q2(q1)+p(q1),
(69)dΨϵ(v)[h]=0,∀h∈Q2,dΨϵ(v)[h]=0,∀h∈P.
Since q2′(q1)[h]∈Q2 and p′(q1)[h]∈P,∀h∈Q1, we deduce that
(70)dΦϵ,N(q1)[h]=dΨϵ(v)[[h]+q2′(q1)[h]+p′(q1)[h]]=dΨϵ(v)[h],∀h∈Q1,
and therefore v:=q1+q2(q1)+p(q1) solves also the (Q1)-equation (36). Write Ψϵ(v):=Ψϵ(2)(v)-ϵ∫T2‍F(φ1,v,ϵ), where
(71)Ψϵ(2)(v)=∫T2‍(1+12ϵ)(∂φ1v)2+ϵ2a+a2ϵ22(∂φ2v)2kkk+(1+ϵa(1+ϵ))(∂φ1v)(∂φ2v)
is a homogeneous functional of degree two. By homogeneity,
(72)Ψϵ(v)=12dΨϵ(2)(v)[v]-ϵ∫T2‍F(φ1,φ2,v,ϵ),
and, according to (69), we have
(73)dΨϵ(2)(q1+q2(q1)+p(q1))[q2(q1)+p(q1)]=ϵ∫T2‍f(φ1,φ2,v,ϵ)(q2(q1)+p(q1)).
Substituting the above equality into (72), we obtain, at v∶=q1+q2(q1)+p(q1),
(74)Φϵ,N(q1)=Ψϵ(q1+q2(q1)+p(q1))=12dΨϵ(2)(v)[q1+q2(q1)+p(q1)]-ϵ∫T2F(φ1,φ2,v,ϵ)=12dΨϵ(2)(q1)[q1]+12ϵ∫T2f(φ1,φ2,v,ϵ)(q2(q1)+p(q1))-ϵ∫T2F(φ1,φ2,v,ϵ)=∫T2(∂φ1q1)2+(∂φ1q1)(∂φ2q1)+ϵ(∫T212(∂φ1q1)2+(2a+a2ϵ2)2(∂φ2q1)2kkkkkik+a(1+ϵ)(∂φ1q1)(∂φ2q1)kkkkkik+12f(φ1,φ2,v,ϵ)(q2(q1)+p(q1))kkkkkik-F(φ1,φ2,v,ϵ)∫T212(∂φ1q1)2)=Ψ0(q1)+ϵ(∫T212(∂φ1q1)2+(2a+a2ϵ2)2(∂φ2q1)2kkkkkik+a(1+ϵ)(∂φ1q1)(∂φ2q1)kkkkkik+12f(φ1,φ2,v,ϵ)(q2(q1)+p(q1))kkkkkik-F(φ1,φ2,v,ϵ)∫T212(∂φ1q1)2+(2a+a2ϵ2)2).
Because Ψ0(q1)≡const, by (52), we can get the bounds of (68).The problem of finding nontrivial solutions of theQ1-equation is reduced to finding nontrivial critical points of the reduced action functional Φϵ,N in B2R. By (66), this is equivalent to finding critical points of the rescaled functional
(75)Φ^ϵ,N=Γ(q1)+Rϵ,N(q1)=(Q(q1)-∫T2a3(φ1,φ2)q144)+Rϵ,N(q1),
where the quadratic form
(76)Q(q)∶=∫T212(∂φ1q1)2+(2a+a2ϵ2)2(∂φ2q1)2kkk+a(1+ϵ)(∂φ1q1)(∂φ2q1)
is positive definite on Q+, negative definite on Q-, and zero definite on Q0. For q1=q++q-+q0∈Q1 and a<0, we have
(77)Q(q+)=∫T212(∂φ1q+)2+(2a+a2ϵ2)2(∂φ2q+)2kki+a(1+ϵ)(∂φ1q+)(∂φ2q+)=∫T2∑(l1,l2)∈Z2l22(-a-a2ϵ22)q^l1,l2eil1φ1eil2φ2=α+(a)2|q+|H12,Q(q0)=0,Q(q-)=∫T2∑(l1,l2)∈Z2l22(1+aϵ)22q^l1,l2eil1φ1eil2φ2=-α-(a)2|q-|H12,
where the positive constants α+(a),α-(a) are bounded away from zero and independent of ϵ. We will prove the existence of critical point of Φ^ϵ,N in B2R of linking type.


## 3.4. Linking Critical Points of the Reduced Action FunctionalΦ^ϵ,N

We cannot directly apply the linking theorem becauseΦ^ϵ,N is defined only in the ball B2R. Therefore, we first extend Φ^ϵ,N to the whole space Q1. We define the extended action function Φ~ϵ,N∈C1(Q1,R) as
(78)Φ~ϵ,N(q1)=Γ(q1)+R~ϵ,N(q1),
where R~ϵ,N(q1):Q1→R is
(79)R~ϵ,N(q1):=λ(|q1|H12R2)Rϵ,N(q1),
and λ:[0,+∞)→[0,1] is a smooth, nonincreasing, cut-off function such that
(80)λ(x)=1,if|x|≤1;λ(x)=0,if|x|≥4,|λ′(x)|<1.
By definition, Φ~ϵ,N(q1)≡Φ^ϵ,N(q1)on BR:={q1∈Q1:|q1|H1≤R} and Φ~ϵ,N(q1)=Γ(q1) outside B2R. Moreover, by (68), there is a constant C3(R)≥C2(R)>0 such that ∀|q|H1≤2R, and
(81)|R~ϵ,N(q1)|≤C3(R)(ϵ+|ϵ|γ-1+1N2),(82)|R~ϵ,N′(q1)[h]|≤C3(R)(ϵ+|ϵ|γ-1+1N2)|h|H1,kkkkkkkkkkkkkkkkkk∀h∈Q1.
Second, we will verify that Φ~ϵ,N(q1) satisfies the geometrical hypotheses of the linking theorem.Lemma 9.
There exist positive constantsρ,β,r1,r2>ρ, and 0<ϵ1(R)≤ϵ0(R),N1(R)≥N0(R), which are independent of (ϵ,N,γ), such that, ∀(|ϵ|/γ)≤ϵ1 and ∀N≥N1(R),(i)
inf⁡q1∈S+⁡Φ~ϵ,N(q1)≥β>0,∀q1∈S+∶={q1∈Q1∩Q+:|q1|H1=ρ},(ii)
∑q1∈∂W-Φ~ϵ,N(q1)≤β/2,∀q1∈∂W-,
whereW- is the rectangle in Q-⊕Q0,
(83)W-∶={q1=q0+q++re+,|q0+q-|≤r1,q-∈Q1∩Q-,kq0∈R,r∈[0,r2]},
and e+ is the unit vector in Q1⊕Q0.Proof.
(i)∀q+∈Q1∩Q+ with |q+|H1=ρ<R, we have
(84)Φ~ϵ,N(q+)=Φ^ϵ,N(q+)=Q(q+)-∫T2a3(φ1,φ2)q+44+Rϵ,N(q+)≥α+(a)2ρ2-κ1ρ4-C3(R)(ϵ+ϵγ+1N2).
Now, we fix ρ>0 small such that (α+(a)/2)ρ2-κ1ρ4≥(α+(a)/4)ρ2. Since C3(R)(ϵ+ϵ/γ+1/N2)≤(α+(a)/8)ρ2, by (84), we can get
(85)Φ~ϵ,N(q+)≥α+(a)8ρ2∶=β>0,∀q+∈Q1∩Q+,kkkkkwith|q+|H1=ρ.
(ii) Let(86)B1∶={q1=q0+q-+r2e+with|q0+q-|H1≤r1,kq-∈Q1∩Q-}⊂∂W-,(87)B2∶={q1=q0+q-+re+with|q0+q-|H1=r1,kq-∈Q1∩Q-,r∈[0,r2]}⊂∂W-,
with r1,r2>2R. For q1=q0+q-+re+∈B1∪B2,
(88)Φ~ϵ,N(q1)=γ(q1)=Q(q1)-∫T2a3(φ1,φ2)(q0+q-+re+)44≤-α-(a)2|q-|H12+r2Q(e+)-∫T2a3(φ1,φ2)(q0+q-+re+)44≤-α-(a)2|q-|H12+r2Q(e+)-α∫T2(q0+q-+re+)4,
because a3(φ1,φ2)/4≥α>0. Now, by Hölder inequality and orthogonality,
(89)∫T2(q0+q-+re+)4≥κ2(∫T2(q0+q-+re+)2)2=κ2(q02+q-2+r2e+2)2≥κ3(q02+r2)2≥κ3(q04+r4),
and by (88) we deduce that
(90)Φ~ϵ,N(q0+q-+re+)≤(κ4r2-κ3r4)-(α-(a)2|q-|H12+κ3q04).
Now, we fix r2 large such that κ4r22-κ3r24≤0, and therefore
(91)Φ~ϵ,N(q1)≤κ4r22-κ3r24≤0,∀q1∈B1.
Next, setting M∶=max⁡[0,r2]⁡κ4r2-κ3r4, we fix r1 large such that
(92)α-(a)2|q-|H12+κ3q04≥M,∀|q-+q0|=r1,
and therefore
(93)Φ~ϵ,N(q1)≤M-(α-(a)2|q-|H12+κ3q04)≤0,∀q1∈B2.
Finally, if q1=q-+q0,
(94)Φ~ϵ,N(q1)=Q(q-)-∫T2a3(φ1,φ2)q144+R~ϵ,N(q1)≤|R~ϵ,N(q1)|≤C3(R)(ϵ+|ϵ|γ-1+1N2).
So, if C3(R)(ϵ+|ϵ|γ-1+(1/N2))≤β/2, we can get Φ~ϵ,N(q1)≤β/2,∀q1∈∂W-.We introduce the minimax classS∶={ψ∈C(W--,Q)∣ψ=Idon⁡∂W-}. According to Proposition 5.9 of [19], the maps of S have an important intersection property as follows.Proposition 10.
(S+ and W- link with respect to ψ). Consider
(95)ψ∈S⟹ψ(W-)∩S+≠∅.One defines the minimax linking level as follows:(96)c~ϵ∶=inf⁡ψ∈S⁡max⁡q1∈W-⁡Φ~ϵ,N(ψ(q1)).
Obviously, by Proposition 10 and Lemma 9,
(97)max⁡q1∈W-⁡Φ~ϵ,N(ψ(q1))≥min⁡q1∈S+⁡Φ~ϵ,N(q1)≥β>0,∀ψ∈S,
and, therefore, c~ϵ>β>0.SinceId∈S, so
(98)c~ϵ≤max⁡q∈W-⁡Φ~ϵ,N((q1))≤max⁡q∈W-⁡(Γ(q1)+R~ϵ,N(q1))≤max⁡q∈W-⁡(α+(a)2|q+|H12+α-(a)2|q-|H12+∫T2κq14)+1≤c~∞≤+∞,
where c~∞ is independent of N,ϵ,γ. By the linking theorem, we deduce the existence of a Palais-Smale sequence (qn)∈Q1 at the level c~ϵ, namely,
(99)Φ~ϵ,N(qn)⟶c~ϵ,Φ~ϵ,N′(qn)⟶0whenn⟶∞.
Third, we will prove that the Palais-Smale sequence (qn) (up to subsequence) converges to some nontrivial critical point q^1≠0 in some open ball of Q1, where Φ~ϵ,N and Φ^ϵ,N coincide. Because the space Q1 is finite dimensional, we only need to prove that the sequence (qn) is bounded.Lemma 11.
There is a constantM>0 independent of R,ϵ,N,γ, such that, for all |ϵ|/γ small enough and N large enough, the Palais-Smale sequence (qn) is bounded; that is, |qn|H1<M. So, there exists a subsequence of the P-S sequence that converges to some critical point q^1≠0, and the functional Φ~ϵ,N possesses a nontrivial critical point q^1∈Q1 with critical value Φ~ϵ,N(q^1)=c~ϵ.Proof.
In the sequel, we will always assume that(ϵ+|ϵ|γ-1+1/N2)<1. Writing Φ~ϵ,N(q)=Γ(q)+R~ϵ,N(q), by (81)-(82), we can get
(100)Φ~ϵ,N(qn)-12Φ~ϵ,N′(qn)[qn]=Γ(qn)-12Γ′(qn)[qn]+(R~ϵ,N(qn)-12R~ϵ,N′(qn)[qn])=(12-14)∫T2a3(φ1)qj4+(R~ϵ,N(qn)-12R~ϵ,N′(qn)[qn])≥α(12-14)∫T2qj4-(ϵ+|ϵ|γ-1+1N2).
Then, by c~ϵ<c~∞<+∞ and Φ~ϵ,N′(qn)→0(n→∞),
(101)c~∞+1+|qn|H1≥κ6∫T2qn4:=κ6|qn|L44.
By Hölder inequality and orthogonality,
(102)c~∞+1+|qn|H1≥κ7(∫T2(q0,n+q-,n+q+,n)2)2=κ7(∫T2q0,n2+q-,n2+q+,n2)2≥κ8(q0,n)4,
and therefore |q0,n|≤(1+|qn|H1)1/4. In the same way, by Hölder inequality and (82),
(103)Φ~ϵ,N′(qn)[q+,n]=α+(a)|q+,n|H12-∫T2a3(φ1)qn3q+,n+R~ϵ,N′(qn)[q+,n]≥α+(a)|q+,n|H12-κ9|q+,n|H1∫T2|qn|3-C3(R)(ϵ+|ϵ|γ-1+1N3/2)|q+,n|H1≥κ10|q+,n|H1(|q+,n|H1-|qn|L43-1).
By (101) and the above inequalities, using Φ~ϵ,N′(qn)→0(n→∞), we conclude that |q+,n|H1≤κ11(1+|qn|H13/4). Estimating analogously, we derive |q-,n|H1≤κ12(1+|qn|H13/4). Finally, we deduce that
(104)|qn|H1=|q0,n|+|q+,n|H1+|q-,n|H1≤κ13(1+|qn|H13/4+|qn|H13/4).
So, we can conclude that |qn|H1≤M for a suitable constant M>0, which is independent of R,ϵ,N,γ. Since Q1 is finite dimensional, {qn} converges, up to subsequence, to some critical point q^ of Φ~ϵ,N with |q^|H1<M. Since Φ~ϵ,N(q^)=c~ϵ≥β>0, we conclude that q^≠0.Proof of Theorem1.
Let us fixR-∶=M+1 and take |ϵ|γ-1≤ϵ2(R-):=ϵ-. According to Lemma 7, we can get, for 0<σN(R-)≤1,0≤s<1/2, a solution (q2(q1),p(q1))∈(Q2(N-)⊕P)∩Hσ,s of the (Q2)-(P)-equations with ∀|q1|H1≤2R. By Lemma 11, the extended functional Φ~ϵ,N(q1) possesses a nontrivial critical point q^ with |q1|H1≤M<R-. Since Φ~ϵ,N coincides with Φϵ,N on the ball BR-, by Lemma 8, there exists a nontrivial weak solution q^1+q2(q^1)+p(q^1)∈Hσ,s of (23). Finally, u=ϵ[q^1+q2(q^1)+p(q^1)]=ϵ[q^ϵ+p(q^1)] solves (7).
According to Lemma6, by the regularizing property of the operator L1, the solution q^ϵ∶=q^1+q2(q^1) of the (Q)-equation belongs to Hσ,s+2∩Q. By the P-equation
(105)((2+ϵ)∂φ12+2(1+a(ϵ+ϵ2))∂φ1∂φ2)p^=-ϵ((2a+a2ϵ2)∂φ22+ΠPf(φ1,φ2,q^ϵ+p,ϵ))p^,
where p^=p(q^1), we can get that
(106)-ϵ((2a+a2ϵ2)∂φ22+ΠPf(φ1,φ2,q^ϵ+p,ϵ))p^∈Hσ′,s,
for 0<σ′<σ, satisfying |ϵ(2a+a2ϵ2)l22|e|l2|σ′<e|l2|σ. For (a,ϵ)∈Bγ, the eigenvalues of operator (2+ϵ)∂φ12+2(1+a(ϵ+ϵ2))∂φ1∂φ2 restricted to P satisfy |(2+ϵ)l12+2(1+aϵ+aϵ2)l1l2|≥|2+ϵ|(γ|l1|/|l2|), ∀l1≠0,l1+l2≠0, and, thus, we deduce that p^∈Hσ′′,s+1, for all 0<σ′′<σ′ (satisfying C|l2|e|l2|σ′′<e|l2|σ′), and |∂φ1p^|σ′′,s=O(|ϵ|/γ). By (105),
(107)(2+ϵ)∂φ12p^=-2(1+a(ϵ+ϵ2))∂φ1∂φ2p^-ϵ((2a+a2ϵ2)∂φ22+ΠPf(φ1,φ2,q^ϵ+p,ϵ))p^,
we get p^∈Hσ-,s+2, and |p^|σ-,s+2=O(|ϵ|γ-1), with 0<σ-<σ′′. Thus, (15) follows with s-=s+2, 0<σ-<σ′′. By (5), u(t,x)=ϵv((1+ϵ)t+x,(1+aϵ2)t+x) is the solution of (1), for all (a,ϵ)∈Bγ. To show that u(t,x) is quasiperiodic, it remains to prove that v depends on both variables (φ1,φ2) independently. According to Lemma 9, ∀|q^1|H1≤R-,Φ~ϵ,N(q^1)≥β. On the other hand, Φ~ϵ,N(q^-+q^0)≤β/2, ∀|q^-+q^0|H1≤R-, so that q^1∉Q-⊕Q0, and, therefore, v depends on φ2. In fact, any solution v of (23) depending only on φ2, that is, the solutions of
(108)(2a+a2ϵ2)d2v(φ2)dφ22+f(φ1,φ2,v(φ2),ϵ)=0,
is v(φ2)≡0. Indeed, by the homogeneity of f(φ1,φ2,v,ϵ), we have
(109)ϵ3f(φ1,φ2,v,ϵ)=f(φ1,φ2,ϵv)=∑k=3∞ak(φ1,φ2)(ϵv)k.
Now consider a smooth function h(φ1) with zero mean, ant it satisfies ∫T‍ak(φ1,φ2)h(φ1)dφ1≠0 for some k. Multiplying (108) by h(φ1) and integrating over [0,π], we have(110)∫02π‍(2a+a2ϵ2)d2v(φ2)dφ22h(φ1)dφ1+∫02π‍f(φ1,φ2,v(φ2),ϵ)h(φ1)dφ1=0.
According to h(φ1) that has zero mean and multiplying above equation by ϵ3, we get
(111)∫02πf(φ1,φ2,ϵv(φ2))h(φ1)dφ1=∑k=3∞(ϵv(φ2))k∫02πak(φ1,φ2)h(φ1)dφ1=0.
The function G(z(φ2))=∑k=3∞bk(φ2)(z(φ2))k, with bk=∫02πak(φ1,φ2)h(φ1)dφ1, is a nontrivial analytic function. Thus, the equation G(ϵv(φ2))=0 cannot have a sequence of zeros accumulating to zero. So, for ϵ small enough, v(φ2)≡0.


## 4. Waves Traveling in Opposite Directions

Substitutingω1=1+ϵ,ω2=1+aϵ into (8), we get
(112)La,ϵv+f(φ1,φ2,v)=0,
where (see (8))
(113)La,ϵ∶=[(1+ϵ+1)∂φ1+(aϵ)∂φ2]∘[ϵ∂φ1+(2+aϵ)∂φ2]=4∂φ1∂φ2+ϵ[(2+ϵ)∂φ12+(2a+a2ϵ)∂φ22kkkkkkkkll+2(a+1+aϵ)∂φ1∂φ2]=L0+ϵL1.
We rescale (112) in order to highlight the relationship between the amplitude and the variation in frequency: v(φ1,φ2)→|ϵ|v(φ1,φ2), and, for convenience, we assume ϵ>0. In the following, we consider the scaled equation
(114)La,ϵv+ϵf(φ1,φ2,v,ϵ)=0,
where f(φ1,φ2,v,ϵ)=a3(φ1,φ2)v3+ϵO(v4).Equation (114) is the Euler-Lagrange equation of the Lagrange action functional Ψϵ∈C1(Hσ,s,R) defined by
(115)Ψϵ(v)∶=∫T22(∂φ1v)(∂φ2v)+ϵ(2+ϵ)2(∂φ1v)2kki+ϵ(2a+a2ϵ)2(∂φ2v)2kki+ϵ(a+1+aϵ)(∂φ1v)(∂φ2v)-ϵF(φ1,v,δ)=Ψ0(v)+ϵΨ1(v,δ),
where F(φ1,φ2,v,ϵ)∶=∫0vf(φ1,φ2,ξ,ϵ)dξ and Ψ0(v)∶=∫T22(∂φ1v)(∂φ2v),
(116)Ψ1(v,δ)∶=∫T2(2+ϵ)2(∂φ1v)2+2a+a2ϵ2(∂φ2v)2kki+(a+1+aϵ)(∂φ1v)(∂φ2v)-F(φ1,φ2,v,ϵ).
In order to find critical points of Ψϵ(v), we use the same method as in Section 3. The operator La,ϵ is diagonal defined on the Banach space Hσ,s under the Fourier basis el1,l2=eil1φ1eil2φ2 with eigenvalue Dl1,l2=-[(2+ϵ)l1+aϵl2][ϵl1+(2+aϵ)l2]. So, we have
(117)La,ϵ[v]=∑(l1,l2)∈Z2Dl1,l2v^l1,l2eil1φ1eil2φ2,∀v∈Hσ,s.
The unperturbed functional Ψ0:Hσ,s→R possesses an infinite-dimensional linear space Q of critical points, which are the solutions of the equation
(118)L0q=4∂φ1∂φ2q=0.
The space Q can be written as
(119)Q={q=∑(l1,l2)∈Z2q^l1,l2eil1φ1eil2φ2∈Hσ,s∣q^l1,l2=0,kkkkforl1l2≠0q=∑(l1,l2)∈Z2q^l1,l2eil1φ1eil2φ2}.
We split Q as
(120)Q=Q++Q0+Q-,
where
(121)Q+∶={q∈Q:q^l1,l2=0,for(l1,l2)∉Λ+}={q+:=q+(φ)∈Hσ,s0},Q0∶={q:q0,0∈R},Q-∶={q∈Q:q^l1,l2=0,for(l1,l2)∉Λ-}={q-:=q-(φ)∈Hσ,s0},Λ+∶={(l1,l2)∈Z2:l1=0,(l1,l2)≠(0,0)},Λ-∶={(l1,l2)∈Z2:l2=0,(l1,l2)≠(0,0)}.
We decompose the space Hσ,s=Q+P, where
(122)P∶={p=∑(l1,l2)∈Z2p^l1,l2eil1φ1eil2φ2∈Hσ,s∣p^l1,l2=0,kkkforl1l2=0p=∑(l1,l2)∈Z2p^l1,l2eil1φ1eil2φ2∈Hσ,s∣p^l1,l2}.
Projecting (114) onto the closed subspaces Q and P, setting v=q+p∈Hσ,s with q∈Q and p∈P, we obtain
(123)(Q)L1[q]+ΠQf(φ1,φ2,q+p,ϵ)=0,(P)La,ϵ[p]+ϵΠPf(φ1,φ2,q+p,ϵ)=0,
where ΠQ:Hσ,s→Q, ΠP:Hσ,s→P are the projectors, respectively, onto Q and P; moreover, they are continuous.In the same way, we decompose the spaceQ=Q1+Q2. Setting q=q1+q2 with q1∈Q1 and q2∈Q2, we finally get
(124)(Q1)L1[q1]+ΠQ1[f(φ1,φ2,q1+q2+p,ϵ)]=0⟺dΨϵ(v)[h]=0,∀h∈Q1,(125)(Q2)L1[q2]+ΠQ2[f(φ1,φ2,q1+q2+p,δ)]=0⟺dΨϵ(v)[h]=0,∀h∈Q2,(126)(P)La,ϵ[p]+ϵΠP[f(φ1,φ2,q1+q2+p,ϵ)]=0⟺dΨϵ(v)[h]=0,∀h∈P.
The operator La,ϵ is diagonal in the Fourier basis {eil1φ1eil2φ2,(l1,l2)∈Z2} with eigenvalues Dl1,l2=-[(2+ϵ)l1+aϵl2][ϵl1+(2+aϵ)l2]. We first prove that La,ϵ restricted to P has a bounded inverse when (a,ϵ) belongs to the uncountable zero-measure set
(127)Dγ∶={(a,ϵ)∈R-×R,(aϵ2+ϵ,ϵ2+aϵ)∈Eγ,k1+ϵ1+aϵ∉Q,1+ϵ≠0,1+aϵ≠0,2+aϵ≠0},
where Eγ is a set of badly approximate numbers defined as
(128)Eγ∶={γ|l2|,|l2+ϵ2l1|>γ|l1|(aϵ2+ϵ,ϵ2+aϵ)∶=(ϵ1,ϵ2)∈(-ϵ0,ϵ0)×(-ϵ0,ϵ0):k|l1+ϵ1l2|>γ|l2|,|l2+ϵ2l1|>γ|l1|}
for ∀l1,l2∈Z∖{0}, and 0<γ<1/4,ϵ0∈(0,1/2).Lemma 12.
For(a,ϵ)∈Dγ, the eigenvalues Dl1,l2 of La,ϵ restricted to P satisfy
(129)|Dl1,l2|=|((2+ϵ)l1+aϵl2)(ϵl1+(2+aϵ)l2)|>γ,kkkkkkkkkkkkkkkkkkkkkkkkkkkk∀l1l2≠0.
As a consequence, the operator La,ϵ:P→P has a bounded inverse La,ϵ-1 and satisfies
(130)|La,ϵ-1[h]|σ,s≤|h|σ,sγ,∀h∈P.Proof.
Denote by[x] the nearest integer close to x and {x}=x-[x]. If both l1≠-[(aϵ/(2+ϵ))l2] and l2≠-[(ϵ/(2+aϵ))l1], then we have
(131)|Dl1,l2|=|(2+ϵ)(l1+aϵ2+ϵl2)|·|(2+aϵ)(l2+ϵ2+aϵl1)|>1.
If l1=-[(aϵ/(2+ϵ))l2], then
(132)|Dl1,l2|=|(2+ϵ)(l1+aϵ2+ϵl2)|·|ϵ(l1+aϵ2+ϵl2)+(2+aϵ)l2-aϵ2+ϵϵl2|≥γ|l2|·|(2+ϵ)(ϵ{aϵ2+ϵl2}+2l2kkkkkkkkkki+aϵ(1-ϵ2+ϵ)l2)|≥γ.
In the same way, if l2=-[(ϵ/(2+aϵ))l1], then we have
(133)|Dl1,l2|≥γ|l1|·|(2+aϵ)(+ϵ(1-a2ϵ2+aϵ)l1)aϵ{ϵ2+aϵl1}+2l1kkkkkkkkkkki+ϵ(1-a2ϵ2+aϵ)l1)|≥γ.Lemma 13.
The operatorL1:Q2→Q2 has a bounded inverse L1-1 which satisfies
(134)|L-1[h]|σ,s≤|h|σ,sN2.Proof.
L
1 is diagonal in the Fourier basis of Q:eil1φ1eil2φ2 with (l1,l2)∈Λ+∪{(0,0)}∪Λ- with eigenvalues
(135)dl1,l2={-(2+ϵ)l12,ifl2=0,-a(2+aϵ)l22,ifl1=0.
The eigenvalues of L1 restricted to Q2(N) verify |dl1,l2|≥N2/C, where the constant C depends on (ϵ,a), and (134) holds.Similarly, the solution ofQ1-equation (124) is the Euler-Lagrange equation of the reduced Lagrangian action functional:
(136)Φϵ,N:B2R⊂Q1⟶R,Φϵ,N(q1)∶=Ψϵ(q1+q2(q1+p(q1))).Lemma 14.
Φ
ϵ
,
N
∈
C
1
(
B
2
R
,
R
) and a critical point q1∈B2R of Φϵ,N is a solution of the bifurcation equation (124). Moreover, Φϵ,N can be written as
(137)Φϵ,N(q1)=const+ϵ(Γ(q1)+Rϵ,N(q1)),
where
(138)Γ(q1)∶=∫T2(2+ϵ)2(∂φ1q1)2kki+(1+a+aϵ)(∂φ1q1)(∂φ2q1)kki+2a+a2ϵ2(∂φ2q1)2-a3(φ1,φ2)q144,Rϵ,N(q1)∶=∫T2a3(φ1,φ2)q144kki-F(φ1,φ2,q1+q2+p,ϵ)kki+12f(φ1,φ2,v,ϵ)(q2+p),
and, for some positive constant C2(R)≥C1(R), we can get
(139)|Rϵ,N(q1)|≤C2(R)(ϵ+ϵγ-1+1N2),|Rϵ,N′(q1)[h]|≤C2(R)(ϵ+ϵγ-1+1N2)|h|H1,kkkkkkkkkkkkkkkkkkkk∀h∈Q1.The problem of finding nontrivial solutions of theQ1-equation is reduced to finding nontrivial critical points of the reduced action functional Φϵ,N in B2R. By (137), this is equivalent to finding critical points of the rescaled functional denoted by Φ^ϵ,N and called the reduced action functional
(140)Φ^ϵ,N(q1)=Γ(q1)+Rϵ,N(q1)≡(Q(q1)-∫T2a3(φ1,φ2)q144+Rϵ,N(q1)),
where the quadratic form
(141)Q(q1)=∫T2(2+ϵ)2(∂φ1q1)2+(1+a+aϵ)(∂φ1q1)(∂φ2q1)+2a+a2ϵ2(∂φ2q1)2
is positive definite on Q+, negative definite on Q-, and zero definite on Q0. For q1=q++q-+q0∈Q1, we have
(142)Q(q+)=∫T2‍(2+ϵ)2(∂φ1q+)2=α+2|q+|H12,Q(q0)=0,Q(q-)=∫T2‍(a+aϵ)2(∂φ1q-)2=-α-2|q-|H12,
where the positive constants α+,α- are bounded away from zero and independent of ϵ. The following steps of finding the nontrivial solutions of the Q1-equation are similar to Lemmas 9 and 11 in Section 3.4.Proof of Theorem2.
We can get the solution of (8) as follows:
(143)v=|ϵ|[q^1+q2(q^1)+p(q^1)]=|ϵ|[q^ϵ+p(q^1)].
According to Lemma 13, by the regularizing property of the operator L1, the solution q^ϵ:=q^1+q2(q^1) of the (Q)-equation belongs to Hσ,s+2∩Q. By the P-equation
(144)(ϵ(2+ϵ)∂φ12+(4+2aϵ+2ϵ+2aϵ2)∂φ1∂φ2)p^=-ϵ((2a+a2ϵ)∂φ22+ΠPf(φ1,φ2,q^ϵ+p,ϵ))p^,
where p^=p(q^1), we can get that
(145)-ϵ((2a+a2ϵ)∂φ22+ΠPf(φ1,φ2,q^ϵ+p,ϵ))p^∈Hσ′,s,
for 0<σ′<σ, satisfying |ϵ(2a+a2ϵ)l22|e|l2|σ′<e|l2|σ. For (a,ϵ)∈Dγ, the eigenvalues of operator
(146)ϵ(2+ϵ)∂φ12+(4+2aϵ+2ϵ+2aϵ2)∂φ1∂φ2
restricted to P satisfy
(147)|ϵ(2+ϵ)l12+(4+2aϵ+ϵ+aϵ2)l1l2|≥|2+ϵ|γ|l1||l2|,kl∀l1≠0,l2≠0,
and, thus, we deduce that p^∈Hσ′′,s+1, for all 0<σ′′<σ′ (satisfying C|l2|e|l2|σ′′<e|l2|σ′), and |∂φ1p^|σ′′,s=O(|ε|/γ). By (144),
(148)ϵ(2+ϵ)∂φ12p^=-(4+2aϵ+2ϵ+2aϵ2)∂φ1∂φ2p^-ϵ((2a+a2ϵ)∂φ22+ΠPf(φ1,φ2,q^ϵ+p,ϵ))p^,
we get p^∈Hσ-,s+2, and |p^|σ-,s+2=O(|ϵ|/γ), with 0<σ-<σ′′. Thus, (19) follows with s-=s+2,0<σ-<σ′′. By (6), u(t,x)=|ϵ|v((1+ϵ)t-x,(1+aϵ)t+x) is the solution of (1), for all (a,ϵ)∈Dγ. Obviously, v depends on both variables (φ1,φ2) independently. So, u(t,x) is a quasiperiodic solution of (1), with frequencies (ω1,ω2)=(1+ϵ,1+aϵ).


## 5. Conclusion

In this paper, for the completely resonant nonlinear wave equations, under periodic boundary conditions, we obtain the existence and regularity of quasiperiodic solutions. The forced terms we consider are quasiperiodic, and, according to the linking theorem, the bifurcation equations are solved by variational method. Moreover, the solutions depending on the the spatial and time variables are coupled and in the form of traveling waves. In [28], Yuan got the existence of quasiperiodic solutions with n∈N  (n≥3) frequencies by KAM theory, in which the form of the solutions is u(t,x)=v(ω1t,ω2t,…,ωnt,x). In the future work, we will investigate the existence of quasiperiodic solutions with the traveling wave form as u(t,x)=v(ω1t+x,ω2+x,…,ωnt+x).



---

*Source: 649270-2014-05-18.xml*