Wenyan CUI Lufang MILi YIN

College of Science,Binzhou University,Binzhou 256600,China

E-mail:yufengxingshi@163.com;milufang@126.com;yinli79@163.com

Abstract In this paper,we consider small perturbations of the KdV-mKdV equation

Key words quasi-periodic solution;KdV-mKdV equation;KAM theory;normal form

1 Introduction and Main Result

The KdV-mKdV equation

is an evolution equation in one space dimension which is described as the wave propagation of the bound particle,sound wave and thermal pulse[18,19].It belongs to the family of KdV equations.Its solutions which including exact solutions,travelling wave solutions,and so on,attracted great attention in the past few years.For instance,Lu-Shi[14]established its exact solution with the aid of symbolic computation system mathematic.More results can refer to[7,8,21,26]and the references therein.

In recent years,the existence of fi nite dimensional tori for in fi nite-dimensional system was wildly investigated in the literature.So far there are two approaches to obtain the periodic and quasi-periodic solutions.One is the Craig-Wayne-Bourgain(CWB)method[1,6,23],another is the in fi nite-dimensional KAM theory which was earliest established by Kuksin and Wayne[9,23].The quasi-periodic solutions obtained by the KAM method have more dynamics properties and linear stability than by the CWB method.So many authors pay attention to the existence of KAM tori for partial di ff erential equations(PDEs).

With regard to the PDE

where Aw is linear vector- fi eld with d:=ordA>0,F(w)is nonlinear vector- fi eld with δ:=ordF,and it is analytic in the neighborhood of the origin w=0.If δ≤ 0,F is named a bounded perturbation,and if δ>0,F is named a unbounded perturbation.

According to a well-known example,due to Lax[20]and Klainerman[11](see also[13]),it is reasonable to assume

in order to guarantee the existence of KAM tori for the PDE.

For the existence of KAM tori of the PDEs with bounded Hamiltonian perturbations has been deeply and widely investigated by many researchers.There have many results in this fi eld in the past few decades.We can’t list all the papers in this fi eld,we give just two survey papers[2,12].Moreover,there are also some results of KAM theory for the PDEs with unbounded Hamiltonian perturbations.The earliest KAM theorem for unbounded perturbations is due to Kuksin[10].In[10],Kuksin proved the persistence of the fi nite-gap solutions alongside the hierarchy of KdV equation with periodic boundary conditions.In 2010,Liu-Yuan[16]obtained a new estimate for the solution of the small-denominators equation with critical unbounded variable coefficients.With the new estimate,a KAM theorem for in fi nite dimensional Hamiltonian including 0< δ

In view of the physical meaning of the family of KdV equations,there have many researchers paid attention to these equations.For example,in[3],Baldi-Berti-Montalto developed KAM theory for quasi-periodically forced KdV equations of the form

This KAM theory is also the fi rst KAM results for quasi-linear or fully nonlinear PDEs.Later,They[4]proved the existence and stability of Cantor families of quasi-periodic solutions of Hamiltonian quasi-linear perturbations of the KdV equation

where

is the most general quasi-linear Hamiltonian nonlinearity.In 2015,Xu-Shi[22]proved the mKdV equation

persist the small amplitude quasi-periodic solutions under periodic boundary.In 2016,Xu-Yan[24]studied small perturbations of the above equation with periodic boundary conditions.They got plenty of time-quasi-periodic solutions under such perturbations.

However,to the best of our knowledge,the existence of quasi-periodic solutions for KdV-mKdV equation aren’t deeply discussed.In this paper,we will study small perturbations of the KdV-mKdV equation

with periodic boundary conditions

To set the stage we introduce the phase space

of real valued functions on T=R/2πZ for any integer N ≥ 0,where

where c is considered as a real parameter.

We consider(1.1)as an in fi nite dimensional Hamiltonian system and subject it to sufficiently small Hamiltonian perturbations.The aim is to show that large families of time-quasiperiodic solutions persist under such perturbations.Our main work is to fi nd the transformation Φ normalizes the Hamiltonian up to order four.For KdV equation,the coordinate transformation is constructed in two step.The fi rst step is to eliminate the third term,the second step is to normalize the fourth order term coming from the fi rst step.For mKdV equation,the coordinate transformation is only normalizes the fourth term.Because equation(1.1)include the third term and fourth term at the same time,we fi rst need to eliminate the third term and normalize the fourth term generated in the fi rst step.On the other hand,we need to normalize the inherent fourth term.This is the important di ff erence between KdV equation and KdV-mKdV equation.In our paper,we obtain that

holds true for k,l,m,n∈Z{0}with k+l+m+n=0,k+l,k+m,k+n 6=0,which is di ff erent from[22].By using this inequality,we normalize the inherent fourth term.Since the fourth order term of normal form contained two part,which bring the difficulty of checking nondegeneracy condition.Luckily,by careful computation and analysis,we get that the frequency matrix satis fi es the non-degeneracy condition.

After overcoming these difficulties,we obtain the following result in the same manner as in[13,22].

Theorem 1.1Consider the nonlinear equation

Then,for in fi nitely many c with[u]=c,there exists an ε0>0 depending only on J={j1

(1)a nonempty Cantor set Πε? Π with meas(ΠΠε)→ 0 as ε → 0,where Π is a compact subset of Rmwith positive Lebesgue measure,

(2)a Lipschitz family of real analytic torus embeddings

where Tm=Rm/2πZm,

(3)a Lipschitz map φ :Πε→ Rm,such that for each(θ,ξ)∈ Tm×Πε,the curve u(t)= Φ(θ+φ(ξ)t,ξ)is a quasi-periodic solution of equation(1.4)winding around the invariant Φ(Tm× {ξ}).Moreover,each such torus is linearly stable.

2 The Birkho ffNormal Form

To write this Hamiltonian system more explicitly as an in fi nite dimensional system,we introduce in fi nitely many coordinates q=(qj)j6=0by writing

and the equations of motion in the new coordinates are given by

Because of the nondegeneracy of the transformed Poisson structure,we de fi ne a symplectic structure

The Hamiltonian expressed in the new coordinates q is determined,we still use the same symbol for the Hamiltonian as a function of q,we obtain

with

In the following,we will normalize the KdV-mKdV Hamiltonian up to order four.

3 The Proof of Main Result

From the transformation Φ in Theorem 2.5,we get the new Hamiltonian of(1.4)

where Λcis real analytic in the neighbourhood V of the origin in,K?Φ satis fi es

For the index set J={j1

Now consider the phase space domain

where the de fi nition of norm k·kpcan refer to Appendix.We will adopt lots of notations and de fi nitions from[13],such as the phase space,weighted norm for the Hamilton vector fi eld,etc..More de fi nitions are presented in Appendix.

To apply the KAM theorem in Appendix,we introduce the parameter domain

In the following we will check Assumptions A,B and C of the KAM Theorem 3.1 in Appendix.

Regarding ? as an in fi nite dimensional column vector with its index j ∈ N?,from(3.2),we know

whereˇ?j=j3is independent of ξ.Furthermore,basing on(3.2),we get

Accordingly,we fi nd

That is,Assumption A is ful fi lled with d=3,δ=1.

In the following we will check Assumption B.

In view of(3.1),we know that ξ 7→ ω is an affine transformation from Π to Rm.Denote

we have detA 6=0,by excluding fi nite zero points c of detA=0.Therefore,the real map ξ 7→ ω(ξ)is a lipeomorphism between Π and its image.This means that the fi rst part of assumption B is ful fi lled with positive M2and L only depend on the set J.Letting

and regarding k and l as m-dimensional and in fi nite dimensional row vector respectively,we have to check

for every k∈Zmand l∈Z∞with 1≤|l|≤2.

Suppose kA+lB=0,for some k∈Zmand 1≤|l|≤2.We let la,lb6=0(a 6=b)be he components of l.By excluding the zero points c of kA+lB=0,then we have kA+lB 6=0.As we know the zero points are uniquely determined by k,la,lb,thus there are countably many zero points.

It remains to check Assumption C.Consider the perturbation

Choose

where γ is taken from the KAM Theorem 3.1 and set M:=M1+M2,which only depends on the set J.When r is small enough,we obtain

which is just the smallness condition(A.5)in KAM Theorem 3.1.Applying Theorem 3.1 in Appendix,the conclusion of Theorem 1.1 is obtained.

Appendix:the KAM Theorem

Consider a small perturbation H=N+P of an in fi nite dimensional Hamiltonian in the parameter dependent normal form

on a phase space

where

where p ≥ 0.The tangential frequencies ω =(ω1,ω2,···,ωm)and normal frequencies ? =(?1,?2,···)are real analytic in the space coordinates and Lipschitz in the parameters,and for each ξ∈ Π its Hamiltonian vector fi eld XP=(Py,?Px,Pv,?Pu)Tde fi nes near T0:=Tm×{y=0}×{u=0}×{v=0}a real analytic map

where

We use the notation iξXPfor XPevaluated at ξ,and likewise in analogous cases.

To give the KAM theorem we need to introduce some domains and norms.For s,r>0,we introduce the complex T0-neighborhoods

and weighted norm for W=(Wx,Wy,Wu,Wv)∈Smq,C,

where|·|denotes the sup-norm for complex vectors.Furthermore,for a map W:U×Π →such as the Hamiltonian vector fi eld XP,we de fi ne the norms

where ?ξζW=iξW ? iζW,and

In a completely analogous manner,the Lipschitz semi-norm of the frequencies ω is de fi ned as

and the Lipschitz semi-norm of:Π→is de fi ned as

for any real number δ.Note that

Suppose the normal form N described above satis fi es the following assumptions.

Assumption A:Frequency AsymptoticsThere exist two real numbers d>1 and δ

are uniformly Lipschitz on Π,or equivalently,the map

is Lipschitz on Π.

for some r>0 and 0<α<1,the following holds.There exist

(i)a Cantor set Πα? Π with meas(ΠΠα)→ 0(α → 0),

(ii)a Lipschitz family of real analytic torus embeddings Φ:Tm×Πα→Smp,(iii)a Lipschitz map φ :Πα→ Rm,

such that for each ξ∈ Πα,the map Φ restricted to Tm× {ξ}is a real analytic embedding of a rotational frequencies φ(ξ)for the perturbed Hamiltonian H at ξ.In other words,

is a real analytic,quasi-periodic solution for the Hamiltonian iξH for every θ∈ Tmand ξ∈ Πα.Moreover,each embedding is real analytic on D(s/2)={|?x|

where

is the trivial embedding for each ξ,and c is a positive constant which depends on the same parameters as γ.

ProofThe proof can be found in[13].