An Averaging Theorem for Perturbed KdV Equation

# An Averaging Theorem for Perturbed KdV Equation

## Abstract

We consider a perturbed KdV equation:

 ˙u+uxxx−6uux=ϵf(x,u(⋅)),x∈T,∫Tudx=0.

For any periodic function , let be the vector, formed by the KdV integrals of motion, calculated for the potential . Assuming that the perturbation is a smoothing mapping (e.g. it is a smooth function , independent from ), and that solutions of the perturbed equation satisfy some mild a-priori assumptions, we prove that for solutions with typical initial data and for , the vector may be well approximated by a solution of the averaged equation.

###### ams:
35Q53, 70K65, 34C29, 37K10, 74H40

## Introduction

We consider a perturbed Korteweg-de Vries (KdV) equation with zero mean-value periodic boundary condition:

 ˙u+uxxx−6uux=ϵf(x,u(⋅)),x∈T=R/Z,∫Tu(x,t)dx=0. (0.1)

Here is a nonlinear perturbation, specified below. For any we denote by the Sobolev space of order , formed by real-valued periodic functions with zero mean-value, provided with the homogeneous norm . Particularly, if we have

 Hp={u∈L2(T):||u||p<∞,∫Tudx=0},||u||2p=∫T∣∣∂pu∂xp∣∣2dx.

For any , the operator defines a linear isomorphism: . Denoting by its inverse, we provide the spaces , , with a symplectic structure by means of the 2-form :

 Ω(u1,u2)=−⟨(∂∂x)−1u1,u2⟩, (0.2)

where is the scalar product in . Then in any space , , the KdV equation (0.1) may be written as a Hamiltonian system with the Hamiltonian , given by . That is, KdV may be written as

 ˙u=∂∂x∇H(u).

It is well-known that KdV is integrable. It means that the function space admits analytic symplectic coordinates , where , such that the quantities , , are actions (integrals of motion), while , , are angles. In the -variables, KdV takes the integrable form

 ˙I=0,˙φ=W(I), (0.3)

where is the frequency vector (see [1, 2]). The integrating transformation , called the nonlinear Fourier transform, for any defines an analytic isomorphism , where

 Extra open brace or missing close brace

It is well established that for a perturbed integrable finite-dimensional system,

 ˙I=ϵf(I,φ),˙φ=W(I)+ϵg(I,φ),ϵ<<1,

where , , on time intervals of order the actions may be well approximated by solutions of the averaged equation:

 ˙J=ϵ⟨f⟩(J),⟨f⟩(J)=∫Tnf(J,φ)dφ,

provided that the initial data are typical (see [3, 4, 5, 6]). This assertion is known as the averaging principle. But in the infinite dimensional case, there is no similar general result. In [7, 8], S. Kuksin and A. Piatniski proved that the averaging principle holds for the randomly perturbed KdV equation of the form:

 ˙u−ϵuxx+uxxx−6uux=√ϵη(t,x),x∈S1,∫udx=∫ηdx=0, (0.4)

where the force is a white noise in , is smooth in and is non-degenerate. Our goal in this work is to justify the averaging principle for the KdV equation with deterministic perturbations, using the Anosov scheme (see [3]), exploited earlier in the finite dimensional situation. The main technical difficulty to achieve this goal comes from the fact that to perform the scheme one has to use a measure in the function space which is quasi-invariant under the flow of the perturbed equation (it is needed to guarantee that a small ’bad’ set which we have to prohibit for a solution of the perturbed equation at a time corresponds to a small set of initial data). For a reason, explained in Section 3, to construct such a quasi-invariant measure we have to assume that the perturbation is smoothing. More precisely, we assume that:

Assumption A. (i) For any , the mapping defined by the perturbation in (0.1):

 P:Hp→Hp+ζ0,u↦f(x,u(⋅)), (0.5)

is analytic. Here is a constant.

(ii) For any and , the perturbed KdV equation (0.1) with initial data

 u(0)=u0∈Hp,

has a unique solution in the time interval , and

 ||u(t)||p⩽C(p,||u0||p,T),|t|⩽Tϵ−1.

We are mainly concerned with the behavior of the actions for . For this end, it is convenient to pass to the slow time and write the perturbed KdV equation (0.1) in the action-angle coordinates :

 dIdτ=F(I,φ),dφdτ=ϵ−1W(I)+G(I,φ). (0.6)

Here , and is the infinite-dimensional torus, endowed with the Tikhonov topology. The two functions and are the perturbation term , written in action-angle variables, see below (1.3) and (1.4). The corresponding averaged equation is

 dJdτ=⟨F⟩(J),⟨F⟩(J)=∫T∞F(J,φ)dφ, (0.7)

where is the Haar measure on . It turns out that the (0.7) is a Lipschitz equation, see below (4.17). We denote by the image of the space under the action-mapping

 πI:v↦I,Ij(v)=12|vj|2,j⩾1.

Clearly, , where is the weighted -space

 hpI={I∈R∞:|I|hpI=|I|p=2∞∑j=1(2πj)2p+1|Ij|<∞}.

and is its positive octant, . This is a closed subset of .

For any , let us denote by the linear operator on the space of sequences which rotates each component by the angle .

Definition 0.1 A Gaussian measure on the Hilbert space is said to be  -admissible (where is the same as in assumption A), if the following conditions are fulfilled:

(i) It is non-degenerate and has zero mean value.

(ii) It has a diagonal correlation operator , where every , and . In particular, is invariant under the rotations .

Such measures can be written as:

 +∞∏j=1(2πj)1+2p2πσjexp{−(2πj)1+2p|vj|22σj}dvj, (0.8)

where , , is the Lebesgue measure on (see [9, 10]). Clearly, they are invariant under the KdV flow (0.3).

The main result of this work is the following theorem:

Theorem 0.2. Fix any and . Let the curve , be a solution of equation (0.1) and , , . If assumption A is fulfilled and is a -admissible Gaussian measure on , then

(i) For any , there exists a Borel subset of and such that , and for we have

 |I(vϵ(τ))−J(τ)|p⩽ρ,for|τ|⩽¯T,vϵ(0)∈Γϵρ, (0.9)

where , , is a solution of the averaged equation (0.7) with the inital data .

(ii) There is a full measure subset of with the following property:

If , then for any the image of the probability measure on under the mapping converges weakly, as , to the Haar measure on .

The assertion (ii) of the theorem means that for any bounded continuous function on ,

 1¯T2−¯T1∫¯T2¯T1g(φ(vϵ(τ)))dτ→∫T∞g(φ)dφ,ϵ→0.

In particular, we have

Proposition 0.3. The assumption A holds if in (0.1) is a smooth function, independent from .

It is unknown for us that if the result of Theorem 0.2 remains true for equation (0.1) with non-smoothing perturbations, e.g. if the right hand side of equation (0.1) is or . So we do not know whether a suitable analogy of the result in [7, 8] holds true if in equation (0.4) the noise vanishes.

The paper has the following structure: Section 1 is about the transformation which integrates the KdV and its Birkhoff normal form. In Section 2 we discuss the averaged equation. We prove that the -admissible Gaussian measures are quasi-invariant under the flow of equation (0.1) in Section 3. Finally in Section 4 and Section 5 we establish the main theorem and Proposition  0.3.

Agreements. Analyticity of maps between Banach spaces and , which are the real parts of complex spaces and , is understood in the sense of Fréchet. All analytic maps that we consider possess the following additional property: for any , a map extends to a bounded analytical mapping in a complex ()-neighborhood of the ball in .

Notation. We use capital letters or to denote positive constants that depend on the parameters , but not on the unknown function . We denote For an infinite-dimensional vector and any we denote . We often identify with a corresponding -vector.

## 1 Preliminaries on the KdV equation

In this section we discuss integrability of the KdV equation (0.1).

### 1.1 Nonlinear Fourier transform for KdV

We provide the -space with the Hilbert basis ,

 es={√2cos(2πsx)s>0,√2sin(2πsx)s<0.

Theorem 1.1. There exists an analytic diffeomorphism and an analytic functional on of the form , where the function is analytic in a suitable neighborhood of the octant in , with the following properties:

(i) The mapping defines an analytic diffeomorphism , for any . This is a symplectomorphism of the spaces (see (0.2) and , where .

(ii) The differential takes the form .

(iii) A curve is a solution of the KdV equation (0.1) if and only if satisfies the equation

 ˙vj=(0−110)∂~K∂Ij(I)vj,vj=(vj,v−j)∈R2,j∈N. (1.1)

Since the maps and are analytic, then for , we have

 ||djΨ(u)||m⩽Pm(||u||m),||djΨ−1(v)||m⩽Qm(|v|m),j=0,1,2,

where and are continuous functions (cf. the agreements).

We denote

 W(I)=(W1,W2,…),Wk(I)=∂~K∂Ik(I),k=1,2,….

Lemma 1.2. For any , if , then

 det((∂Wi∂Ij)1⩽i,j⩽n)≠0.

Let be the Banach space of all real sequences with the norm

 |l|−1=supn⩾1n−1|ln|<∞.

Denote , where .

Lemma 1.3. The normalized frequency map

 ~W:I↦~W(I)=W(I)−κ

is real analytic as a map from to .

The coordinates are called the Birkhoff coordinates, and the form (1.1) of KdV is its Birkhoff normal form. See [1] for Theorem 1.1 and Lemma 1.3. A detailed proof of Lemma 1.2 can be found in [2].

### 1.2 Equation (0.1) in the Birkhoff coordinates.

For we denote:

 Ψk:Hm→R2,Ψk(u)=vk,

where . Let be a solution of equation (0.1). We get

 ˙vk=dΨk(u)(ϵf(x,u)+V(u)),k⩾1, (1.2)

where . Since is an integral of motion of KdV equation (0.1), we have

 ˙Ik=ϵ(dΨk(u)f(x,u),vk):=ϵFk(v). (1.3)

Here and below indicates the scalar product in .

For define if , and if . Using equation (1.1), we get

 ˙φk=Wk(I)+ϵ|vk|−2(dΨk(u)f(x,u),v⊥k),ifvk≠0, (1.4)

where . Denoting for brevity, the vector field in equation (1.4) by , we rewrite the equation for the pair as

 ˙Ik(t)=ϵFk(v)=ϵFk(I,φ),˙φk(t)=Wk(I)+ϵGk(v). (1.5)

We set

 F(I,φ)=(F1(I,φ),F2(I,φ),…).

In the following lemma and are some fixed continuous functions.

Lemma 1.4. For , we have for any

1. The function is analytic in each space .

2. For any , , the function is bounded by .

3. For any , the function is bounded by .

4. The function is bounded by , and for any and , the function is analytic on .

Proof: Items (i) and (ii) follow directly from Theorem 1.1. Items (iii) and (iv) follow from item (i) and the chain-rule:

 ∂Fk∂φj=√2Ij(∂Fk∂v−jcos(φj)−∂Fk∂vjsin(φj)),∂Fk∂Ij=(√2Ij)−1(∂Fk∂vjcos(φj)+∂Fk∂v−jsin(φj)).□

From this lemma we know that equation (1.5) may have singularities at . We denote

 ΠI:hp→hpI,ΠI(v)=I(v),ΠI,φ:hp→hpI×T∞,ΠI,φ(v)=(I(v),φ(v)).

Abusing notation, we will identify with .

Definition 1.5. For , we say that a curve , , is a regular solution of equation (1.5), if there exists a solution of equation (0.1) such that and

 ΠI,φ(Ψ(u(t)))=(I(t),φ(t)),|t|⩽T.

If is a regular solution of (1.5) and , then by assumption A we have

 |I(t)|p=|v(t)|2p⩽C(p,M0,T),|t|⩽Tϵ−1. (1.6)

## 2 Averaged equation

For a function on a Hilbert space , we write if

 |f(u1)−f(u2)|⩽P(R)||u1−u2||,||u1||,||u2||⩽R, (2.1)

for a suitable continuous function which depends on . Clearly, the set of functions is an algebra. By the Cauchy inequality, any analytic function on belongs to (see agreements). In particularly, for any ,

 Wk(I)∈Liploc(hpI),p⩾1,andFk(v)∈Liploc(hp),p⩾0.

In the further analysis, we systematically use the fact that the functional only weakly depends on the tail of the vector . Now we state the corresponding results. Let and , . Denoting by , the projection

 ΠM:h0→h0,(v1,v2,…)↦(v1,…,vM,0,…),

we have . Accordingly,

 |f(v)−f(ΠMv)|⩽P(|v|p1)(2πM)−(p1−p). (2.2)

The torus acts on the space by linear transformations , , where . Similarly, the torus acts on by linear transformations with .

For a function and a positive integer we define the average of in the first angles as the function

 ⟨f⟩N(v)=∫TNf((ΦθN⊕Id)(v))dθN,

and define the averaging in all angles as

 ⟨f⟩(v)=∫T∞f(Φθ(v))dθ,

where is the Haar measure on . The estimate (2.2) readily implies that

 |⟨f⟩N(v)−⟨f⟩(v)|⩽P(R)(2πN)−(p1−p),|v|p1⩽R.

Let , then is a function independent of , and is independent of . Thus can be written as .

Lemma 2.1. (See [7]). Let , then

1. The functions and satisfy (2.1) with the same function as and take the same value at the origin.

2. These two functions are smooth (analytic) if is. If is smooth, then is a smooth function with respect to vector , for any . If is analytic in the space , then is analytic in the space .

We recall that a vector is non-resonant if

 ω⋅k≠0,∀k∈Zn∖{0}.

Denote by the set of all Lipschitz functions on .

Lemma 2.2. Let for some . Then for any non-resonant vector we have

 limT→∞1T∫T0f(x0+ωt)dt=⟨f⟩,

uniformly in . The rate of convergence depends on , and .

Proof: Let us write as the Fourier series . Since the Fourier series of a Lipschitz function converges uniformly (see [11]), for any we may find such that for all . Now it is enough to show that

 ∣∣1T∫T0fR(x0+ωt)dt−f0∣∣⩽ϵ2,∀T⩾Tϵ, (2.3)

for a suitable , where . Observing that

 ∣∣1T∫T0eik⋅(x0+ωt)dt∣∣⩽2T|k⋅ω|,

for each nonzero . Therefore the l.h.s of (2.3) is smaller than

 2T(inf|k|⩽R|k⋅ω|)−1∑|k|⩽R|fk|.

The assertion of the lemma follows.

## 3 Quasi-invariance of Gaussian measures

Fix any integer , and let be a -admissible Gaussian measure on the Hilbert space . In this section we will discuss how this measure evolves under the flow of the perturbed KdV equation (0.1). We follow a classical procedure based on finite dimensional approximations (see e.g. [12, 10]).

We suppose the assumption A holds. Let us write the equation (0.1) in the Birkhoff normal form, using the slow time :

 ddτvj=ϵ−1JWj(I)vj+Xj(v),j∈N, (3.1)

where and .

For any , we consider the -dimensional subspace of with coordinates . On , we define the following finite-dimensional systems:

 ddτ→ωj=ϵ−1JWj(I(ωn))→ωj+Xj(ωn),1⩽j⩽n, (3.2)

where and .

We denote and . By assumption A and Theorem 1.1, for any the mapping

 X:hp→hp+ζ0,v↦X(v) is analytic. (3.3)

Theorem 3.2. For any , converges to as in , where and are, respectively, solutions of (3.1) and (3.2) with initial data and .

Proof:  Fix any . From (1.6) we know that there exists a constant such that if , then

 |v(τ)|p⩽M1,τ∈[0,T]. (3.4)

The equation (3.2) yields that

 ddτ|ωn|2p=2n∑j=1j1+2p→ωj⋅Xj(ωn):=χn(ωn). (3.5)

We define

 χ(v):=2∞∑j=1j1+2pvj⋅Xj(v).

By (3.3), we know that there exists a constant such that

 |χn(ωn)|⩽C1,|ωn|p⩽2M1,∀n∈N. (3.6)

Denote , then if , then

 |ωn(τ)|p⩽2M1,τ∈[−¯τ,¯τ],∀n∈N. (3.7)

Lemma 3.3. In the space , we have the convergence

 ωn(⋅)→v(⋅)asn→∞.

Proof: Denote , and . Since , using equations (3.1) and (3.2), for , we get

 ddτ|→ξj|2=2(→ξj)t[ϵ−1J(Wj(Iv)vj−Wj(Iωn)→ωj)+Xj(v)−Xj(ωn)]=2ϵ−1[Wj(Iv)−Wj(Iωn)]vj⋅(→ωj)⊥+2(→ξj)t⋅(Xj(v)−Xj(ωn)).

By Lemma 1.3 and Cauchy’s inequality, we know that

 ∣∣Wj(I(v))−Wj(I(ωn))∣∣⩽C2(M1)j|v−ωn|p−1.

Using (3.3) we get that

 ddτ|v−ωn|2p−1⩽C3(ϵ,M1)|v−ωn|2p−1+an(v),τ∈[−¯τ,¯τ],

where

 an(v)=∞∑j=n+1j2p−1vj⋅Xj(v).

Obviously, as uniformly for .

The lemma now follows directly from Gronwall’s Lemma.

Lemma 3.4. If strongly in and , , as , then

 limn→∞|v(τ)−ωn(τn)|p=0.

Proof: From (3.5) we know that for any ,

 |ω