On long time dynamics of perturbed KdV equations
Consider a perturbed KdV equation:
where the nonlinear perturbation defines analytic operators in sufficiently smooth Sobolev spaces. Assume that the equation has an -quasi-invariant measure and satisfies some additional mild assumptions. Let be a solution. Then on time intervals of order , as , its actions can be approximated by solutions of a certain well-posed averaged equation, provided that the initial datum is -typical.
The KdV equation on the circle, perturbed by smoothing perturbations, was studied in . There an averaging theorem that describes the long-time behavior for solutions of the perturbed KdV equation was proved. In this work, we suggest an abstract theorem which applies to a large class of -perturbed KdV equations which have -quasi-invariant measures; the latter notion is explained in the main text. We show that the systems considered in , satisfy this condition, and believe that it may be verified for many other perturbations of KdV. More exactly, we consider a perturbed KdV equation with zero mean-value periodic boundary condition:
where is a nonlinear perturbation, specified below. For any we introduce the Sobolev space of real valued function on with zero mean-value:
Here and , are the Fourier coefficients of with respect to the trigonometric base
i.e. . It is well known that KdV is integrable. It means that the function space admits analytic coordinates , where , such that the quantities and , , are action-angle variables for KdV. In the -varibles, KdV takes the integrable form
We introduce the weighted -space ,
and the mapping :
Obviously, is continuous, and its image is the positive octant of .
We wish to study the long-time behavior of solutions for equation (0.1). Accordingly, we fix some
Assumption A: (i) For any , there exists a unique solution of (0.1) with . It satisfies
(ii) There exists a such that for , the perturbation term defines an analytic mapping
We are mainly concerned with the behavior of the actions on time interval . For this end, we write the perturbed KdV (0.1), using slow time and the -variables:
Here is the vector filed of KdV and is the perturbation term, written in the -variables. In the action-angle variables this equation reads:
Here and , where is the infinite-dimensional torus, endowed with the Tikhonov topology. The two functions and represent the perturbation term , written in the action-angle variables, see below (1.3) and (1.4). We consider an averaged equation for the actions:
where is the Haar measure on . It turns out that defines a Lipschitz vector filed in (see (3.11) below). So equation (0.6) is well-posed, at least locally. We want to study the relation between the actions of solutions for equation (0.5) and solutions of equation (0.6), for .
Let , , be the flow-maps of equation (0.4) on and denote
1) A measure on is called regular if for any analytic function on such that , we have .
2) A measure on is said to be -quasi-invariant for equation (0.4) on the ball if it is regular, and there exists a constant such that for any Borel set , we have
Similarly, these definitions can be carried to measures on the space and the flow-maps of equation (0.1) on .
Fix any . Suppose that assumption A holds and equation (0.1) has an -quasi-invariant measure on . Then
(i) For any and any , there exists , and a Borel subset such that
and for , we have that if , then
Here , is the unique solution of the averaged equation (0.6) with any initial data , satisfying , and
(ii) Let be the probability measure on , defined by the relation
where . Then the averaged measure
converges weakly, as , to the Haar measure on
1) Assume that an -quasi-invariant measure depends on , i.e. . Then the same conclusion holds with replaced by , if satisfies some consistency conditions. See subsection 3.3.
2) Item (ii) of Assumption A may be removed if the perturbation is hamiltonian. See the end of subsection 3.1.
Toward the existence of -quasi-invariant measures, following , consider a class of Gaussian measures on the Hilbert space :
where , , is the Lebesgue measure on . We recall that (0.10) is a well-defined probability measure on if and only if (see ). It is regular in the sense of Definition 0.1 and is non-degenerated in the sense that its support equals to (see [2, 3]). From (0.2), it is easy to see that this kind of measures are invariant for KdV.
For any , we say the measure is -admissible if the in (0.10) satisfies for all . It was proved in  that if Assumption A holds and
the operator defined by (see (0.4)) analytically maps the space to the space with some ,
then every -admissible measure is -quasi-invariant for equation (0.1) on .
However, the conditions is not easy to verify due to the complexity of the nonlinear Fourier transform. Fortunately, there exists another series of Gibbs-type measures (see (4.3) below) known to be invariant for KdV, explicitly constructed on the space in . We will show in Section 4 that these measures are -quasi-invariant for equation (0.1) if Assumption A holds with . We point out straight away that this condition is not optimal (see Remark 4.11).
The paper is organized as follows: Section 1 is about some important properties of the nonlinear Fourier transform and the action-angle form of the perturbed KdV (0.1). We discuss the averaged equation in Section 2. The Theorem 0.2 is proved in Section 3. Finally we will discuss the existence of -quasi-invariant measures in Section 4.
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 . We call such analytic maps uniformly analytic.
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
(see ) 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) For any , the mapping defines an analytic diffeomorphism .
(ii) The differential is the operator .
(iii) A curve is a solution of the KdV equation (0.1) if and only if satisfies the equation
The coordinates are called the Birkhoff coordinates, and the form (1.1) of KdV is its Birkhoff normal form
Since the maps and are analytic, then for , we have
where and are continuous functions.
A remarkable property of the nonlinear Fourier transform is its quasi-linearity. It means:
This is the KdV-frequency map. It is non-degenerate:
(see , appendix 6) For any , if , then
Let be the Banach space of all real sequences with the norm
Denote , where .
(see , Thoerem 15.4) The normalized frequency map
is real analytic as a map from to .
1.2. Equation (0.1) in the Birkhoff coordinates.
For we denote:
where . Let be a solution of equation (0.1). Passing to the slow time and denoting to be , we get
where and . Since the action is an integral of motion for the KdV equation (0.1), we have
Here and below indicates the scalar product in .
For defines the angle if and if . Using equation (1.2), we get
where . Denoting for brevity, the vector field in equation (1.4) by , we rewrite the equation for the pair as
For any , define a map as
In the following lemma and are some fixed continuous functions.
For and each we have:
(i) The function is analytic in each space .
(ii) For any , the function is bounded by .
(iii) For any , the function is bounded by .
(iv) The function is bounded by , and for any and , the function is smooth on .
Abusing notation, we will identify with .
Note that if is a regular solution, then each is a -function, while may be discontinuous at points , where .
2. Averaged equation
For a function on a Hilbert space , we write if
for a suitable continuous function which depends on . By the Cauchy inequality, any analytic function on belongs to (see Agreements). In particularly, for any ,
Let for some and , . Denoting by , , the projection
we have . Accordingly,
The torus acts on the space by the linear transformations , , where . For a function , we define the averaging in angles as
where is the Haar measure on . Clearly, the average is independent of . Thus can be written as .
Extend the mapping to a complex mapping , using the same formulas (0.3). Obviously, if is a complex neighbourhood of , then is a complex neighbourhood of .
(See , Lemma 4.2) Let , then
(i) The function satisfy (2.1) with the same function as and take the same value at the origin.
(ii) This function is smooth (analytic) if is. If is analytic in a complex neighbourhood of , then is analytic in the complex neighbourhood of .
For any , we consider the mapping defined by
For every , the mapping is analytic as a map from the space to .
The mapping extends analytically to a complex neighbourhood of (see Agreements). Then by (1.3), the functions , are analytic in . Hence it follows from Lemma 2.1 that for each , the function is analytic in the complex neighbourhood of . By (1.6), the mapping is locally bounded on . It is well known that the analyticity of each coordinate function and the locally boundness of the maps imply the analyticity of the maps (see, e.g. ). This finishes the proof of the corollary. ∎
We recall that a vector is called non-resonant if
Denote by the set of all Lipschitz functions on . The following lemma is a version of the classical Weyl theorem (for a proof, see e.g. Lemma 2.2 in ).
Let for some . For any and any non-resonant vector , there exists such that if , then
uniformly in .
3. Proof of the main theorem
In this section we prove Theorem 0.2.
Assume . So
Without loss of generality, we assume that . Fix any . Let
We pass to the slow time . Let be a regular solution of the system (1.5) with . We will also write it as when we want to stress the dependence on . Then by assumption A, there exists such that
By (1.6), we know that
where the constant depends only on .
We denote , , and , for any .
3.1. Proof of assertion (i)
By (2.2), there exists such that
From now on, we always assume that
and identify with
By Lemma 1.5, we have
By (2.2) we get
where is the -norm.
Below we define a number of sets, depending on various parameters. All of them also depend on , and , but this dependence is not indicated. For any and , we define a subset
as the collection of all such that for every , we have,
Let be the flow generated by regular solutions of the system (1.5). We define two more groups of sets.
Here and below stands for the Lebesgue measure in . We will indicate the dependence of the set on and as , when necessary.
By continuity, is a closed subset of and is an open subset of . Repeating a version of the classical averaging argument (cf. ), presented in the proof of Lemma 4.1 in , we have the following averaging lemma:
For , the -component of any regular solution of (1.5) with initial data in can be written as
where the function is bounded on by