Reducibility of the dispersive Camassa-Holm equation with unbounded perturbations

In this paper, we are mainly concerned with the reducibility of the quasi-periodically time dependent linear dynamical system

$$h_{t}-J\circ\big{(}(a_{0}(\omega t,x)+m_{0})+(a_{2}(\omega t,x))_{x}\partial_{x}+(a_{2}(\omega t,x)+m_{2})\partial_{xx}\big{)}h=0,$$

where $x\in\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z},$ $m_{0},m_{2}\in\mathbb{R},$ $a_{0},a_{2}$ are small functions, and $\omega\in\mathcal{O}_{0}\subset\mathbb{R}^{\nu}$ is a diophantine frequency vector. The above system arises from linearizing the dispersive Camassa-Holm (CH) equation with unbounded perturbations on the circle at a small amplitude quasi-periodic function.

There have been a number of literatures to study reducibility of ordinary differential equations (ODEs) and partial differential equations (PDEs). Indeed, the concept of reducibility originates from ODEs (see [11, 26, 38, 40] and the references therein). The problem of reducibility of quasi-periodic linear systems in the PDE’s context has received much attention, mostly in a perturbative regime in these years [5, 6, 27, 37, 49].

A quasi-periodic linear system

$$\partial_{t}h=L(\omega t)h$$

(1.1)

is said to be reducible, if there exists a bounded invertible change of coordinates depending on time quasi-periodically: $h=\Upsilon(\omega t)g$ such that the transformed system is a linear one with constant coefficients, namely,

$$\partial_{t}g=D_{\omega}g,\quad D_{\omega}:={\rm diag}_{j\in\mathbb{Z}}{d_{j}},\quad d_{j}\in\mathbb{C}.$$

Notice that, if $d_{j}$ $(\forall j\in\mathbb{Z})$ is purely imaginary, then the system is linear stable.

In the periodic case, i.e. $\omega\in\mathbb{R}$, the classical Floquet theory shows that any time periodic linear system (1.1) is reducible. In the quasi-periodic case, this is not always true, see e.g., [13]. However, if $L(\omega t)$ is a so-called almost-reducible quasi-periodic vector field in the sense that it can be reduced to the one with constant coefficients up to a small remainder, viz., the linear differential equation of the form

$$\partial_{t}\tilde{g}=D_{\omega}\tilde{g}+\varepsilon P(\omega t)\tilde{g},$$

where $P(\omega t)$ is a linear quasi-periodically forced operator with non-constant coefficients and $\varepsilon$ is the size of the perturbation, then the quasi-periodic system is reducible by perturbative reducibility algorithm. In general, it is assumed that $\varepsilon$ is small enough, $\omega$ and $d_{j}$ satisfy the second-order Melnikov non-resonance conditions involving the differences of the eigenvalues of the operator $D_{\omega}$.

Among the literatures that related to reducibility, a strong motivation of a vast of them is the reducible KAM theory. As is well known, the KAM theory for PDEs is to find a family of approximately invariant tori of perturbed autonomous integrable equation. It is a natural extension of the classical KAM theory for finite dimensional phase spaces. For PDEs in one spacial dimension with bounded perturbations and Dirichlet boundary conditions, we refer to [43, 45, 51, 53], and for those with periodic boundary conditions, we quote [16, 23] for instance. In higher spacial dimension, we mention [10, 12, 28, 35, 52] among others in which the authors have to overcome the difficulties produced by the multiple eigenvalues. In all these aforementioned results, the perturbations are bounded. In the case of unbounded perturbations, we mention [8, 9, 42, 44, 48, 54] among others for semi-linear PDEs, and [2, 3, 4, 7, 32, 36] for quasi-linear or fully nonlinear PDEs. In particular, these results with regard to the KAM theory involve the quasi-linear perturbations of the Airy, KdV equations, etc.

The CH equation

$$u_{t}-u_{xxt}=6uu_{x}-4u_{x}u_{xx}-2uu_{xxx}$$

(1.2)

describes the unidirectional propagation of shallow water waves over a flat bottom [14, 22, 34, 41], where $u(t,x)$ is a function of time $t$ and a single spatial variable $x$. Equation (1.2) is proved to be completely integrable since it admits the Lax-pair and bi-Hamiltonian structure [14, 15, 33, 34, 46]. We refer to [21, 47] and references therein for more literature for the study of the well-posedness of the CH-equation. A tremendous amount of work has been done on strong nonlinear effects of the CH-equation, such as peakon, multi-peakons [1, 14] and wave-breaking phenomena [17, 18, 19, 20, 21, 47].

Note that $\int_{{T}}u(t,x){\rm{d}}x$ is a constant for soluitions of the CH-equation (1.2), the set

$${\mathcal{G}}_{c}:=\{u:\int_{{T}}u(t,x){\rm{d}}x=c\}$$

is invariant under the flow governed by (1.2). Consequently, the dynamics of equation (1.2) on the invariant subsets ${\mathcal{G}}_{c}$ with $c\neq 0$ are equivalent to the ones of the equation

$$u_{t}-u_{xxt}-6cu_{x}+2cu_{xxx}=6uu_{x}-4u_{x}u_{xx}-2uu_{xxx}$$

on the invariant subset ${\mathcal{G}_{0}}$. We assume $c=1$ without loss of generality and impose the unbounded perturbations of the form

$$N(u,u_{x},u_{xx},u_{xxx})=\partial_{x}[(\partial_{u}f)(u,u_{x})-\partial_{x}((\partial_{u_{x}}f)(u,u_{x}))],$$

where $f(u,u_{x})=C^{\infty}(\mathbb{R}\times\mathbb{R},\mathbb{R})$, and $f=f_{\geq 3}(u,u_{x})$ denotes a function with a zero of order at least three at the origin. Then the equation we are concerned with is the perturbed dispersive CH-equation

$$u_{t}-u_{xxt}-6u_{x}+2u_{xxx}=6uu_{x}-4u_{x}u_{xx}-2uu_{xxx}+N(u,u_{x},u_{xx},u_{xxx}),$$

(1.3)

under periodic boundary condition: $x\in\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$.

The Degasperis-Procesi equation [24] exhibits several similar properties as the CH equation, for example, wave breaking phenomena, peakon and soliton solutions [22, 25]. However, the Hamiltonians and symplectic structures are intrinsically different. In [31], the authors developed the KAM theory for Hamiltonian perturbations of the Degasperis-Procesi equation.

At the best of our knowledge, up to now there is no work to study the KAM theory and the reducibility of the CH equation with a small perturbation, which was also emphasized in [31]. In fact, the reducibility of the linearized equations at a small quasi-periodic approximate solutions is the fundamental step for the existence and linear stability of quasi-periodic solutions (KAM tori) for nonlinear PDEs via the Nash-Moser iterative algorithm. This motivates us to study the reducibility of the CH-equation.

Let us first introduce the Hamiltonian setting for the CH-equation and its linearized equation at a small quasi-periodic function.

Equation (1.3) can be formulated as a Hamiltonian PDE of the form $u_{t}=J\nabla H(u)$, where $J:=(1-\partial_{xx})^{-1}\partial_{x}$ and $\nabla H(u)$ is the $L^{2}({{T}},{{R}})$ gradient of the Hamiltonian

$$H(u)=\int_{{T}}\big{(}u^{3}+uu^{2}_{x}+3u^{2}+u^{2}_{x}+f(u,u_{x})\big{)}{\rm{d}}x,$$

defined on the real phase space $H^{1}_{0}(\mathbb{T}):=H^{1}\cap\mathcal{G}_{0}$. The symplectic structure is provided by a non-degenerate 2-form $\varrho$ defined by

$$\varrho(u,v):=\int_{\mathbb{T}}(J^{-1}u)v{\rm d}x,\quad u,v\in H^{1}_{0}.$$

Note that $J$ can be written as

$$J=\Lambda\partial_{x},\quad\Lambda:=(1-\partial_{xx})^{-1}.$$

The functional space be concerned is the Sobolev space $H^{s}({{T}}^{\nu+1};{{R}}),$ $\nu\geq 0,s\in\mathbb{R},$ equipped with the norm

$${\|u(\varphi,x)\|}^{2}_{s}:=\sum\limits_{j\in\mathbb{Z},l\in\mathbb{Z}^{\nu}}{|u_{lj}|}^{2}{\langle l,j\rangle}^{2s}<\infty,$$

where $\langle l,j\rangle:=\max\{1,|j|,|l|\}$, $|l|:=\max\limits_{i=1,\cdots,\nu}|l_{i}|.$ If $u(\varphi,x)$ depends on a parameter $\omega\in\mathcal{O}$, where $\mathcal{O}$ is a compact subset of $\mathbb{R}^{\nu}$, we define the sup-norm and Lipschitz semi-norm of $u$ respectively as follows:

$$\|u\|_{s}^{sup}:=\|u\|_{s}^{sup,\mathcal{O}}:=\sup\limits_{\omega\in\mathcal{O}}\|u\|_{s},$$

$$\|u\|_{s}^{lip}:=\|u\|_{s}^{lip,\mathcal{O}}:=\sup\limits_{\begin{subarray}{c}\omega,\omega^{\prime}\in\mathcal{O},\\ \omega\neq\omega^{\prime}\end{subarray}}\|\Delta_{\omega,\omega^{\prime}}u\|_{s},$$

where

$$\Delta_{\omega,\omega^{\prime}}u:=\frac{u(\omega)-u(\omega^{\prime})}{\omega-\omega^{\prime}}.$$

For $\gamma>0$, the weighted Lipchitz norm of $u$ is defined as

$$\|u\|_{s}^{\gamma,\mathcal{O}}:=\|u\|_{s}^{sup,\mathcal{O}}+\gamma\|u\|_{s-1}^{lip,\mathcal{O}}.$$

Let $m:\mathcal{O}\rightarrow\mathbb{R}$, the sup-norm, Lipschitz semi-norm and weighted Lipchitz norm of $m$ are defined respectively as

$$|m|^{sup}:=|m|^{sup,\mathcal{O}}:=\sup\limits_{\omega\in\mathcal{O}}|m|,$$

$$|m|^{lip}:=|m|^{lip,\mathcal{O}}:=\sup\limits_{\begin{subarray}{c}\omega,\omega^{\prime}\in\mathcal{O},\\ \omega\neq\omega^{\prime}\end{subarray}}|\frac{m(\omega)-m(\omega^{\prime})}{\omega-\omega^{\prime}}|,$$

$$|m|^{\gamma,\mathcal{O}}:=|m|^{sup,\mathcal{O}}+\gamma|m|^{lip,\mathcal{O}}.$$

In the sequel, we fix $s_{0}:=[(\nu+1)/2]+2$. For all $s\geq s_{0}$, the Sobolev space $H^{s}$ is a Banach algebra.

Fix $\nu\in\mathbb{N}\setminus\{0\}$, $L>0$, let $\gamma\in(0,1)$, the frequency vector of oscillations $\omega=(\omega_{1},\dots,\omega_{\nu})$ satisfies the diophantine condition, i.e., $\omega\in\mathcal{O}_{0}\subset\Omega$, where

$$\mathcal{O}_{0}:=\big{\{}\omega\in\Omega:|\omega\cdot l|\geq\frac{2\gamma}{\langle l\rangle^{\nu}},l\in\mathbb{Z}^{\nu}\setminus\{0\}\big{\}},\quad\langle l\rangle:=\max\{|l|,1\},$$

(1.4)

$$\Omega:=\{\omega\in\mathbb{R}^{\nu}:\omega\in[L,2L]^{\nu}\}.$$

(1.5)

A quasi-periodic function with $\nu$ frequencies is defined by an embedding

$$\mathbb{T}^{\nu}\ni\varphi\mapsto\mathfrak{J}(\varphi,x),\quad\varphi=\omega t$$

with the frequency vector $\omega\in\mathcal{O}_{0}$. Then a small-amplitude quasi-periodic solution $u(t,x)$ of equation (1.3) can be represented as

$$u(t,x)=\varepsilon\mathfrak{J}(\varphi,x),\quad\varepsilon\ll 1,\quad\mathfrak{J}(\varphi,x)\in C^{\infty}(\mathbb{T}^{\nu+1},\mathbb{R}).$$

The linearized equation of (1.3) at the quasi-periodic solution $u(t,x)$ is

$\displaystyle\begin{aligned} h_{t}=&\;J(\partial_{u}\nabla H)[h]\\ =&J\circ\big{(}(a_{0}(\omega t,x)+6)+a_{2,x}(\omega t,x)\partial_{x}+(a_{2}(\omega t,x)-2)\partial_{xx}\big{)}h,\end{aligned}$

(1.6)

where $a_{i}(\varphi,x)\in C^{\infty}(\mathbb{T}^{\nu+1},\mathbb{R})$, $i=0,2$, depend on the the parameter $\omega\in\mathcal{O}_{0}$ in a Lipschitz way, as well as on the quasi-periodic function $\mathfrak{J}$. The operator associated with the above equation is

	$\displaystyle\mathcal{L}^{*}=$	$\displaystyle\omega\cdot\partial_{\varphi}-J\circ\big{(}(a_{0}(\omega t,x)+6)$		(1.7)
		$\displaystyle\quad+a_{2,x}(\omega t,x)\partial_{x}+(a_{2}(\omega t,x)-2)\partial_{xx}\big{)}.$

It is noticed that this operator is a Hamiltonian operator, and exhibits the reversible structure.

One can check that the operator $L^{*}(\omega t)$ has the form $L^{*}(\omega t)=-(a_{2}-2)\partial_{x}+{\rm Op}(r)$ where ${\rm Op}(r)$ is a pseudo differential operator of order $-1$ (see Definition A.2). Therefore, it is a new type of operator different from the ones for the KdV and Degasperis-Procesi equations, etc.

In the following, we consider a class of generalized linear operators of the form

$$\mathcal{L}=\omega\cdot\partial_{\varphi}-J\circ\big{(}(a_{0}(\varphi,x)+m_{0})+a_{2,x}(\varphi,x)\partial_{x}+(a_{2}(\varphi,x)+m_{2})\partial_{xx}\big{)}.$$

(1.8)

Now we make the following assumptions for the operators $\mathcal{L}$:

(A1). $a_{i}(\varphi,x)\in C^{\infty}(\mathbb{T}^{\nu+1},\mathbb{R})$, $i=0,2$, are even functions, $m_{0},m_{2}\in\mathbb{R}$.

(A2). There exists a positive constant $\mu$ sufficiently large such that

$$\|\mathfrak{J}\|_{s_{0}+\mu}^{\gamma,\mathcal{O}_{0}}\leq 1.$$

(1.9)

(A3). Assume that $a_{i}$, $i=0,2$, depend on $\omega\in\mathcal{O}_{0}$ in a Lipschitz way, satisfying

$$\|a_{i}\|_{s}^{\gamma,\mathcal{O}_{0}}\leq_{s}\varepsilon\|\mathfrak{J}\|_{s+\eta_{0}}^{\gamma,\mathcal{O}_{0}},\quad i=0,2,\quad\forall s\geq s_{0}$$

(1.10)

for some $\eta_{0}>0$.

(A4). Assume that $a_{i}$, $i=0,2$, depend on the quasi-periodic function $\mathfrak{J}$. Let $\mathfrak{J}_{1},\mathfrak{J}_{2}\in C^{\infty}(\mathbb{T}^{\nu+1},\mathbb{R})$ satisfying (1.9) and

$$\|\Delta_{12}a_{i}\|_{p}\leq_{p}\varepsilon\|\mathfrak{J}_{1}-\mathfrak{J}_{2}\|_{p+\eta_{0}},\quad i=0,2,$$

for any $s_{0}\leq p\leq s_{0}+\mu-\eta_{0}(\mu>\eta_{0})$, where $\Delta_{12}v:=v(\mathfrak{J}_{1},\varphi,x)-v(\mathfrak{J}_{2},\varphi,x)$ for any $v(\varphi,x)=v(\mathfrak{J},\varphi,x)\in C^{\infty}(\mathbb{T}^{\nu+1},\mathbb{R})$.

The main goal in this paper is to prove that the linear operators (1.8) arising from the linearized CH-equation with unbounded perturbations at a small and sufficiently smooth quasi-periodic function on $\mathbb{T}$ are reducible.

It is well known that the most common algebraic structures that ensure the existence of quasi-periodic motions are the Hamiltonian or reversible structures. The system considered here exhibits both of them. From the perspective of the Hamiltonian structure, we have to find symplectic transformations induced by the flow of the vector field generated by a Hamiltonian function. The flow is given by $\partial_{\tau}A^{\tau}u=B^{\tau}A^{\tau}u$, where $B^{\tau}$ is a pseudo differential operator of order $\leq-1$ (see Definition A.2) so as to guarantee the transformations are bounded. The solution $A^{\tau}u$ of the flow is of the form: $C+OPS^{-1}$ for some $C\in\mathbb{R}$, while $OPS^{-1}$ is a pseudo differential operator of order $-1$. Unfortunately, $A^{\tau}u$ can not change the leading order terms of the operator (1.8). Thus in order to prove the reducibility of the system, we utilize the reversible structure and choose the transformations preserving the reversible structure.

It is remarked that for the Degasperis-Procesi equation, the operator $J$ associated with the symplectic structure is a pseudo differential operator of order 1, which is also true for the KdV equation. Nevertheless, for the CH-equation, $J=(1-\partial_{xx})^{-1}\partial_{x}$ is a pseudo differential operator of order $-1$. Such difference needs us to develop a different strategy to verify the corresponding reducibility.

The existence and stability of the solution for the Cauchy problem of the linearized equation (1.6) are also the direct results of this paper (see Corollary 1.1). The stability entails the purely-imaginary eigenvalues of the diagonal operator. Indeed, the reversible structure ensures that the eigenvalues of the diagonal matrices are all purely-imaginary.

The reducibility result for the linearized CH-equation (1.3) at a small and sufficiently smooth quasi-periodic function is as follows.

The following corollary indicates the dynamical consequence of Theorem 1.1.

In the case of $m_{2}=-2,m_{0}=6$, Theorem 1.1 is a conclusion of the following result.

We explain the key ideas and the outline of this paper as follows.

In Section 2, we introduce some fundamental definitions, notations and lemmas. Particularly, we introduce the classes of Lip-$-1$-modulo-tame linear operators (see Definition 2.6) and study their properties which will be used in Section 4. The reduction procedure is split into two sections.

In Section 3, we perform the regularization procedure. This step consists in conjugating $\mathcal{L}(\omega t)$ in (1.8) to the linear operator which is a small bounded regularizing perturbation of a constant coefficient. This is achieved by applying a suitable quasi-periodically change of variables depending on time so that the highest order (order 1) term has a constant coefficient. Meanwhile, the term of order 0 is eliminated. Then we extract the terms that is not "small" from the ones of order $-1$, to constitute the diagonal part. This is the content of Theorem 3.1.

In Section 4, the KAM reducibility scheme is proposed. After the preliminary reduction of the order of derivatives, we can perform a KAM reducibility scheme to complete the diagonalization, see Theorem 4.1. We use the Lip-$-1$-modulo-tame constants to estimate the size of the remainders along the iteration. This is convenient since the class of Lip-$-1$-modulo-tame operators are closed under the composition (Lemma 2.5), the solution map of the homological equation (Lemma 4.1) and the projections (Lemma 2.7). As a matter of fact, the properties mentioned above also hold for Lip-$0$-modulo-tame operators (see Definition 2.5). The reason to adopt Lip-$-1$-modulo-tame operators instead of Lip-$0$-modulo-tame operators is that we require that $\langle j\rangle r_{j}^{\infty}$ is small enough in order to prove the relations between the bad sets $P_{ljj^{\prime}}$ and $Q_{l,j}$ in (5.1).

In Section 5, we give the measure estimate for the set $\mathcal{O}_{0}-\mathcal{O}_{\infty}$. Therefore, the operator in (1.8) can be reduced to the one with constant coefficients for almost all $\omega$ in the sense of Lebesgue measure. At the same time, Corollary 1.1 is verified.

Finally, in the appendices, we introduce some classes of linear operators which are mentioned in the paper, such as pseudo-differential operators, the classes of operators $\mathfrak{L}_{\rho,p}$. The classical results for the change of variable are provided as well.

In this section, we elaborate some conceptions, notations and lemmas, which will be used in the subsequent sections.

It is obvious that the Sobolev norms of $u$ and $\underline{u}$ are equal, namely,

$$\|u\|_{s}=\|\underline{u}\|_{s}.$$

The following lemma is an important property of $\|\cdot\|_{s}$ and $\|\cdot\|_{s}^{\gamma,\mathcal{O}}$.

In the following, we introduce a class of operators, whose matrix representations are Töplitz in time, and the relevant properties.

In terms of matrix coefficients, the above definitions are equivalent to the following:

(1). $A$ is real $\Leftrightarrow A_{-j}^{-j^{\prime}}(-l)=\overline{A_{j}^{j^{\prime}}(l)}$.

(2). $A$ is reversible $\Leftrightarrow A_{-j}^{-j^{\prime}}(-l)=-A_{j}^{j^{\prime}}(l)$.

(3). $A$ is reversibility-preserving $\Leftrightarrow A_{-j}^{-j^{\prime}}(-l)=A_{j}^{j^{\prime}}(l)$.

In the following, we collect several properties of the Lip-$-1$-modulo-tame operators.