Stability of smooth multi-solitons for the Camassa–Holm equation

Abstract

Consideration in this paper is the stability of exact smooth multi-solitons for the Camassa–Holm equation. By constructing a suitable Lyapunov functional, it is found that the smooth multi-solitons are non-isolated constrained minimizers satisfying a suitable variational nonlocal elliptic equation and the dynamical stability issue is reduced to study of the spectrum of explicit linearized systems. Our approach in the spectral analysis consists in an invariant for the multi-solitons and new operator identities motivated by the bi-Hamilton structure of the Camassa–Holm equation. The key ingredient in the spectral analysis is to use integrable property of the recursion operator of the Camassa–Holm equation. It is demonstrated here that orbital stability of shape of smooth single soliton implies that the shapes of all smooth multi-solitons are dynamically stable under small disturbances in a suitable Sobolev space.

Appendix

The tools presented in this section have been developed in [34, 36, 46]. It is noted that the work of Neves and Lopes [46] was devoted to the case of the double solitons and [36] extends their results to the case of N-solitons with N an arbitrary integer. For the sake of completeness, we give the most relevant elements of the statement only and refer to [34, 36, 46] for the details of the proof and further discussion.

1.1 Iso-inertial family of operators

We will be working with linearized operators around a multi-soliton, which fit in the following more generic framework.

Consider the abstract evolution equation

$$\begin{aligned} \frac{du}{dt} =f(u), \end{aligned}$$

(4.23)

for $u : \mathbb {R}\rightarrow X$, and recall that the following framework was set in [34, 46]. Let $X_2\subset X_1\subset X$ be Hilbert spaces and $V : X_1\rightarrow \mathbb {R}$ be such that the following assumptions are verified.

(H1) $X_2\subset X_1\subset X$ are continuously embedded. The embedding from $X_2$ to $X_1$ is denoted by i.

(H2) The functional $V: X_1\rightarrow \mathbb {R}$ is ${\mathcal {C}}^3$.

(H3) The function $f: X_2\rightarrow X_1$ is ${\mathcal {C}}^2$.

(H4) For any $u\in X_2$, we have $V'(i(u))f(u)=0.$ Moreover, given $u \in {\mathcal {C}}^1(\mathbb {R}, X_1)\cap {\mathcal {C}}(\mathbb {R}, X_2)$ a strong solution of (4.23), we assume that there exists a self-adjoint operator $L(t) : D(L) \subset X\rightarrow X$ with domain D(L) independent of t such that for $h, k \in Z$, where $Z\subset D(L)\cap X_2$ is a dense subspace of X, we have $\displaystyle \langle L(t)h,k\rangle =V''(u(t))(h,k).$ We also consider the operators $B(t):D(B)\subset X\rightarrow X$ such that for any $h\in Z$ we have $B(t)h=-f'(u(t))h.$ Then we assume moreover that

(H5) The closed operators B(t) and $B^*(t)$ have a common domain D(B) which is independent of t. The Cauchy problems

$$\begin{aligned} \frac{du}{dt} =B(t)u,\quad \frac{dv}{dt} =B^*(t)v, \end{aligned}$$

are well-posed in X for positive and negative times.

We then have the following result (see [34, 46]).

Proposition 4.1

Let $u\in {\mathcal {C}}^1(\mathbb {R}, X_1)\cap {\mathcal {C}}(\mathbb {R}, X_2)$ be a strong solution of (4.23) and assume that (H1)-(H5) are satisfied. Then the following assertions hold.

$\bullet$ Invariance of the set of critical points. If there exists $t_0 \in \mathbb {R}$ such that $V'(u(t_0))=0$, then $V'(u(t))=0$ for any $t\in \mathbb {R}$.

$\bullet$ Invariance of the inertia. Assume that u is such that $V'(u(t))=0$ for all $t\in \mathbb {R}$. Then the inertia in(L(t)) of the operator L(t) representing $V''(u(t))$ is independent of t.

1.2 Calculation of the inertial

Given an t-dependent family of operators whose inertia we are interested in, Proposition 4.1 allows to choose for a specific t to perform the calculation of the inertia. This is however in most situations not sufficient, as we would like to let t go to infinity and relate the inertia of our family with the inertia of the asymptotic objects that we obtain. This is what is allowed in the following framework.

Let X be a real Hilbert space. Let $N \in \mathbb {N}$ and $(\tau _n^j)$ be sequences of isometries of X for $j = 1,\ldots , N$. For brevity in notation, we denote the composition of an isometry $\tau _n^k$ and the inverse of $\tau _n^j$ by

$$\begin{aligned} \tau _n^{k/j}:=\tau _n^k(\tau _n^j)^{-1}. \end{aligned}$$

Let $A, (B^j)_{j=1,\ldots ,N}$ be linear operators and $(R_n)$ be a sequence of linear operators. Define the sequences of operators based on $(B^j)$ and $(\tau _n^j)$ by

$$\begin{aligned} B_n^j=(\tau _n^j)^{-1}B_j(\tau _n^j). \end{aligned}$$

Define the operator $L_n: D(A)\subset X \rightarrow X$ by

$$\begin{aligned} L_n=A+\sum _{j=1}^NB_n^j+R_n. \end{aligned}$$

We make the following assumptions.

(A1) For all $j=1,\ldots , N$ and $n\in \mathbb {N}$, the operators $A, A + B^j , A + B_n^j$ and $L_n$ are self-adjoint with the same domain D(A).

(A2) The operator A is invertible. For all $j=1,\ldots , N$ and $n\in \mathbb {N}$, the operator A commutes with $\tau _n^j$ (i.e. $A=(\tau _n^j)^{-1}A(\tau _n^j)$).

(A3) There exists $\delta >0$ such that for all $j=1,\ldots , N$ and $n\in \mathbb {N}$, the essential spectrum of $A, A + B_j , A + B_n^j$ and $L_n$ are contained in $(\delta , +\infty )$.

(A4) For every $\lambda \in \cap _{j=1}^N\rho (A+B^j)$ and for all $j=1,\ldots , N$ the operators $A(A+B^j-\lambda I)^{-1}$ are bounded.

(A5) In the operator norm, $\Vert R_nA^{-1}\Vert \rightarrow 0$ as $n\rightarrow +\infty$.

(A6) For all $u\in D(A)$ and $j,k=1,\ldots ,N$ and $j\ne k$ one has

$$\begin{aligned} \lim _{n\rightarrow +\infty }\Vert \tau _n^{j/k}B^k\tau _n^{k/j}\Vert _X=0. \end{aligned}$$

(A7) For all $u\in X$ and $j,k=1,\ldots ,N$ and $j\ne k$ we have $\tau _n^{j/k}u\rightharpoonup 0$ weakly in X as $n\rightarrow \infty$.

(A8) For all $j=1,\ldots , N$, the operators $B^jA^{-1}$ is compact.

Theorem 4.1

Assume that assumptions (A1)-(A8) hold and let $\lambda <\delta$. The following assertions hold.

$\bullet$ If $\lambda \in \cap _{j=1}^N\rho (A+B^j)$, then there exists $n_\lambda \in \mathbb {N}$ such that for all $n>n_\lambda$ we have $\lambda \in \rho (L_n)$.

$\bullet$ If $\lambda \in \cup _{j=1}^N\sigma (A+B^j)$, then there exists $\varepsilon _0>0$ such that for all $0<\varepsilon <\varepsilon _0$ there exists $n_\varepsilon \in \mathbb {N}$ such that for all $n > n_\varepsilon$ we have

$$\begin{aligned} dim(Range(P_{\lambda ,\varepsilon }(L_n)))=\sum _{j=1}^Ndim(Range(P_{\lambda ,\varepsilon }(A+B^j))), \end{aligned}$$

where $P_{\lambda ,\varepsilon }(L)$ is the spectral projection of L corresponding to the circle of center $\lambda$ and radius $\varepsilon$.

Corollary 4.2

Under the assumptions of Theorem 4.1, if there exists $n_L$ such that for all $n > n_L$ we have

$$\begin{aligned} dim(ker(L_n))\ge \sum _{j=1}^Ndim(ker(A+B^j)), \end{aligned}$$

then for all $n > n_L$ we have

$$\begin{aligned} in(L_n)=\sum _{j=1}^Nin(A+B^j). \end{aligned}$$

Moreover, a non-zero eigenvalue of $L_n$ cannot approach 0 as $n\rightarrow \infty$.

Theorem 4.1 and Corollary 4.2 were proved in [46] in the case $N=2$. For the proof of general $N \in \mathbb { N}$ cases, we refer to [36] for details.