Global attraction to solitons for 2D Maxwell–Lorentz

equations with spinning particle

E.A. Kopylova111 Supported partly by Austrian Science Fund (FWF) P34177

Faculty of Mathematics of Vienna University
Institute for Information Transmission Problems RAS

A.I. Komech

Institute for Information Transmission Problems RAS

###### Abstract

We consider the 2D Maxwell–Lorentz system which describes a rotating particle
coupled to the Maxwell field. The system admits stationary soliton-type
solutions. We prove the attraction to solitons for any finite energy solution
relying on the conservation of angular momentum.

##  1 Introduction

The 2D Maxwell–Lorentz equations reads [6, 15]

$\left\\{\begin{array}[]{rcl}\dot{E}(x,t)&=&J\nabla
B(x,t)-[\dot{q}(t)-\omega(t)J(x-q(t))]\rho(x-q(t))\\\
\dot{B}(x,t)&=&-\nabla\cdot(JE(x,t)),\qquad\nabla\cdot E(x,t)=\rho(x-q(t))\\\
m\ddot{q}(t)&=&\langle
E(x,t)+B(x,t)[J\dot{q}(t)+\omega(t)(x-q(t))],\rho(x\\!-\\!q(t))\rangle\\\
I\dot{\omega}(t)&=&\langle(x\\!-\\!q(t))\cdot\big{[}JE(x,t)\\!-B(x,t)\dot{q}(t)],\rho(x\\!-\\!q(t))\rangle\end{array}\right|,\quad
x\in\mathbb{R}^{2}.$ (1.1)

Here $(E(x,t),B(x,t))\in\mathbb{R}^{2}\oplus\mathbb{R}^{2}$ is the Maxwell
field, $\rho(x-q(t))\in\mathbb{R}$ is the charge distribution of an extended
particle centered at a point $q(t)\in R^{2}$, $m$ is the mass of the particle,
$I$ is its moment of inertia, and $\omega(t)$ is the angular velocity of the
particle rotation. The brackets $\langle\,,\,\rangle$ denote the inner product
in the Hilbert space $L^{2}:=L^{2}(\mathbb{R}^{2})\otimes\mathbb{R}^{2}$, and

$J=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}.$ (1.2)

We restrict ourselves to the situation, where the spinning particle is located
at the origin, i.e. $q(t)\equiv 0$. This can be achieved by assuming the
(anti-) symmetry conditions $E(-x)=-E(x)$, $B(-x)=B(x)$ for the initial
fields. Then this property persists for all times. In this case the
Maxwell–Lorentz equations (1.1) simplify to the linear system

$\left\\{\begin{array}[]{rcl}\dot{E}(x,t)&=&J\nabla
B(x,t)+\omega(t)Jx\rho(x),\\\ \dot{B}(x,t)&=&-\nabla\cdot
JE(x,t),\qquad\nabla\cdot E(x,t)=\rho(x)\\\ I\dot{\omega}(t)&=&\langle x\cdot
JE(x,t),\rho(x)\rangle\end{array}\right|.$ (1.3)

We assume that real-valued charge density $\rho(x)$ is a smooth rapidly
decreasing spherically symmetric function, i.e.,

$\rho\in{\cal S}(\mathbb{R}^{2}),\qquad\rho(x)=\rho_{1}(|x|).$ (1.4)

Moreover, we impose that the total charge vanishes (neutrality of the
particle):

$\int\rho(x)\,dx=0.$ (1.5)

The system admits stationary solutions (solitons)
$(E_{\omega}(x),B_{\omega}(x),\omega)$ rotating with constant angular velocity
$\omega\in\mathbb{R}$. The condition $\hat{\rho}(0)=0$ provides that the
solitons belong to the finite energy phase space (2.9).

One main result is the long-time convergence in weighed energy norms of any
finite energy solution with the angular momentum $M$ to the soliton with the
same angular momentum.

Our proof relies on the regularity of the resolvent $({\bf H}-\lambda)^{-1}$
at the threshold $\lambda=0$ , where

${\bf H}:=\left(\begin{array}[]{cccc}0&1\\\
\Delta-\frac{1}{I}|J\varrho\rangle\langle J\varrho|&0\\\
\end{array}\right),\quad\varrho(x)=x\rho(x)$ (1.6)

is the operator defined on the space (2.9). We calculate the Puiseux expansion
of the resolvent up to $\lambda^{2}$ by suitable development of the
Agmon–Jensen–Kato theory [1, 4, 5, 12]. The condition (1.5) provides that the
resolvent belongs to $C^{2}$ at a vicinity of $\lambda=0$. This fact implies
the decay $|t|^{-2}$ of the solution to (4.4) in weighted energy norms (3.3)
for sufficiently large moment of inertia $I$. This decay rate allows us to
obtain scattering asymptotics (3.6) in the global energy norm (without a
weight).

Note that the decay $|t|^{-2}$ in the weighted energy norms (3.3) does not
contradict the well known slow local decay $|t|^{-1}$ of solutions of the 2D
wave equation. Indeed, the slow decay $|t|^{-1}$ corresponds to the
singularity of the resolvent $(\Delta-\lambda)^{-1}\sim\log\lambda$ at the
threshold $\lambda=0$. The corresponding resonances are the constant
functions. However, the decay in the weighted norms (3.3) is better since the
norm includes spatial derivatives of the solution (which vanish on the
resonances), but does not include the values of the solution.

Let us comment on previous related results. The global attraction to
stationary states was established in [9, 11] for 3D wave system with a moving
particle without rotation. In [10], the global attraction to stationary states
was established for 3D Maxwell–Lorentz system with a relativistic particle
without rotation in presence of an external confining potential. The global
convergence to solitons of such system without external potential was proved
in [2]. The results [2, 10] provide the first rigorous proof of the radiation
damping in classical electrodynamics. These results were obtained under the
Wiener-type condition on the charge density $\rho(x)$.

The global convergence to rotating solitons was established in [3] for
solutions to the 3D Maxwell–Lorentz system with $q(t)\equiv 0$ in the case of
sufficiently small charge density $\rho$. The global attraction to the set of
all solitons for this system was established in [12] under a considerably
weaker Wiener-type condition than in [2]. The survey of the results for 3D
system can be found in [7, 8].

In [6], we have proved the orbital stability of moving and rotating solitons
for the 2D Maxwell–Lorentz system (1.1). The proof relies on the reduction of
the system by the canonical transformation to a comoving frame and the
conservation of the corresponding linear and angular momenta.

##  2 The Maxwell potentials

In this section we recall the Hamiltonian structure for the system (1.3)
expressed in the Maxwell potentials $A(x,t)=(A_{1}(x,t),A_{2}(x,t))$ and
$\Phi(x,t)$:

$B(x,t)=\nabla\cdot(JA)=\nabla_{1}A_{2}(x,t)-\nabla_{2}A_{1}(x,t),\qquad
E(x,t)=-\dot{A}(x,t)-\nabla\Phi(x,t).$ (2.1)

We choose the Coulomb gauge

$\nabla\cdot A(x,t)=0.$ (2.2)

Then the first two equations of (1.3) are equivalent to the system

$\left\\{\begin{array}[]{rcl}-\ddot{A}(x,t)-\nabla\dot{\Phi}(x,t)&=&-\Delta
A(x,t)+\omega(t)Jx\rho(x),\\\ -\Delta\Phi(x,t)&=&\rho(x)\end{array}\right|.$
(2.3)

The second equation of (2.3) can be solved explicitly:

$\Phi(x,t)=\Phi(x)=-\frac{1}{2\pi}\int\log|x-y|\rho(y)dy.$ (2.4)

In the Fourier transform $\hat{\Phi}_{0}(k)=\hat{\rho}(k)/k^{2}$. Hence,
$\Phi(x)\in{\cal S}(\mathbb{R}^{2})$ due to (1.5) and spherical symmetry
(1.4). Now the last equation of (1.3) becomes

$I\dot{\omega}(t)=-\langle
J(\dot{A}(x,t)+\nabla\Phi(x),\varrho(x)\rangle=-\langle
J\dot{A}(x,t),\varrho(x)\rangle,\quad\varrho(x):=x\rho(x)$ (2.5)

since the spherical symmetry (1.4) implies

$\langle J\nabla\Phi(x),\varrho(x)\rangle=\langle x\cdot
J\nabla\Phi(x),\rho(x)\rangle=-\langle\nabla\cdot
Jk\hat{\Phi}(k),\hat{\rho}(k)\rangle=-\int(k_{2}\nabla_{1}-k_{1}\nabla_{2})\frac{\hat{\rho}^{2}(k)}{k^{2}}dk=0.$

The angular momentum is defined as in [6]:

$M=I\omega-\langle A,J\varrho\rangle.$ (2.6)

By (2.5) and (2.1),

$\dot{M}(t)=I\dot{\omega}(t)-\langle\dot{A}(x,t),J\varrho(x)\rangle\\\
=-\langle J\dot{A},\varrho(x)\rangle-\langle\dot{A},J\varrho(x)\rangle=0.$

Thus, the angular momentum is conserved: $M(t)=M$, $t\in\mathbb{R}$. Hence,
the system (1.3) can be rewritten as the linear inhomogeneous system

$\left\\{\begin{array}[]{llll}\dot{A}(x,t)=\Pi(x,t)\\\ \dot{\Pi}(x,t)=\Delta
A(x,t)-\omega(t)J\varrho(x)\end{array}\right|,$ (2.7)

where

$\omega(t):=\frac{1}{I}\big{(}M+\langle A(t),J\varrho\rangle\big{)}.$ (2.8)

Let us introduce a phase space for the system (2.7). Denote by $\dot{H}^{1}$
the closure of $C_{0}^{\infty}(\mathbb{R}^{2})\otimes\mathbb{R}^{2}$ with
respect to the norm $\|A\|_{\dot{H}^{1}}=\|\nabla A\|_{L^{2}}$, where
$L^{2}:=L^{2}(\mathbb{R}^{2})\otimes\mathbb{R}^{2}$. Define the energy phase
space

${\cal E}=\dot{H}^{1}\oplus L^{2}.$ (2.9)

Obviously, (2.7) is the canonical Hamilton system with the Hamilton functional

$H(Y)=\frac{1}{2}\int\big{(}\Pi^{2}+|\nabla
A|^{2}\big{)}dx+\frac{I\omega^{2}}{2},\quad Y:=(A,\Pi)\in{\cal E},$ (2.10)

where $\frac{I\omega^{2}}{2}=\frac{1}{2I}(M+\langle
A(x),J\varrho(x)\rangle)^{2}$ in accordance with (2.8). The Hamiltonian (2.10)
is well defined and Fréchet differentiable on the Hilbert phase space (2.9).
The well-posedness for the system (2.7) is proved in [6].

######  Proposition 2.1.

Let (1.4) holds, and let $Y_{0}=(A_{0},\Pi_{0})\in{\cal E}$. Then
(i) there exists a unique solution $Y(t)\in C(\mathbb{R},{\cal E})$ to the
Cauchy problem for (2.7);
(ii) the energy is conserved: $H(Y(t))=H(Y_{0})$ for $t\in R$;

###  2.1 Solitons

The solitons of the system (2.7) are stationary solutions

$Y_{\omega}=(A_{\omega}(x),0).$ (2.11)

Substituting into (2.7), we get

$\Delta A_{\omega}(x)=\omega J\varrho(x).$ (2.12)

In the Fourier representation

$\hat{A}_{\omega}(k)=-\frac{\omega J\hat{\varrho}(k)}{k^{2}}.$ (2.13)

By (1.5), $\hat{A}_{\omega}(k)\sim k$, as $|k|\sim 0$. Hence, $\nabla
A_{\omega}\in L^{2}(\mathbb{R}^{2})$. The angular momentum $M_{\omega}$ of the
soliton (2.11) is expressed by

$M_{\omega}=I\omega+\omega\langle\frac{J\tilde{\varrho}}{k^{2}},J\tilde{\varrho}\rangle=\omega(I+\varkappa_{0}),\quad\varkappa_{0}=\int\frac{|\hat{\varrho}(k)|^{2}dk}{k^{2}}.$
(2.14)

##  3 Main result

To state our main result we need some notations. Denote

$\varkappa(\lambda)=\int\frac{|\hat{\varrho}(k)|^{2}dk}{k^{2}+\lambda^{2}},\quad{\rm
Re{\hskip 1.42262pt}}\lambda>0.$ (3.1)

The function $\varkappa(\lambda)$ is analytic in the halfplane ${\rm Re{\hskip
1.42262pt}}\lambda>0$. By (1.4) and by the Sokhotski-Plemelj formulas, there
exists the limit

$\varkappa(i\mu+0)=\lim_{\varepsilon\to+0}\varkappa(i\mu+\varepsilon),\quad\mu\in\mathbb{R},$

where $\varkappa(+0)=\varkappa(0)$ by (1.5). By Corollary 6.3,
$\varkappa(i\mu+0)\in C^{2}(\mathbb{R}\setminus 0)\cup C^{1}(\mathbb{R})$. We
will suppose that

$I+\varkappa(i\mu+0)\neq 0,\qquad\mu\in\mathbb{R}.$ (3.2)

Note that the condition holds obviously for $\mu=0$. It holds for all
$\mu\in\mathbb{R}$ if the moment of inertia $I$ is sufficiently large.

Let $H^{s}_{\beta}=H^{s}_{\beta}(\mathbb{R}^{2})$ be the weighted Sobolev
spaces with the finite norm

$\|f\|_{H^{s}_{\beta}}=\|\langle
x\rangle^{\beta}\langle\nabla\rangle^{s}f\|_{L^{2}(\mathbb{R}^{2})},\quad\langle
x\rangle=(1+|x|^{2})^{1/2}.$

Denote $L^{2}_{\beta}=H^{0}_{\beta}$.

######  Definition 3.1.

i) ${\cal E}_{\beta}$, $~{}\beta\in R$, is the Hilbert space of vector fields
$(A,\Pi)$ with finite norm

$\|(A,\Pi)\|_{\beta}:=\|(A,\Pi)\|_{{\cal E}_{\beta}}=\|\nabla A\|_{{\bf
L}^{2}_{\beta}}+\|\Pi\|_{{\bf L}^{2}_{\beta}}.$ (3.3)

ii) ${\cal E}_{\beta}^{+}$ is the Hilbert space of vector fields $(A,\Pi)$
with finite norm

$\|(A,\Pi)\|_{{\cal E}_{\beta}^{+}}=\|A\|_{{\bf L}^{2}_{\beta}}+\|\nabla
A\|_{{\bf L}^{2}_{\beta}}+\|\Pi\|_{{\bf L}^{2}_{\beta}}.$

In these notations, ${\cal E}={\cal E}_{0}$.

######  Theorem 3.2.

Let conditions (1.4), (1.5) and (3.2) hold, and $Y_{0}\in{\cal E}_{\beta}^{+}$
with $\beta>5/2$. Let $Y(t)\in C(\mathbb{R},{\cal E})$ be the solution to
(2.7) with initial data $Y_{0}$ and the corresponding angular momentum $M$. i)
Then the solution converges to the soliton (2.11) with the same angular
momentum $M$:

$|\omega(t)-\omega_{*}|={\cal
O}(|t|^{-2}),\quad\|Y(t)-Y_{\omega_{*}}\|_{-\beta}={\cal O}(|t|^{-2}),\quad
t\to\infty,$ (3.4)

where the frequency $\omega_{*}$ is defined by (2.14):

$\omega_{*}:=\frac{M}{I+\varkappa(0)}.$ (3.5)

ii) The scattering asymptotics hold

$Y(x,t)=Y_{\omega_{*}}(x)+W(t)\Psi+r(x,t),\quad t\to\pm\infty.$ (3.6)

Here $W(t)$ is the dynamical group of the free wave equation, $\Psi\in{\cal
E}$, and the remainder $r(x,t)$ decays to zero in the global energy norm:

$\|r(t)\|_{\cal E}={\cal O}(|t|^{-1}),\quad t\to\pm\infty.$ (3.7)

##  4 Extraction of a soliton

We split a solution to the system (2.7) as the sum

$Y(t)=Y_{\omega_{*}}+Z(t),\qquad
Y_{\omega_{*}}=\left(\\!\\!\begin{array}[]{c}A_{\omega_{*}}(x)\\\
0\end{array}\\!\\!\right),\quad
Z(t)=\left(\\!\\!\begin{array}[]{c}\Lambda(x)\\\
\Pi(x)\end{array}\\!\\!\right).$ (4.1)

Substituting into (2.7), we obtain

$\left\\{\begin{array}[]{rcl}\dot{\Lambda}(x,t)&=&\Pi(x,t)\\\
\dot{\Pi}(x,t)&=&\Delta
A_{\omega_{*}}(x)+\Delta\Lambda(x,t)-\omega(t)J\varrho(x)\end{array}\right|.$
(4.2)

Using equation (2.12) with $\omega=\omega_{*}$, and equation (2.8) with
$\omega=\omega(t)$, we get

$\dot{\Pi}=\omega_{*}J\varrho+\Delta\Lambda-\omega
J\varrho=\Delta\Lambda-\frac{1}{I}(\langle\Lambda+A_{\omega_{*}},J\varrho\rangle-\langle
A_{\omega_{*}},J\varrho\rangle)J\varrho=\Delta\Lambda-\frac{1}{I}\langle\Lambda,J\varrho\rangle
J\varrho.$ (4.3)

Now the system (4.2) reads

$\dot{Z}(t)={\bf H}Z(t),\quad t\in\mathbb{R},$ (4.4)

where the operator ${\bf H}$ is defined by (1.6).

######  Lemma 4.1.

i) The system (4.4) formally can be written as the Hamilton system

$\dot{Z}(t)=JD{\cal H}(Z(t)),~{}~{}~{}~{}~{}~{}~{}t\in\mathbb{R},$ (4.5)

where $D{\cal H}$ is the Fréchet derivative of the Hamilton functional

${\cal
H}(Z)=\frac{1}{2}\int\Big{[}|\Psi|^{2}+|\nabla\Lambda|^{2}\Big{]}dx+\frac{1}{2I}\langle\Lambda,J\varrho\rangle^{2},\quad
Z=(\Lambda,\Psi)\in{\cal E}.$ (4.6)

ii) Energy conservation holds for solutions $Z(t)\in C^{1}(\mathbb{R},{\cal
E})$:

${\cal H}(Z(t))=\mathop{\rm const}\nolimits,\qquad t\in\mathbb{R}.$ (4.7)

##  5 Solution of equation (4.4)

We apply the Fourier-Laplace transform

$\tilde{Z}(\lambda)=\int_{0}^{\infty}e^{-\lambda
t}Z(t)dt,~{}~{}~{}~{}~{}~{}~{}{\rm Re{\hskip 1.42262pt}}\lambda>0.$ (5.1)

According to Proposition 2.1, $Z(t)=Y(t)-Y_{\omega_{*}}$ is bounded in ${\cal
E}$. Hence the integral (5.1) converges and is analytic for ${\rm Re{\hskip
1.42262pt}}\lambda>0$, and

$\|\tilde{Z}(\lambda)\|_{{\cal E}}\leq\frac{C}{{\rm Re{\hskip
1.42262pt}}\lambda},~{}~{}~{}~{}~{}~{}~{}{\rm Re{\hskip 1.42262pt}}\lambda>0.$
(5.2)

Applying the Fourier-Laplace transform to (4.4), we get

$\lambda\tilde{Z}(\lambda)={\bf H}\tilde{Z}(\lambda)+Z_{0},\quad{\rm Re{\hskip
1.42262pt}}\lambda>0.$ (5.3)

The solution $\tilde{Z}(\lambda)$ to (5.3) is given by

$\tilde{Z}(\lambda)=-({\bf H}-\lambda)^{-1}Z_{0},\quad{\rm Re{\hskip
1.42262pt}}\lambda>0.$ (5.4)

where the resolvent $({\bf H}-\lambda)^{-1}$ exists for ${\rm Re{\hskip
1.42262pt}}\lambda>0$ by the following lemma.

######  Lemma 5.1.

The operator-valued function $({\bf H}-\lambda)^{-1}:{\cal E}\to{\cal E}$ is
analytic for ${\rm Re{\hskip 1.42262pt}}\lambda>0$.

###### Proof.

First, the energy conservation (4.7) and non negativity of ${\cal H}(Z)$ imply
that ${\rm Ker}({\bf H}-\lambda)=0$. Indeed, otherwise there is a nonzero
solution $Z(t)=e^{\lambda t}Z_{\lambda}$ for which the energy cannot be
conserved. Second, ${\bf H}-\lambda={\bf H}_{0}-\lambda+T$, where ${\bf
H}_{0}=\begin{pmatrix}0&1\\\ \Delta&0\\\ \end{pmatrix}$, and $T$ is a finite-
dimensional operator. Operator $({\bf H}_{0}-\lambda)^{-1}$ is bounded in
${\cal E}$ for every ${\rm Re{\hskip 1.42262pt}}\lambda>0$. Hence, ${\bf
H}-\lambda$ is invertible by the Fredholm theory. ∎

We rewrite (5.3) as

$\left\\{\begin{array}[]{rcl}\tilde{\Pi}(x,\lambda)-\lambda\tilde{\Lambda}(x,\lambda)&=&-\Lambda_{0}(x)\\\
\Delta\tilde{\Lambda}(x,\lambda)-\lambda\tilde{\Psi}(x,\lambda)&=&-\Pi_{0}(x)+\tilde{\nu}(\lambda)J\varrho(x)\end{array}\right|,\qquad\tilde{\nu}(\lambda)=\frac{1}{I}\langle\tilde{\Lambda}(\lambda),J\varrho\rangle.$
(5.5)

Equivalently, in the Fourier space,

$\left\\{\begin{array}[]{rcl}\hat{\tilde{\Pi}}(k,\lambda)-\lambda\hat{\tilde{\Lambda}}(k,\lambda)&=&-\hat{\Lambda}_{0}(k)\\\
-k^{2}\hat{\tilde{\Lambda}}(k,\lambda)-\lambda\hat{\tilde{\Pi}}(k,\lambda)&=&-\hat{\Pi}_{0}(k)+\tilde{\nu}(\lambda)J\hat{\varrho}(k)\end{array}\right|,\quad
k\in\mathbb{R}^{2}{\hskip 1.42262pt}.$ (5.6)

We have

$\left(\begin{array}[]{cc}-\lambda&1\\\
-k^{2}&-\lambda\end{array}\right)^{-1}=\frac{1}{k^{2}+\lambda^{2}}\left(\begin{array}[]{cc}-\lambda&-1\\\
k^{2}&-\lambda\end{array}\right).$

Hence,

$\displaystyle\hat{\tilde{\Lambda}}$ $\displaystyle=$
$\displaystyle\frac{\lambda\hat{\Lambda}_{0}+\hat{\Pi}_{0}-\tilde{\nu}J\hat{\varrho}}{k^{2}+\lambda^{2}}=\frac{\hat{K}_{0}}{k^{2}+\lambda^{2}}-\frac{\tilde{\nu}J\hat{\varrho}}{k^{2}+\lambda^{2}},\qquad\hat{K}_{0}(\lambda):=\lambda\hat{\Lambda}_{0}+\hat{\Pi}_{0}.$
(5.7)

Now

$I\tilde{\nu}(\lambda)=\langle\hat{\tilde{\Lambda}},J\hat{\varrho}\rangle=\langle\frac{\hat{K}_{0}}{k^{2}+\lambda^{2}},J\hat{\varrho}\rangle-\tilde{\nu}\langle\frac{J\hat{\varrho}}{k^{2}+\lambda^{2}},J\hat{\varrho}\rangle=\langle\frac{\hat{K}_{0}}{k^{2}+\lambda^{2}},J\hat{\varrho}\rangle-\tilde{\nu}\varkappa(\lambda),$

where $\varkappa(\lambda)$ is defined in (3.1). Thus,

$\tilde{\nu}(\lambda)=\frac{\langle\frac{\hat{K}_{0}}{k^{2}+\lambda^{2}},J\hat{\varrho}\rangle}{I+\varkappa(\lambda)}.$
(5.8)

##  6 2D Schrödinger resolvent

Denote by $R(\zeta)=(-\Delta-\zeta)^{-1}$ the resolvent of 2D Laplacian.
Recall the properties of $R(\zeta)$ [1, 5]:
i) $R(\zeta)$ is strongly analytic function of
$\zeta\in\mathbb{C}\setminus[0,\infty)$ with the values in
$H^{-1}(\mathbb{R}^{2})\to H^{1}(\mathbb{R}^{2})$.
ii) For $\zeta\in(0,\infty)$, the convergence holds

$R^{(k)}(\zeta\pm i\varepsilon)\to R^{(k)}(\zeta\pm
i0):=R_{\pm}^{(k)}(\zeta),\quad\varepsilon\to+0,\quad k=0,1,2$ (6.1)

in $H^{-1}_{\sigma}(\mathbb{R}^{2})\to H^{1}_{-\sigma}(\mathbb{R}^{2})$ with
$\sigma>k+1/2$.
iii) For $s=0,1,2$ and $|l|\leq 2$, the asymptotics hold

$\|R_{\pm}^{(k)}(\zeta^{2})\|_{H^{s}_{\sigma}\to H^{s+l}_{-\sigma}}={\cal
O}(|\zeta|^{-(1-l)}),\quad\zeta\to\infty,\quad k=0,1,2,\quad\sigma>1/2+k.$
(6.2)

Now we obtain asymptotics of $R_{\pm}(\zeta^{2})$ for $\zeta\to 0$. Denote by
$P_{\pm}(\zeta)$ an integral operator with the kernel

$P_{\pm}(\zeta,z):=-\frac{\log(\zeta|z|)}{2\pi}+h_{\pm},~{}~{}{\rm
where}~{}~{}h_{\pm}=\pm\frac{i}{4}+\frac{\gamma}{2\pi}+\frac{\log 2}{2\pi}.$
(6.3)

Here $\gamma$ is Euler’s constant. The following lemma is an adapted version
of [14, Lemma 5].

######  Lemma 6.1.

The following asymptotics hold

$\|\partial^{k}_{\zeta}\big{(}R_{\pm}(\zeta^{2})-P_{\pm}(\zeta)\big{)}\|_{L^{2}_{\beta}\to
L^{2}_{-\beta}}={\cal
O}(\zeta^{\frac{3}{2}-k}),\quad\zeta\to+0,\quad\beta>5/2,\quad k=0,1,2.$ (6.4)

We give the proof in Appendix A for the sake of completeness. Denote by
$g(\lambda)$ for ${\rm Re{\hskip 1.42262pt}}\lambda>0$, the integral operator
with the integral kernel

$g(\lambda,z):=F^{-1}_{k\to z}[\frac{1}{k^{2}+\lambda^{2}}].$ (6.5)

By (6.1), there exists

$g(i\mu+\varepsilon)\to
g(i\mu+0)=R_{\mp}(\mu^{2}),\quad\varepsilon\to+0,\quad\pm\mu>0$ (6.6)

in $H^{-1}_{\sigma}(\mathbb{R}^{2})\to H^{1}_{-\sigma}(\mathbb{R}^{2})$ with
$\sigma>1/2$. The representation and asymptotics (6.2) imply that

$\|g^{(k)}(i\mu+0)\|_{H^{s}_{\sigma}\to H^{s+l}_{-\sigma}}={\cal
O}(|\mu|^{-1+l}),\quad\mu\to\infty,\quad k=0,1,2,\quad\sigma>1/2+k$ (6.7)

for $s=0,1,2$ and $|l|\leq 2$. Lemma 6.1 implies the following asymptotics.

######  Lemma 6.2.

Let $\rho\in{\cal S}(\mathbb{R}^{2})$ satisfy (1.4) and (1.5). Then

$g(i\mu+0)\varrho=g_{0}\varrho+g_{R}(\mu)\varrho,\quad\pm\mu>0,$ (6.8)

where $g_{0}:H^{-1}_{\beta}\to H^{1}_{-\beta}$, $\beta>1$ is an integral
operator with the kernel $g_{0}(x-y)=-\frac{1}{2\pi}\log|x-y|$, and

$\|g_{R}^{(k)}(\mu)\|_{L^{2}_{\beta}\to L^{2}_{-\beta}}={\cal
O}(|\mu|^{\frac{3}{2}-k}),\qquad\mu\to 0,\quad k=0,1,2,\quad\beta>5/2.$ (6.9)

###### Proof.

For $\rho$ satisfying (1.4) and (1.5), the formulas (6.3) imply that

$\displaystyle[P_{\pm}(\mu)\varrho](y)$ $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\int\log|x-y|\varrho(x)dx-(\frac{\log|\mu|}{2\pi}-h_{\pm})\int\varrho(x)dx$
$\displaystyle=$ $\displaystyle-\frac{1}{2\pi}\int\log|x-y|\varrho(x)dx.$

Hence, (6.8)–(6.9) follows by (6.4) and (6.6). ∎

The lemma and asymptotics (6.4) imply

######  Corollary 6.3.

Let $\rho\in{\cal S}(\mathbb{R}^{2})$ satisfy (1.4) and $f\in L^{2}_{\beta}$
with $\beta>5/2$. Then

$\langle\frac{\hat{\varrho}}{k^{2}+(i\mu+0)^{2}},\hat{f}\rangle=\langle\frac{\hat{\varrho}}{k^{2}},\hat{f}\rangle+{\cal
O}(|\mu|^{3/2}),\qquad\mu\to 0.$ (6.10)

The asymptotics can be differentiated two times.

##  7 Asymptotics of $\tilde{\nu}$

First we derive asymptotics of $\nu(i\mu+0)$ as $\mu\to 0$.

######  Lemma 7.1.

Let conditions (1.4)–(1.5) hold, and $Z_{0}=(\Lambda_{0},\Psi_{0})\in{\cal
E}_{\beta}^{+}$ with $\beta>5/2$. Then

$\tilde{\nu}(i\mu+0)=\Big{(}-\langle\frac{J\hat{\varrho}}{k^{2}},\overline{\hat{\Psi}}_{0}\rangle+i\mu\langle\frac{J\hat{\varrho}}{k^{2}},\overline{\hat{\Lambda}}_{0}\rangle\Big{)}\frac{1}{I+\varkappa(0)}+{\cal
O}(|\mu|^{3/2}),\quad\mu\to 0,$ (7.1)

where
$\varkappa(0)=\displaystyle\int\frac{|\hat{\varrho}(k)|^{2}dk}{k^{2}}>0$. The
asymptotics allows two differentiations.

###### Proof.

Definition (3.1) of $\varkappa$ and Corollary 6.3 imply that

$\varkappa(i\mu+0)=\langle\frac{\hat{\varrho}}{k^{2}+(i\mu+0)^{2}},\hat{\varrho}\rangle=\varkappa(0)+{\cal
O}(\mu^{2}\log|\mu|),\quad\mu\to 0.$

Further, Corollary 6.3 implies that

$\displaystyle\langle\frac{\hat{K}_{0}(i\mu+0)}{k^{2}+(i\mu+0)^{2}},J\hat{\varrho}\rangle$
$\displaystyle=$ $\displaystyle
i\mu\langle\frac{J\hat{\varrho}}{k^{2}+(i\mu+0)^{2}},\overline{\hat{\Lambda}}_{0}\rangle-\langle\frac{J\hat{\varrho}}{k^{2}+(i\mu+0)^{2}},\overline{\hat{\Psi}}_{0}\rangle$
$\displaystyle=$
$\displaystyle-\langle\frac{J\hat{\varrho}}{k^{2}},\overline{\hat{\Psi}}_{0}\rangle+i\mu\langle\frac{J\hat{\varrho}}{k^{2}},\overline{\hat{\Lambda}}_{0}\rangle+{\cal
O}(\mu^{2}\log|\mu|),\quad\mu\to 0.$

Hence, (7.1) follows by definition (5.8) of $\tilde{\nu}$. ∎

Now we estimate $\tilde{\nu}(i\mu+0)$ for large $\mu\in\mathbb{R}$.

######  Lemma 7.2.

Let condition (1.4) hold, and $Z_{0}=(\Lambda_{0},\Psi_{0})\in{\cal
E}_{\beta}^{+}$ with $\beta>5/2$. Then for sufficiently large $B>0$, the bound
holds

$|\tilde{\nu}^{(k)}(i\mu+0)|\leq
C|\mu|^{-2},\quad\mu\in\mathbb{R},\quad|\mu|\geq B,\quad k=0,1,2.$ (7.2)

###### Proof.

Applying (6.7) with $s=l=0$, $k=0,1,2$ and $\sigma>1/2+k$, we obtain

$\displaystyle|\varkappa^{(k)}(i\mu+0)|$ $\displaystyle=$
$\displaystyle|\langle g^{(k)}(i\mu+0)\varrho,\varrho\rangle|\leq
C\|g^{(k)}(i\mu+0)\varrho\|_{L^{2}_{-\sigma}}\|\varrho\|_{L^{2}_{\sigma}}$
(7.3) $\displaystyle\leq$ $\displaystyle
C_{1}|\mu|^{-1}\|\varrho\|_{L^{2}_{\sigma}}^{2}\leq
C_{2}|\mu|^{-1},\quad|\mu|\geq B.$

Further, for $\mu\geq B$, (6.7) implies

$\displaystyle|\langle g^{(k)}(i\mu+0)\Psi_{0},J\varrho\rangle|$
$\displaystyle\leq$ $\displaystyle
C\|g^{(k)}(i\mu+0)\Psi_{0}\|_{H^{-1}_{-\beta}}\|\varrho\|_{H^{1}_{\beta}}\leq
C_{1}|\mu|^{-2}\|\Psi_{0}\|_{L^{2}_{\beta}},$ $\displaystyle|\langle
g^{(k)}(i\mu+0)\Lambda_{0},J\varrho\rangle|$ $\displaystyle\leq$
$\displaystyle
C\|g^{(k)}(i\mu+0)\Lambda_{0}\|_{H^{-2}_{-\beta}}\|\varrho\|_{H^{2}_{\beta}}\leq
C_{1}|\mu|^{-3}\|\Lambda_{0}\|_{L^{2}_{\beta}},\quad k=0,1,2.$

Hence,

$|\partial^{k}_{\mu}\langle\frac{\hat{K}_{0}}{k^{2}+(i\mu+0)^{2}},J\hat{\varrho}\rangle\leq
C|\mu|^{-2}\|Z_{0}\|_{{\cal E}_{\beta}},\quad|\mu|\geq B,\quad k=0,1,2,$ (7.4)

by definition (5.7) of $\hat{K}_{0}$. Finally, (7.2) follow from (5.8) and
(7.3)–(7.4). ∎

######  Corollary 7.3.

Lemmas 7.1 and 7.2 together with condition (3.2) imply that
$\tilde{\nu}(i\mu+0)\in C^{2}(\mathbb{R})$.

##  8 Attraction to soliton in weighted Sobolev norms

###  8.1 Time decay of $\nu(t)$

Here we prove time decay of $\nu(t)$.

######  Lemma 8.1.

Suppose that the condition (3.2) holds, and $Z_{0}\in{\cal E}_{\beta}^{+}$
with $\beta>5/2$. Then $\nu(t)$ is continuous and

$|\nu(t)|\leq\frac{C(\rho,\beta)}{(1+t)^{2}},\qquad t>0.$ (8.1)

###### Proof.

The component $\nu(t)$ is given by

$\nu(t)=\frac{1}{2\pi}\int e^{i\mu
t}\tilde{\nu}(\mu+i0)d\mu=\frac{1}{2\pi}\int_{-\infty}^{0}e^{i\mu
t}\tilde{\nu}(\mu+i0)d\mu+\frac{1}{2\pi}\int_{0}^{\infty}e^{i\mu
t}\tilde{\nu}(\mu+i0)d\mu,\quad t>0.$

Now Corollary 7.3, asymptotics (7.1) and bound (7.2) imply (8.1) by the double
partial integration. ∎

###  8.2 Time decay of $Z(t)$

######  Lemma 8.2.

Suppose that the condition (3.2) holds, and $Z_{0}\in{\cal E}_{\beta}^{+}$
with $\beta>5/2$. Then

$\|Z(t)\|_{-\beta}\leq\frac{C(\rho,\beta)}{(1+t)^{2}},\qquad t>0.$ (8.2)

###### Proof.

We rewrite (4.4) as

$\dot{Z}(t)={\bf H}_{0}Z(t)-\begin{bmatrix}0\\\
\nu(t)J\varrho\end{bmatrix},\qquad{\bf H}_{0}=\begin{pmatrix}0&1\\\
\Delta&0\end{pmatrix}.$ (8.3)

To prove time decay of $Z(t)$, we apply the Duhamel representation

$Z(t)=W(t)Z_{0}-\int_{0}^{t}W(t-s)\begin{bmatrix}0\\\
\nu(s)J\varrho\end{bmatrix}ds,\quad t\geq 0.$ (8.4)

For the group $W(t)$ the dispersion decay holds (see Appendix B)

$\|W(t)X\|_{-\beta}\leq C(1+t)^{-2}\|X\|_{\beta},\quad\beta>2,\quad t\geq 0.$
(8.5)

Applying (8.5) to (8.4), and using (8.1), we obtain (8.2). ∎

######  Corollary 8.3.

The splitting (4.1) and the decay (8.2) imply the convergence

$Y(t)\stackrel{{\scriptstyle{\cal
E}_{-\beta}}}{{-\\!\\!\\!-\\!\\!\\!-\\!\\!\\!\to}}Y_{\omega_{*}},\qquad
t\to\infty.$ (8.6)

##  9 Asymptotics with dispersive wave in global energy norm

Here we prove our main Theorem 3.2. For the considered solution $Y(t)$ and for
the soliton (2.11) with angular velocity $\omega_{*}$, (2.6) together with
(8.2) imply that

$\displaystyle\omega(t)$ $\displaystyle=$
$\displaystyle\frac{1}{I}\big{(}M+\langle
A(t),J\varrho\rangle\big{)}=\omega_{*}+\frac{1}{I}\big{(}\langle
A(t),J\varrho\rangle-\langle A_{\omega_{*}},J\varrho\rangle\big{)}$ (9.1)
$\displaystyle=$
$\displaystyle\omega_{*}+\frac{1}{I}\langle\Lambda(t),J\varrho\rangle=\omega_{*}+{\cal
O}(t^{-2}),\quad t\to\infty.$

Further, for the difference $Z(t)=Y(t)-Y_{\omega_{*}}(t)$, equation (2.7)
implies

$\dot{Z}(t)={\bf A}_{0}Z(x.t)-R(t),\qquad R(t):=\begin{pmatrix}0\\\ \
(\omega(t)-\omega_{*})J\varrho(x)\end{pmatrix}.$

Then similarly to (8.4),

$\displaystyle Z(t)$ $\displaystyle=$ $\displaystyle
W(t)Z_{0}-\int_{0}^{t}W(t-s)R(s)ds$ $\displaystyle=$ $\displaystyle
W(t)\Big{(}Z_{0}-\int_{0}^{\infty}W(-s)R(s)ds\Big{)}+\int_{t}^{\infty}W(t-s)R(s)ds=W(t)\Psi_{+}+r_{+}(t).$

Here $\Psi_{+}\in{\cal E}$ and $\|r_{+}(t)\|_{\cal E}={\cal O}(t^{-1})$ by
(9.1) since $W(t)$ is the unitary group in the space ${\cal E}$.

##  Appendix A Proof of Lemma 6.1

Recall that (cf. [14, Section 2])

$R_{\pm}(\zeta^{2},z)=\pm\frac{i}{4}H_{0}^{\pm}(\zeta|z|),$ (A.1)

where for $H_{0}(s)$ the asymptotics hold (see [13])

$H_{0}^{\pm}(s)=1\pm\frac{2i}{\pi}\big{(}\log\frac{s}{2}+\gamma\big{)}+{\cal
O}(s^{2}\log s),\quad s\to 0.$

The asymptotics imply that

$\partial_{\zeta}^{k}R_{\pm}(\zeta^{2},z)=\partial_{\zeta}^{k}P_{\pm}(\zeta,z)+{\cal
O}\big{(}(\zeta|z|)^{2-k}\log(\zeta|z|)\big{)},\quad\zeta|z|\to 0,\quad
k=0,1,2,$ (A.2)

where $P_{\pm}(\zeta,z)=-\frac{\log(\zeta|z|)}{2\pi}+h_{\pm}$ as in (6.3).
Moreover,

$|\partial_{\zeta}^{k}R_{\pm}(\zeta^{2},z)|=\frac{1}{4}|\partial_{\zeta}^{k}H_{0}^{\pm}(\zeta|z|)|\leq\frac{C|z|^{k}}{\sqrt{\zeta|z|}},\quad\zeta|z|\geq
1,\quad k=0,1,2.$ (A.3)

Let $\chi_{\zeta}(z)$ be characteristic function of the ball
$|z|\leq\frac{1}{\zeta}$. We have

$\displaystyle R_{\pm}(\zeta^{2},z)-P_{\pm}(\zeta,z)$ $\displaystyle=$
$\displaystyle\chi_{\zeta}(z)\big{(}R_{\pm}(\zeta^{2},z)-P_{\pm}(\zeta,z)\big{)}+(1-\chi_{\zeta}(z))\big{(}R_{\pm}(\zeta^{2},z)-P_{\pm}(\zeta,z)\big{)}$
(A.4) $\displaystyle=$ $\displaystyle Q_{\pm}(\zeta,z)+S_{\pm}(\zeta,z).$

For $\zeta|z|\leq 1$, (A.2) implies

$\displaystyle|Q_{\pm}(\zeta,z)|$ $\displaystyle\leq$ $\displaystyle
C(\zeta|z|)^{2}\log(\zeta|z|)\leq C(\zeta|z|)^{3/2},$
$\displaystyle|\partial_{\zeta}Q_{\pm}(\zeta,z)|$ $\displaystyle\leq$
$\displaystyle C|z|\zeta|z|\log(\zeta|z|)\leq
C|z|(\zeta|z|)^{1/2}=C\zeta^{1/2}|z|^{3/2},$
$\displaystyle|\partial^{2}_{\zeta}Q_{\pm}(\zeta,z)|$ $\displaystyle\leq$
$\displaystyle C|z|^{2}\log(\zeta|z|)\leq
C\frac{|z|^{2}}{\sqrt{\zeta|z|}}=C\zeta^{-1/2}|z|^{3/2}.$ (A.5)

Further, for $\zeta|z|\geq 1$, (6.3), (A.1) and (A.3) imply

$\displaystyle|S_{\pm}(\zeta,z)|$ $\displaystyle\leq$ $\displaystyle
C\Big{(}1+\log(\zeta|z|)+\frac{1}{\sqrt{\zeta|z|}}\Big{)}\leq
C_{1}(\zeta|z|)^{3/2},$ $\displaystyle|\partial_{\zeta}S_{\pm}(\zeta,z)|$
$\displaystyle\leq$ $\displaystyle
C\Big{(}\frac{|z|}{\zeta|z|}+\frac{C|z|}{\sqrt{\zeta|z|}}\Big{)}\leq
C|z|(\zeta|z|)^{1/2}=C\zeta^{1/2}|z|^{3/2},$
$\displaystyle|\partial^{2}_{\zeta}S_{\pm}(\zeta,z)|$ $\displaystyle\leq$
$\displaystyle
C\Big{(}\frac{|z|^{2}}{(\zeta|z|)^{2}}+\frac{C|z|^{2}}{\sqrt{\zeta|z|}}\Big{)}\leq
C\frac{|z|^{2}}{\sqrt{\zeta|z|}}=C\zeta^{-1/2}|z|^{3/2}.$ (A.6)

Finally, (A.5) and (A.6) imply (6.4) by (A.4).

##  Appendix B Dispersive decay

Here we prove the decay (8.5). Note that

$W(z,t)=\left(\begin{array}[]{ll}\dot{G}(z,t)&G(z,t)\\\
\ddot{G}(z,t)&\dot{G}(z,t)\end{array}\right),\quad
G(z,t)=\frac{1}{2\pi}\frac{\theta(t-|z|)}{\sqrt{t^{2}-|z|^{2}}},\quad
z\in\mathbb{R}^{2},\quad t>0.$ (B.1)

Step i) For any $\varepsilon\in(0,1)$, the bounds hold

$\displaystyle|\partial^{\alpha}_{z}\partial^{j}_{t}G(z,t)|$
$\displaystyle\leq$ $\displaystyle C(\varepsilon)t^{-2},\quad
1\leq|\alpha|+j\leq 2,\quad|z|\leq\varepsilon t,\quad t\geq 1.$ (B.2)

Formula (B.1) imply the Huygen’s principle for the group $W(t)$: $W(z,t)=0$
for $|z|>t$. The energy conservation gives that

$\|W(t)Z\|_{\cal E}=\|Z\|_{\cal E},\quad t\geq 0,\qquad Z\in{\cal E}.$ (B.3)

Step ii) We fix an arbitrary $\varepsilon\in(0,1)$. For any $t\geq 1$ we split
the function $Z(x)$ in two terms, $Z=Z_{1,t}+Z_{2,t}$ such that

$Z_{1,t}(x)=0~{}~{}\mbox{for}~{}|x|>\varepsilon
t/2,~{}~{}~{}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}Z_{2,t}(x)=0~{}~{}\mbox{for}~{}|x|<\varepsilon
t/4,$ (B.4)

with

$\|Z_{1,t}\|_{\cal E}+\|Z_{2,t}\|_{\cal E}\leq C\|Z\|_{\cal E},\quad t\geq 1.$
(B.5)

Step iii) Using (B.3)–(B.5), we obtain for any $\sigma\geq 0$

$\|W(t)Z_{2,t}\|_{{\cal F}_{-\sigma}}\leq\|W(t)Z_{2,t}\|_{{\cal
F}_{0}}=\|Z_{2,t}\|_{{\cal F}_{0}}\leq
C(\varepsilon)t^{-\sigma}\|Z_{2,t}\|_{{\cal F}_{\sigma}}\leq
C(\varepsilon)t^{-\sigma}\|Z\|_{{\cal F}_{\sigma}},\,\,\,t\geq 1.$ (B.6)

Step iv) Denote by $\zeta$ the operator of multiplication by the function
$\zeta({|x|}/{t})$ such that $\zeta=\zeta(s)\in C_{0}^{\infty}(\mathbb{R})$,
$\zeta(s)=1$ for $|s|<\varepsilon/4$, $\zeta(s)=0$ for $|s|>\varepsilon/2$.
Obviously, $|\partial_{x}^{\alpha}\zeta({|x|}/{t})|\leq C(\varepsilon)<\infty$
for $|\alpha|\leq 1$ and $t\geq 1.$ Furthermore, $1-\zeta({|x|}/{t})=0$ for
$|x|<\varepsilon t/4$. Hence, using (B.3), we get for $\sigma\geq 0$

$||(1-\zeta)W(t)Z_{1,t}||_{{\cal F}_{-\sigma}}\leq
C(\varepsilon)t^{-\sigma}\|W(t)Z_{1,t}\|_{{\cal F}_{0}}\leq
C_{1}(\varepsilon)t^{-\sigma}\|Z\|_{{\cal F}_{\sigma}},\quad t\geq 1.$ (B.7)

Step v) It remains to estimate $\zeta W(t)Z_{1,t}$. Let $\chi_{\varepsilon
t/2}$ be the characteristic function of the ball $|x|\leq\varepsilon t/2$. We
will use the same notation for the operator of multiplication by this
characteristic function. By (B.4),

$\zeta W(t)Z_{1,t}=\zeta W(t)\chi_{\varepsilon t/2}Z_{1,t}.$ (B.8)

The norm of the operator $\zeta W(t)\chi_{\varepsilon t/2}:{\cal
F}_{\sigma}\rightarrow{\cal F}_{-\sigma}$ is equivalent to the norm of the
operator $\langle x\rangle^{-\sigma}\zeta W(t)\chi_{\varepsilon t/2}(y)\langle
y\rangle^{-\sigma}:{\cal F}_{0}\rightarrow{\cal F}_{0}$. The norm of the later
operator does not exceed the sum in $\alpha$ and $j$, $1\leq|\alpha|+j\leq 2$
of the norms of integral operators

$\langle x\rangle^{-\sigma}\partial_{t}^{j}\partial_{x}^{\alpha}[\zeta
W(t)\chi_{\varepsilon t/2}(y)\langle
y\rangle^{-\sigma}]:L^{2}(\mathbb{R}^{2})\oplus L^{2}(\mathbb{R}^{2})\to
L^{2}(\mathbb{R}^{2})\oplus L^{2}(\mathbb{R}^{2}).$ (B.9)

The estimates (B.2) imply that these operators are of the Hilbert-Schmidt type
for $\sigma>2$, and their Hilbert-Schmidt norms do not exceed $Ct^{-2}$.
Hence, (B.5) and (B.8) imply that for $\sigma>2$

$\|\zeta W(t)F_{1,t}\|_{{\cal F}_{-\sigma}}\leq
C(\varepsilon)t^{-2}\|F_{1,t}\|_{{\cal F}_{\sigma}}\leq
C(\varepsilon)t^{-2}\|F_{0}\|_{{\cal F}_{\sigma}},\quad t\geq 1.$ (B.10)

Finally, the estimates (B.6), (B.7) and (B.10) with $\sigma=\beta$ imply
(8.5).

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