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\begin{document}
\title{Calculation of NMR Properties of Solitons in Superfluid $^3$He-A}
\author{R. H\"anninen}
\affiliation{Low Temperature Laboratory, Helsinki University of
Technology, P.O.Box 2200, FIN-02015 HUT, Finland } 
\author{E. V. Thuneberg}
\affiliation{Low Temperature Laboratory, Helsinki University of
Technology, P.O.Box 2200, FIN-02015 HUT, Finland } 
\affiliation{Department of Physical Sciences,
P.O.Box 3000, FIN-90014 University of Oulu, Finland}

\date{\today}

\begin{abstract}
Superfluid $^3$He-A has domain-wall like structures, which are called
solitons. We calculate numerically the structure of a splay soliton. We
study the effect of solitons on the NMR spectrum by calculating the
frequency shifts and the amplitudes of the soliton peaks for both
longitudinal and transversal oscillations of magnetization. The effect of
dissipation coming from normal-superfluid conversion and spin diffusion
is calculated.  The calculations are in good agreement with experiments
except a problem in the transversal resonance frequency of the splay
soliton, or in magnetic field dependence of reduced resonance
frequencies.
\end{abstract}

\pacs{67.57.Fg, 67.57.Lm}
%\pacs{67.57.Pq (superfluid 3He-impurities)}
%\pacs{74.20.Fg (BCS theory and its development)}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{INTRODUCTION}

Nuclear magnetic resonance (NMR) has turned out to be very useful
for studying the superfluid phases of liquid
$^3$He. The two superfluid phases, A and B, are distinguished in the
NMR spectrum by different frequency shifts of the absorption
peaks.\cite{Leggett74} In addition to these ``bulk'' peaks, one
often observes additional ``satellite'' peaks. These are caused by
topological objects and textures that appear in the superfluid order
parameter. Especially in superfluid $^3$He-A, several different objects
have been identified based on the frequency
shifts of the satellite peaks.\cite{vollhardtwolfle,Volovik,Lounas} The
simplest of these are solitons. They are domain-wall like structures
where a planar object separates two different but degenerate bulk
states.

The satellite peaks in $^3$He-A were first observed in
measurements in the mid 70's.\cite{Avenel,GouldLee} 
Soon after the theory of solitons of $^3$He was developed by Maki
and Kumar.\cite{MakiKumar77,MakiKumar} Their
calculation gave a striking agreement with the measured frequency
shifts of the satellite peaks at temperatures close to the superfluid
transition temperature $T_{\rm c}$.  This initial success had the
consequence that further studies of solitons went to other
directions\cite
{BruinsmaMaki,GouldBartolacBozler,KharadzeMaki,NakaharaMaki} and,
unfortunately, no more precise calculations were conducted.

A soliton has two basic structures, ``twist''
and ``splay'', which correspond to the cases of a magnetic field ($B\gg
1$  mT) perpendicular and parallel, respectively, to the
plane of the soliton wall. Both these structures can be studied using
small oscillations of the magnetization that are either transversal or
longitudinal relative to the static field.

There are several points that can be improved in the previous
calculations.
(i) The structure of the splay soliton was calculated only by using
variational approximation. (ii) The calculations were limited to
temperatures near
$T_{\rm c}$. (iii) The effect of
different parameter values was not studied. (iv) The amplitudes and
widths of the satellite peaks, as well as peaks of higher order
were not studied. (v) Dissipation was neglected. 
It is just these points  that we
address in this paper. An additional motivation is that the
study of solitons opens the way to detailed understanding of more
complicated topological objects such as vortices.

We find that, using dissipationless theory, the agreement between
theoretical and experimental frequency shifts is generally very good.
However, we find a puzzling disagreement in the transverse oscillation
frequency of the splay soliton. This disagreement has gone unnoticed
because no detailed comparison between theory and experiment has been
published. Further we find that the theory is particularly inflexible to
explain this discrepancy away. Taking into account dissipation, in
particular spin diffusion, changes these conclusions. On the one hand,
the  disagreement in the splay soliton frequency is reduced. On the
other hand, we find considerable extra shift of resonance frequencies in
high fields, which has not been reported experimentally. We also
point out that the parallel resonance
of a splay soliton has not been studied experimentally.
Measurement of these quantities would be important to test our
understanding of the basic properties of superfluid
$^3$He.

We start in Sec.~\ref{s.hyrdo} with a short introduction to the 
hydrodynamic theory and NMR in $^3$He-A. In Sec.~\ref{s.eqstruc} we
solve numerically the structure of the splay soliton. The frequency and
the absorption of the principal satellite peak are determined in
Secs.~\ref{s.princnmr} and \ref{s.intes} ignoring dissipative 
effects. In Sec.~\ref{s.intes} we calculate the frequencies and
absorption of the higher modes. In Sec.~\ref{s.dissipation} we take
into account dissipation and calculate the effect of the spin diffusion
and normal-superfluid relaxation on the absorption spectrum.

%We start in Sec.~\ref{s.hyrdo} with a short introduction to the 
%hydrodynamic theory and NMR in $^3$He-A. In Sec.~\ref{s.eqstruc} we
%solve numerically the structure of the splay soliton. The frequency and
%the absorption of the principal satellite peak are determined in
%Secs.~\ref{s.princnmr} and \ref{s.intes} ignoring dissipative
%effects. In Sec.~\ref{s.spindif} we take into account the spin
%diffusion and calculate it's effect on the absorption spectrum. The
%effect of the higher modes is analyzed in Sec.~\ref{s.intes}. Other
%relaxation mechanisms are finally considered  in Sec.~\ref{s.otherrel}. 

  
 

\section{HYDRODYNAMIC THEORY}\label{s.hyrdo}

Here we briefly present some main points of the hydrodynamic theory and
NMR in $^3$He-A. The order parameter of superfluid $^3$He-A is a
$3\times 3$ tensor of the form\cite{Leggett,vollhardtwolfle}   
\begin{equation}
A_{\mu j}=\Delta\hat{d}_\mu(\hat{m}_j+{\rm i}\hat{n}_j) ,
\label{e.ordpar} 
\end{equation} 
where $\hat{\bf d}$, $\hat{\bf m}$ and $\hat{\bf n}$ are unit vectors
and $\hat{\bf m}\perp \hat{\bf n}$. It is conventional to define
$\hat{\bf l}=\hat{\bf m}\times\hat{\bf n}$, which gives the axis of
the orbital angular momentum of a Cooper pair. The unit vector
$\hat{\bf d}$ defines the axis along which the spin of the Cooper pair
vanishes. In a static magnetic field, the structure of a soliton can
be determined by finding a local minimum
for the free energy\cite{vollhardtwolfle,Cross,Fetterrev}   
\begin{equation}
F_{\rm static}=\int d^3r(f_{\rm d}+f_{\rm g}+f_{\rm h}).
\label{e.fstatic}
\end{equation}
Here $f_{\rm d}$ comes from the dipole-dipole interaction between
nuclear moments, 
\begin{equation}
f_{\rm d}=-\frac{1}{2}\lambda_{\rm d}(\hat{\bf d}\cdot\hat{\bf l})^2,
\label{e.fdip}
\end{equation}
$f_{\rm h}$ from coupling to the external field ${\bf B}$,
\begin{equation}
f_{\rm h}=\frac{1}{2}\lambda_{\rm h}(\hat{\bf d}\cdot{\bf B})^2\ ,
\label{e.fmag}
\end{equation}
and $f_{\rm g}$ from the gradient of the order parameter,
\begin{eqnarray}
2f_{\rm g} &=& \rho_\perp{\bf v}_s^2 
+ (\rho_\parallel-\rho_\perp)(\hat{\bf l}\cdot{\bf v}_s)^2 
+ 2C{\bf v}_s\cdot\mbox{\boldmath$\nabla$}\times\hat{\bf l} 
\nonumber \\ 
&-& 2C_0(\hat{\bf l}\cdot{\bf v}_s)(\hat{\bf
l}\cdot\mbox{\boldmath$\nabla$}\times\hat{\bf l})  
+ K_{\rm s}(\mbox{\boldmath$\nabla$}\cdot\hat{\bf l})^2 
+ K_{\rm t}(\hat{\bf l}\cdot\mbox{\boldmath$\nabla$}
\times\hat{\bf l})^2 \nonumber \\
&+& K_{\rm b}\vert\hat{\bf l}\times(\mbox{\boldmath$\nabla$}
\times\hat{\bf l})\vert^2 
+ K_5\vert(\hat{\bf l}\cdot\mbox{\boldmath$\nabla$})\hat{\bf d}\vert^2 
+ K_6 \sum_{i,j}[(\hat{\bf l}\times\mbox{\boldmath$\nabla$})_i
\hat{\bf d}_j]^2. 
\label{e.fgrad}
\end{eqnarray} 
The gradient energy (\ref{e.fgrad}) also includes the 
kinetic energy arising from the superfluid velocity ${\bf v}_s =
(\hbar/2m_3) \sum_{i} m_i
\mbox{\boldmath$\mbox{\boldmath$\nabla$}$}n_i$, where $m_3$ is the
mass of a $^3$He atom. However, in the following we limit to the case
of zero superfluid velocity.  The parameters appearing in the gradient
energy have been calculated in the weak coupling approximation by
Cross\cite{Cross} and D\"orfle,\cite{Dorfle} the latter including more
Fermi-liquid parameters. For numerical values see  Refs.\
\onlinecite{Hook} and \onlinecite{web}. The characteristic scales are
given by the dipole length $\xi_{\rm d} =
(\hbar/2m_3)\sqrt{\rho_\parallel/\lambda_{\rm d}} \approx 10\     
\mu{\rm m}$ and the dipole field $B_{\rm d} =\sqrt{\lambda_{\rm
d}/\lambda_{\rm h}} \approx 2\ {\rm mT}$. 

We consider a static field ${\bf B}_0 = B_0 \hat{\bf z}$.
We assume that the equilibrium 
$\hat{\bf d}$, denoted by $\hat{\bf d}_0$, lies in the plane  
perpendicular to
${\bf B}_0$: 
\begin{eqnarray}
\hat{\bf d}_0 = \hat{\bf x}\cos\theta + \hat{\bf y}\sin\theta.
\label{e.d0}
\end{eqnarray}
This situation is always achieved in a large field $B_0
\gg B_{\rm d}$, where $\hat{\bf d}$ is forced to the plane by $f_{\rm
h}$ (\ref{e.fmag}), but in some cases this happens in low fields as
well.  Minimization of the total energy (\ref{e.fstatic}) gives for
$\theta$ the equation 
\begin{equation}
\mathcal{D}\theta+(\hat{\bf l} \cdot \hat{\bf d}_0)
 (\hat{\bf l}\times\hat{\bf d}_0)_z=0, 
\label{e.dtheta}\end{equation} 
where the operator $\mathcal{D}$ is defined by
\begin{equation}
\mathcal{D}f=-\frac{K_6}{\lambda_{\rm d}}\nabla^2f
%-\frac{K_5-K_6}{\lambda_{\rm d}}\mbox{\boldmath$\nabla$}\cdot\lbrack\hat{\bf l} 
%(\hat{\bf l}\cdot\mbox{\boldmath$\nabla$}) f\rbrack.
-\frac{K_5-K_6}{\lambda_{\rm d}}
\mbox{\boldmath$\nabla$}\cdot\left\lbrack\hat{\bf l}  (\hat{\bf l}\cdot
\mbox{\boldmath$\nabla$}) f\right\rbrack.
\label{e.ddef}
\end{equation} 

In a dynamic magnetic state one has to include the spin magnetization
$\gamma{\bf S}$ as a new variable in addition to $\hat{\bf d}$ and
$\hat{\bf l}$. The effective energy density has the
form\cite{Leggett74,vollhardtwolfle}
\begin{equation}
f_{\rm eff}=\frac{\mu_0\gamma^2}{2}{\bf S} \cdot
\stackrel{\leftrightarrow}{\chi}^{-1}\cdot{\bf S} 
- \gamma{\bf S}\cdot{\bf B}+f_{\rm d}+f_{\rm g} ,
\label{e.feff}
\end{equation}
where $\gamma$ is the gyromagnetic ratio and
$\stackrel{\leftrightarrow}{\chi}$ the susceptibility tensor. This
leads to the equations of motion 
\begin{subequations}\label{e.leggett}
\begin{eqnarray}
\dot{\bf S}&=&\gamma{\bf S}\times{\bf B}-\hat{\bf
d}\times\frac{\delta f}{\delta\hat{\bf d}}    \\ 
\dot{\hat{\bf d}}&=&\hat{\bf d}\times\gamma\left({\bf
B}-\frac{\mu_0\gamma}{\chi}{\bf S}\right)  , 
\end{eqnarray}\end{subequations}
where $f=f_{\rm d}+f_{\rm g}$ and $\chi$ is the
susceptibility in the normal state. The motion of
$\hat{\bf l}$ is strongly limited by viscosity and therefore we assume
that $\hat{\bf l}$ is independent of time.\cite{LT2} Equations
(\ref{e.leggett}) describe dissipationless dynamics. The inclusion of
dissipative terms is postponed to Sec.~\ref{s.dissipation}. The field
${\bf B}$ is the sum of the static field ${\bf B}_0$ and a small 
radio-frequency field
${\bf B}'$ that oscillates at angular frequency $\omega$.
Throughout this paper we limit to study the  linear response of ${\bf
S}$ and
$\hat{\bf d}$ to
${\bf B}'$. We parametrize the deviation of
$\hat{\bf d}$ with two parameters $d_\theta$ and $d_z$: 
\begin{eqnarray}
 \hat{\bf d} = \hat{\bf d}_0+(\hat{\bf z}\times \hat{\bf
d}_0)d_\theta+\hat{\bf z}d_z.
\label{e.drf}\end{eqnarray}
For ${\bf S}$  we parametrize the deviation ${\bf S}'$ from the
equilibrium
${\bf S}_0=\chi{\bf B}_0/\mu_0\gamma$ by $S'_z=S_z-S_0$ and circular
components 
$S^{\pm}=S_x\pm\imagu S_y$. Similar definitions are used for
other vectors as well. We linearize Eqs.\ (\ref{e.leggett})  and 
assume the time dependence ${\bf S}'(t)={\bf
S}'\exp(-\imagu\omega t)$ etc. Using the equilibrium condition
(\ref{e.dtheta}) we get
\begin{subequations}\label{e.setcons}
\begin{eqnarray}
{\omega}S^{\pm} &=& \pm \omega_0S^{\pm}
\mp\lambda_{\rm d}e^{\pm\imagu\theta}
(\mathcal{D}+U_\perp)d_z
+\imagu \lambda_{\rm d}e^{\pm 2\imagu\theta}l_zl_\mp
d_\theta
\mp \chi B_0B_\pm \\
%%%
{\omega}d_z &=&
\frac{\mu_0\gamma^2}{2\chi}
\left(S^{-}e^{\imagu\theta}-S^{+}e^{-\imagu\theta}\right) +
\frac{\gamma}{2}
\left(B_{+}e^{-\imagu\theta}-B_{-}e^{\imagu\theta}\right)  \\
%%%
\omega S'_z &=&
-\imagu\lambda_{\rm d}(\mathcal{D}+U_\parallel) d_\theta
+\imagu\lambda_{\rm d}(\hat{\bf d}_0\times\hat{\bf l})_z l_z d_z
 \\
%%%
\omega d_\theta &=&
\imagu \frac{\mu_0\gamma^2}{\chi}S'_z -\imagu\gamma
B'_z.
\end{eqnarray}
\end{subequations}
Here $\omega_0=\gamma B_0$ is
the Larmor frequency.
The potentials $U_\parallel$ and $U_\perp$ are defined by
\begin{eqnarray}
U_\parallel &=& 1-l_z^2-2(\hat{\bf l}\times\hat{\bf d}_0)_z^2 
 \\
U_\perp &=&1-2\hat{l}_z^2-(\hat{\bf l}\times\hat{\bf d}_0)_z^2
-\frac{K_6}{\lambda_{\rm d}}
(\mbox{\boldmath$\nabla$}\theta)^2-\frac{K_5-K_6}{\lambda_{\rm
d}}(\hat{\bf l}\cdot\mbox{\boldmath$\nabla$}\theta)^2. 
\label{e.pot}
\end{eqnarray} 

In order to simplify the Eqs.\ (\ref{e.setcons}), let us consider the
special case $l_z\equiv 0$. In this case the equations
separate into independent blocks for longitudinal and
transverse oscillations of the magnetization. The resonance
frequencies  are determined by two independent Schr\"odinger-type
equations for
$d_\theta$ and
$d_z$:
\begin{eqnarray}
(\mathcal{D}+U_\parallel) d_\theta &=&
\alpha_\parallel d_\theta \label{e.schrod1}\\ 
(\mathcal{D}+U_\perp) d_z &=&
\alpha_\perp d_z .  
\label{e.schrod2}
\end{eqnarray}  
The eigenvalues 
$\alpha_{\parallel,k}$ and $\alpha_{\perp,k}$ of these equations
are related to the resonance frequencies as 
\begin{eqnarray}
\omega_{\parallel,k}^2 &=&
\Omega^2\alpha_{\parallel,k}\label{e.freqll}\\
\omega_{\perp,k}^2
&=&
\omega_0^2 + \Omega^2\alpha_{\perp,k}.
\label{e.freqperpgen}
\end{eqnarray}
Here $\Omega = (\mu_0\gamma^2\lambda_{\rm d}/\chi)^{1/2}$ is the
longitudinal resonance frequency of the A phase.  
The corresponding
eigenfunctions of Eqs.\ (\ref{e.schrod1}) and (\ref{e.schrod2}) are
denoted by
$\psi_{\parallel,k}$  and $\psi_{\perp,k}$, respectively. They are
assumed to be  normalized: 
$\int d^3r\,\vert\psi_{\parallel,k}\vert^2=1$ and
$\int d^3r\,\vert\psi_{\perp,k}\vert^2=1$. Because we have temporarily
neglected dissipative processes, the power absorption
$P(\omega)$ consists of delta peaks, $P(\omega)=\sum_k
I_k\delta(\omega-\omega_k)$.

Instead of assuming $l_z\equiv 0$, and alternative approach to Eqs.\
(\ref{e.setcons}) is to study the
high field limit $\omega_0\gg\Omega$. More precisely, one can
calculate the resonance frequencies as a power series of $\Omega^2$ and
 neglect terms of order $\Omega^4$ and
higher. In this approximation all the three components $S'_z$, $S^+$,
and
$S^-$ decouple. The eigenvalue equations and frequencies are
the same as above (\ref{e.schrod1})-(\ref{e.freqperpgen}) except that
Eq.\ (\ref{e.freqperpgen}) is valid only to leading order in $\Omega$: 
\begin{eqnarray}
\omega_{\perp,k}
&=&\pm\left[
\omega_0 +
\frac{\Omega^2}{2\omega_0}\alpha_{\perp,k}
+O(\frac{\Omega^4}{\omega_0^3})\right].
\label{e.freqperp}
\end{eqnarray} 
For the rest of this section we assume the high field
limit $\omega_0\gg\Omega$.

In the case of dipole locking, $\hat{\bf l}({\bf r}) =
\hat{\bf d}({\bf r})$, the lowest ``bulk'' eigenvalues are
$\alpha_{\rm \parallel,b} = \alpha_{\rm \perp,b} = 1$. In this case
only the bulk eigenstate gives rise to absorption 
$I_{\rm b}=Vi$, where for the two modes
\begin{eqnarray}
i_{\parallel}&=& \frac{\pi}{4\mu_0} {B'_z}^2 
\chi  \Omega^2\\
i_{\perp}&=& \frac{\pi}{4\mu_0} (B_x^2+B_y^2) \chi \omega_0^2, 
\label{e.abspower}
\end{eqnarray} 
and $V=\int d^3r$ is the volume of the liquid. 

In the presence of dipole unlocking also other eigenstates contribute
to the absorption. Their intensities are given
by\cite{BruinsmaMaki,VSF84,vollhardtwolfle} 
\begin{eqnarray}
I_{\parallel,k} &=& i_{\parallel}\,\alpha_{\parallel,k}\,
Q_{\parallel,k} = i_{\parallel}\,\alpha_{\parallel,k}
\left\vert\int d^3r
\psi_{\parallel,k}({\bf r})\right\vert^2  \label{e.intensity1}\\
I_{\perp,k} &=&i_{\perp} Q_{\perp,k} =
i_{\perp}\left\vert\int d^3r\psi_{\perp,k}({\bf r})\exp[{\rm
i}\theta({\bf r})]\right\vert^2. 
\label{e.intensity2}
\end{eqnarray} 
Here the $Q_k$'s satisfy the sum rules
\begin{eqnarray}
\sum_{k=0}^{\infty} Q_{\parallel,k} &=&\sum_{k=0}^{\infty}
Q_{\perp,k} = V  \label{e.sumrule0}\\
\sum_{k=0}^{\infty} \alpha_{\parallel,k} Q_{\parallel,k} &=& \int d^3r\,
U_\parallel   \label{e.sumrule1a}\\
\sum_{k=0}^{\infty} \alpha_{\perp,k} Q_{\perp,k} &=& \int d^3r
\left[(\hat{\bf l} \cdot \hat{\bf d}_0)^2 - \hat{l}_z^2 \right] 
\label{e.sumrule1b}\\
\sum_{k=0}^{\infty} \alpha_{\parallel,k}^2 Q_{\parallel,k} &=& \int
d^3r\, U_\parallel^2  \label{e.sumrule2a}\\
\sum_{k=0}^{\infty} \alpha_{\perp,k}^2 Q_{\perp,k} &=& \int
d^3r\left\{
 \left[(\hat{\bf l} \cdot \hat{\bf d}_0)^2 - \hat{l}_z^2\right]^2 +
(\hat{\bf l}\times\hat{\bf d}_0)_z^2 (\hat{\bf l} \cdot \hat{\bf
d}_0)^2 \right\} ,
\label{e.sumrule2b}
\end{eqnarray}
and so on.
The sum rules can be derived using the orthogonality
properties of the eigenfunctions. [In Eq.\ (\ref{e.sumrule2b}) one also
needs Eq.\ (\ref{e.dtheta}).] The lowest order rules
(\ref{e.sumrule0})-(\ref{e.sumrule1b}) apparently are equivalent to the
sum rules presented by Leggett.\cite{Leggett73}


\section{EQUILIBRIUM STRUCTURE OF A SPLAY SOLITON}\label{s.eqstruc}

The minimum of the dipole energy (\ref{e.fdip}) can
be achieved in two ways: either $\hat{\bf l}$ and $\hat{\bf d}$ are
parallel or they are antiparallel. This double degeneracy gives
rise to domain walls or solitons. On one side of the wall $\hat{\bf l}
= \hat{\bf d}$ and on the other side $\hat{\bf l}=-\hat{\bf d}$. We
will study cases where the static
field ${\bf B}_0$ is either perpendicular or parallel
to the plane of the soliton wall. The soliton structures in these
cases are known to have composite twist and splay
structures, respectively\cite{MakiKumar}. The structure and the
fundamental NMR frequency of a twist soliton have  been solved
analytically.\cite{MakiKumar77,MakiKumar,vollhardtwolfle} Here we
concentrate on the case of a splay soliton, which has previously
been studied only by variational
methods.\cite{MakiKumar,VollhardtMaki,BruinsmaMaki,exp} Its structure
is illustrated in Fig.~\ref{splay}. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!tb]
\begin{center} 
\includegraphics[width=8cm]{ldvect3.eps}
\caption{The structure of a splay soliton, where the $\hat{\bf l}$
field (light arrows) has splay shape. The structure is homogeneous
in the $y$-$z$ plane of the soliton wall. Both $\hat{\bf d}$ and
$\hat{\bf l}$ are perpendicular to the static field ${\bf B}_0 = B_0
\hat{\bf z}$.}  
\label{splay} 
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the cases of a twist or a splay soliton, both $\hat{\bf l}$ and
$\hat{\bf d}$  are in the plane perpendicular to ${\bf B}_0$.
We parametrize 
\begin{eqnarray} 
\hat{\bf l}=\hat{\bf x}\cos\eta+\hat{\bf y}\sin\eta  
\label{e.ldvect}
\end{eqnarray} 
and $\hat{\bf d}$ as above (\ref{e.d0}). For a splay soliton the angles
$\theta$ and
$\eta$ are taken as functions of $x$ only. We use the boundary
conditions
$\eta(+\infty)=\theta(+\infty)$  and
$\eta(-\infty)=\theta(-\infty)+\pi$. Substituting the vector fields
(\ref{e.d0}) and (\ref{e.ldvect}) into the total energy
(\ref{e.fstatic}) gives that the energy per unit area is    
\begin{eqnarray} 
f_{\rm s}&=&\frac{1}{2}\int dx [(K_{\rm s}\sin^2\eta
+ K_{\rm b}\cos^2\eta)\left(\frac{d\eta}{dx}\right)^2 \nonumber \\
&& + (K_6\sin^2\eta +K_5\cos^2\eta)\left(\frac{d\theta}{dx}\right)^2 -
\lambda_{\rm d}\cos^2(\eta-\theta) ] \ .  
\label{e.fcsplay0}
\end{eqnarray}
The energy functional (\ref{e.fcsplay0}) was discretized using equal
intervals ($x_i=-L/2+Li/N$, $i=0 \ldots N$) and the values
of $\eta$ and $\theta$ at these discrete points were taken to be
the minimizing variables. The
boundary conditions were taken into account by linear initial
approximations
$\eta = \frac{\pi}{2}-
\frac{2\pi}{3L}x$ and $\theta = \frac{\pi}{3L}x$. The
minimization can be done by a simple relaxation method. The
resulting functions $\eta(x)$ and $\theta(x)$ at different
temperatures are shown in Fig.~\ref{angles}. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!tb]
\begin{center} 
\includegraphics[width=8cm]{angles3.eps}
\caption{The functions $\eta(x)$ and $\theta(x)$ for a splay-soliton 
at temperatures
$T/T_{\rm c} = 0.01, 0.25, 0.50, 0.75, 1.00$. The parameter values
correspond to weak coupling at 3.4 MPa ($F_1^s=14.45$ and
$F_1^a=-1$).}   
\label{angles} 
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{PRINCIPAL NMR FREQUENCY}\label{s.princnmr}

We now apply the theory of Sec.\ \ref{s.hyrdo} to calculate NMR
properties of solitons. For solitons $\hat l_z\equiv 0$, which
decouples the longitudinal and transverse modes at arbitrary $\omega_0$
in Eqs.\ (\ref{e.setcons}).  Thus the eigenvalues $\alpha_{\perp,k}$ are
related to resonance frequencies as given in Eq.\   
(\ref{e.freqperpgen}) at any field. 

The main signature of solitons in the
NMR spectrum comes from the lowest eigenvalue of  Schr\"odinger-like
equations (\ref{e.schrod1}) and (\ref{e.schrod2}). This lowest
frequency can be calculated, for example, using a variational
formulation: 
\begin{eqnarray}
\alpha_{\parallel,\perp}&=& \min_\psi \frac{\int d^3r\left\lbrack
K_6\vert\mbox{\boldmath$\nabla$}\psi\vert^2 +(K_5-K_6)\vert\hat{\bf
l}\cdot\mbox{\boldmath$\nabla$}\psi\vert^2 + \lambda_{\rm d}
 U_{\parallel,\perp}
\vert\psi\vert^2 \right\rbrack }{ \lambda_{\rm d}\int
d^3r\vert\psi\vert^2 } .
\label{e.varform}
\end{eqnarray}
This was discretized and the values of $\psi(x_i)$ were taken to be
the minimizing variables. The form for the initial approximation used
was $\psi(x)=\cosh^n(qx)$ and  
%where $n=-0.314\cdot T_{\rm c}/T$ and $q=1.411$.
the boundary conditions ${d\psi}/{dx}=0$ were assumed at the end
points. The length $L$ was increased until its effect disappeared. 
One solution is shown in Fig.\ \ref{wavefunc}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!tb] 
\begin{center} 
\includegraphics[width=8cm]{potwell_splay.eps}
\caption{The potential $U_\perp$ (solid line), the lowest eigenvalue
$\alpha_\perp$  and the eigenfunction $\psi_\perp$ (dashed line,
vertical scale arbitrary) for transversal resonance of a splay soliton
at
$T\approx T_{\rm c}$. The absorption thickness $s_\perp$ gives the
effective width of the wave function in NMR.  } 
\label{wavefunc} 
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In
Fig.\ \ref{R2} we plot the temperature dependence of the lowest
resonance frequencies as well as some experimental data. For
completeness we  also include the analytical results for the twist
soliton:\cite{MakiKumar,vollhardtwolfle} 
\begin{eqnarray}
\alpha_\parallel^{\rm twist}&=&\frac{1}{2K_{\rm t}} \lbrack
\sqrt{(9K_{\rm t}+K_6)(K_{\rm t}+K_6)}-3K_{\rm t}-K_6\rbrack \\
\alpha_\perp^{\rm twist}&=&\frac{K_6}{K_6+K_{\rm t}} .
\label{R2twist}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!tb] 
\begin{center} 
\includegraphics[width=8cm]{lambdaexp.eps}
\caption{The reduced resonance frequencies $\alpha_\parallel$
(\ref{e.freqll}) and $\alpha_\perp$ (\ref{e.freqperpgen}) as
functions of temperature. Splay soliton transverse resonance: four
upper solid lines and experimental points $\circ$\cite{GouldLee},
$\triangle$\cite{GouldBartolacBozler}, and $\diamondsuit$.\cite{exp}
Splay soliton longitudinal resonance: four lower solid lines. Twist
soliton transverse resonance: four upper dash-dotted lines and
experimental points $+$\cite{Hakonen} and $\square$.\cite{exp} Twist
soliton longitudinal resonance: four lower dash-dotted lines and
experimental points $\times$\cite{Avenel} and
$\triangledown$.\cite{GouldLee} In each case there are four
theoretical lines, which in order of decreasing $\alpha$ correspond to
$F_1^a = 0$ \& weak coupling, $F_1^a = 0$ \& gap enhancement,
$F_1^a = -1$ \& weak coupling, and $F_1^a = -1$ \& gap enhancement.
Good agreement between theory and experiment is achieved with $F_1^a =
-1$ and weak coupling except for the high-temperature transversal splay
data. The effect of strong coupling corrections is shown at
$T/T_{\rm c}=1$ where the lower ends of the bars correspond to
$\gamma=2$ and the upper ends to $\gamma=4$.   
} 
\label{R2} 
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The reduced frequency shifts $\alpha_\parallel$
(\ref{e.freqll}) and
$\alpha_\perp$ (\ref{e.freqperpgen}) depend only on the ratios of
the hydrodynamic coefficients $K_{\rm b}$,
$K_{\rm s}$, $K_{\rm t}$, $K_{5}$, and $K_{6}$. (Note that the absolute
magnitudes of $K_i$'s and $\lambda_{\rm d}$ define length and energy
scales that do not affect $\alpha_\parallel$ or $\alpha_\perp$.)  In
the weak coupling approximation the ratios of $K_i$'s are functions of
an infinite set of Fermi-liquid parameters $F_j^s$ and $F_j^a$, with
$j= 1, 3, 5$, etc.\cite{Dorfle} Here we neglect all the coefficients
with $j>1$ since they are unknown. We assume the melting pressure, but
the pressure dependence is weak. For example the maximum difference in
between $\alpha$  at 2.6 and 3.4 MPa is one per cent at
temperatures higher than $0.5T_{\rm c}$. For $F_1^s$ we use the value
by Greywall,\cite{Greywall} but the results are almost independent of
it. For example, the variation 
$F_1^s=14.5\pm 1$  shows up only in the fourth decimal of $\alpha$
at $T=0.8T_{\rm c}$.     The
value
$F_1^a=-1$ was taken from Ref.~\onlinecite{Greywall83} and it is also
consistent with Ref.~\onlinecite{Candela}. 
In order to see the effect of $F_1^a$ we
also used $F_1^a=0$. It can be seen in Fig.\ \ref{R2} that this shifts
the resonance frequency up at temperatures below $T_{\rm c}$. 

There are no quantitative calculations of strong-coupling effects in the
A phase at a general temperature. In order to get some idea how strong
coupling could affect the soliton frequencies, we use a ``trivial
strong coupling'' model developed by Serene and
Rainer\cite{SereneRainer} for the B phase. In this model the weak
coupling energy gap is multiplied by a factor that depends on the
temperature and on the jump of the specific heat $\Delta C_{\rm
B}/C_{\rm n}$ at $T=T_{\rm c}$. This dependence is 
tabulated in Ref.\
\onlinecite{SereneRainer}. We adapt this model to the A phase by
calculating the multiplying factor using the same table but substituting
$\frac{6}{5}\Delta C_{\rm A}/C_{\rm n}$ in the place of
$\Delta C_{\rm B}/C_{\rm n}$, and using extrapolation when needed. We
take
$\Delta C_{\rm A}/C_{\rm n}$ from measurements by 
Greywall.\cite{Greywall} Even if this seems to be a plausible simple
way to correct the weak coupling results, the best fit with experiments
is obtained by using the bare weak coupling values. It can be seen in
Fig.\ \ref{R2} that the gap enhancement, which affects only
intermediate temperatures, has a smaller effect than the change of
$F_1^a$. 

In the limiting case $T\rightarrow T_{\rm c}$ the reduced frequencies
$\alpha_{\parallel}$ and
$\alpha_{\perp}$ are independent of any parameters appearing
in the weak-coupling model, including also the gap enhancement. This is
a consequence of the Ginzburg-Landau expansion that gives to the
parameters $K_i$ the ratios $K_{\rm b} : K_{\rm s} : K_{\rm t} : K_5 :
K_6 = \gamma : 1 : 1 : 2 : (\gamma+1)$. The only free parameter here is
$\gamma$, which in the weak coupling (with or without gap
enhancement) has the value
$\gamma = 3$. This value is
changed only when nontrivial strong-coupling corrections are included.
The weak coupling plus model by Serene and
Rainer\cite{SereneRainer2} gives an estimate $\gamma \approx 3.12$. 
Fig.\ \ref{R2} shows the reduced frequencies corresponding to
$\gamma=2$ and
$\gamma=4$. We see that this variation changes the splay-soliton
$\alpha_\perp$ less than 3\% but for the twist soliton the effect is ten
times larger.

There is rather good agreement between the experiment and the theory
corresponding to weak coupling and $F_1^a=-1$. In particular, the
longitudinal twist data is a strong indication that the deviation from
the weak coupling value $\gamma=3$ is small, as predicted by Serene and
Rainer.\cite{SereneRainer2} The only major difference between theory and
experiment exists in the transverse splay-soliton frequency at high
temperatures. It seems very difficult to improve the agreement in this
case by any change of the parameters in the theory above since
$\alpha_\perp^{\rm splay}$ at
$T_{\rm c}$ is effectively fixed. On the experimental side, one
possibility is that the field is not precisely in the plane of the
soliton in the measurements. This would add a small twist component to
the splay soliton and thus shift up the frequency.\cite{MakiKumar}
Another possibility is that the inclusion of relaxation mechanisms
could shift the calculated resonance frequency, as will be discussed in
Sec.\ \ref{s.dissipation}.

At $T_{\rm c}$  we find the eigenvalues
$\alpha_\parallel=0.388$ and $\alpha_\perp=0.672$ for $\gamma=3$.
These differ slightly from the variational results by Maki and
Kumar\cite{MakiKumar} that are 0.403 and 0.677, respectively. 


\section{ABSORPTION AND HIGHER MODES}\label{s.intes}

Here we calculate the intensity of the principal soliton peak and
analyze the absorption at other frequencies. For planar objects
the intensities (\ref{e.intensity1}) and
(\ref{e.intensity2}) are most conveniently expressed in the form of an
absorption thickness $s_{k}$, which equals $Q_{k}$ divided by the area
of the planar object,
\begin{eqnarray}
s_{k} = \frac{Q_{k}}{A},
\label{e.abslendef}
\end{eqnarray} 
for each mode $k$.  The absorption thicknesses
in the lowest eigenstate are plotted in Fig.~\ref{abslen}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!tb]
\begin{center} 
\includegraphics[width=8cm]{abslen.eps}
\caption{Absorption (\protect\ref{e.abslendef}) of the principal soliton
peak in four cases. The parameter values correspond to weak coupling
at 3.4 MPa ($F_1^s=14.45$ and $F_1^a=-1$). 
The numerical values at $T_{\rm c}$ are $s_\parallel^{\rm splay} =
6.63\xi_{\rm d}$, $s_\perp^{\rm splay} = 7.35\xi_{\rm d}$,
$s_\parallel^{\rm twist}=8.64\xi_{\rm d}$, and $s_\perp^{\rm
twist}=12.02\xi_{\rm d}$.  }   
\label{abslen} 
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The absorption for a twist soliton can be calculated
analytically. Substituting
the eigenfunction $\psi=c_1\mathop{\rm sech}^\mu az$ into 
Eqs.\ (\ref{e.intensity1}) and (\ref{e.intensity2}) 
and using $\theta=c_2-\mu_\perp\mathop{\rm
sgn}(z)\arccos(\mathop{\rm sech} az)$ (Refs.\
\onlinecite{MakiKumar} and \onlinecite{vollhardtwolfle}) we find
\begin{eqnarray}
s_\parallel^{\rm twist} &=& 
\frac{ \sqrt{\pi}\Gamma^2(\frac{\mu_\parallel}{2})
\Gamma(\mu_\parallel+\frac{1}{2}) }{a \Gamma(\mu_\parallel) 
\Gamma^2(\frac{\mu_\parallel+1}{2})} \label{st1}\\  
s_\perp^{\rm twist} &=&
\frac{\sqrt{\pi}4^{1-\mu_\perp}\Gamma(\mu_\perp)}
{a\Gamma(\mu_\perp+\frac{1}{2})}. \label{st2}
\end{eqnarray}
Here $\Gamma$ is the gamma function and 
\begin{eqnarray}
a&=&\sqrt{\frac{\lambda_{\rm d}(K_6+K_{\rm t})}{K_6 K_{\rm t}}}\\
\mu_\parallel&=&\frac{1}{2}\left(\sqrt{\frac{9K_{\rm t}+K_6}{K_{\rm
t}+K_6}} -1 \right)  \\
\mu_\perp&=&\frac{K_{\rm t}}{K_6+K_{\rm t}} .
\label{e.mudef}\end{eqnarray}

In order to get all the resonance modes, we discretize the space as
discussed above ($x_i=-L/2+Li/N$, $i=0 \ldots N$, for twist soliton
replace $x$ with $z$). This means that equations (\ref{e.schrod1}) and
(\ref{e.schrod2}) turn into a matrix eigenvalue problem. This can be
solved by standard library routines for matrices of reasonable size.
This gives all the frequencies, including the fundamental one
(\ref{e.varform}). However, the higher modes extend over the whole
interval, and it is important to fix boundary conditions for them. We
require zero derivative of $\psi$ at $x=\pm L/2$. This can be justified
by considering a lattice of solitons. The general eigenfunctions $\psi$
of Eqs.\ (\ref{e.schrod1}) and (\ref{e.schrod2}) are of the Bloch form,
but only strictly periodic functions lead to nonzero absorption in Eqs.\
(\ref{e.intensity1}) and (\ref{e.intensity2}).  The unit cell of a
soliton lattice $x=-L/2 \ldots 3L/2$ consists of two solitons located
around
$x=0$ and $x=L$.  The
vectors
$\hat{\bf l}$ and
$\hat{\bf d}$ have the symmetries $\hat{\bf l}(L/2+x) = \hat{\bf
l}(L/2-x)$ and
$\hat{\bf d}(L/2+x) = \hat{\bf d}(L/2-x)$. This implies that the
eigenfunctions can be classified as symmetric or antisymmetric (with
respect to
$x=L/2$). The symmetric solutions have $d\psi/dx(\pm L/2)=0$, and the
antisymmetric solutions can be neglected since they do not contribute
to the absorption. 

The results for the frequencies and intensities of the higher modes are
shown in Fig.~\ref{highmode}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!tb] 
\begin{center} 
\includegraphics[width=8cm]{highmodepicnew.eps}
\caption{The higher modes in the case of transversal resonance of a
splay soliton.  The length used in the calculation is
$L=50\xi_{\rm d}$ in the main frame whereas the inset also contains
the points for $L=250\xi_{\rm d}$. In both cases the lattice spacing
of the discretized lattice is $0.1\xi_{\rm d}$. Other parameters are
weak coupling at 3.4 MPa ($F_1^s=14.45$, $F_1^a=-1$) and temperature
$T=0.95T_{\rm c}$.}
\label{highmode} 
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As expected, the higher modes depend on the size $L$ of the system
used in the calculation. The modes seem to appear as pairs above the
bulk peak. The bulk peak also seems to consist of two peaks that do
not appear exactly at $\alpha = 1$ due to finite $L$. In fact, the
finite $L$ corresponds to a periodic lattice of solitons, that should
give rise to peaks that are analogous to the Bragg peaks in the
x-ray scattering from periodic solids. Such peaks are indeed seen
experimentally in the case of a vortex sheet, where a nearly periodic
arrangement is automatically generated.\cite{Parts} 
The observation of the peaks depends essentially on dissipative
effects, which broadens the peaks. This will be discussed in the
following section.  



\section{THE EFFECT OF DISSIPATION}\label{s.dissipation}

There are two sources of dissipation that are important for solitons.
The nonequilibrium between normal and superfluid magnetizations is
the main relaxation mechanism in homogeneous superfluid
$^3$He.\cite{CE,Combescot76,LT} In inhomogeneous situations the
nonequilibrium between normal magnetizations at different locations
causes relaxation via spin diffusion. We study these two mechanism
below, the latter in the case that the inhomogeneity arises from the
soliton. We neglect extrinsic effects like the
inhomogeneity caused by nonuniform magnetic field. We also do not
consider dissipation caused by orbital motion. According to the
estimation in Ref.\ \onlinecite{LT2}, this is important only at
temperatures very near $T_{\rm c}$, which region is vanishingly small
on the scale of Fig.\ \ref{R2}.

Our treatment of the dissipation is purely phenomenological. For
normal-superfluid relaxation we use the model developed by Leggett and
Takagi, which is extensively described in
Ref.~\onlinecite{LT}. For the relaxation of spatially nonequilibrium
magnetization we use simple diffusion equation.  This approach can be
correct only when the mean free path of the quasiparticles is smaller
than the thickness of the soliton, and therefore it necessarily fails at
low temperatures, where the mean free path diverges. Our treatment also
neglects the tensor character of the spin diffusion coefficient, as
well as the anisotropic energy gap with varying anisotropy axis
$\hat{\bf l}({\bf r})$, which leads to  Andreev reflection of the
quasiparticles.

In order to treat the relaxation, the spin
polarization ${\bf S}$ is divided into a superfluid (pair) part ${\bf
S}_p$ and a normal (quasiparticle) part ${\bf S}_q= {\bf S}-{\bf
S}_p$. Both parts have separate equations of motion   
\begin{subequations}\label{e.ltequ2}\begin{eqnarray}
\dot{\bf S}_q &=& \gamma{\bf S}_q \times \left( {\bf B} -
\mu_0\gamma\frac{F_0^a}{\chi_{0}} {\bf S}_p \right) +
\frac{1}{\tau}\left[ (1-\lambda) {\bf S}_p - \lambda {\bf S}_q \right]
+ \kappa \nabla^2 {\bf S}_q\label{e.ltequ2a}
 \\ 
\dot{\bf S}_p &=& \gamma{\bf S}_p \times \left( {\bf B} -
\mu_0\gamma \frac{F_0^a}{\chi_{0}} {\bf S}_q \right) -
\frac{1}{\tau}\left[ (1-\lambda) {\bf S}_p - \lambda {\bf S}_q \right]
-\hat{\bf d}
\times \frac{\delta f}{\delta\hat{\bf d}}  \\ 
\dot{\hat{\bf d}} &=& \gamma\hat{\bf d}\times \left[ {\bf B} -
\mu_0\gamma\frac{F_0^a}{\chi_{0}}{\bf S}_q
-\mu_0\gamma
\left(\frac{F_0^a}{\chi_{0}}+\frac{1}{\lambda\chi_0}\right)
{\bf S}_p
\right] .  
\end{eqnarray}\end{subequations}
Except the spin diffusion term $\kappa \nabla^2 {\bf
S}_q$, these equations are the same as in Ref.\ \onlinecite{LT}. Here
$\chi_0$ is the susceptibility in the absence of Fermi liquid effects.
As above, $\chi$ is the susceptibility in the normal state, so that
$\chi/\chi_0 = 1/(1+F_0^a)$. The function $\lambda(T/T_{\rm c})$ is
defined as the equilibrium fraction of the superfluid magnetization,
and is given by Eq.\ (4.23) in Ref.~\onlinecite{LT}. The Leggett-Takagi
relaxation time
$\tau$ describes local relaxation of ${\bf
S}_p$ and ${\bf S}_q$ to their equilibrium values. 
The spin diffusion appears via term $\kappa \nabla^2 {\bf S}_q$ in the
equation for $\dot{\bf S}_q$ (\ref{e.ltequ2a}). In the normal state
the spin diffusion constant $\kappa$ is related to the spin diffusion
time $\tau_{\rm D}$ by\cite{Leggett70}  
\begin{equation}
\kappa = \frac{1}{3}v_{\rm F}^2 (1+F_0^a)\tau_{\rm D},
\label{e.diffcoeff}
\end{equation}
where $v_{\rm F}$ is the Fermi velocity. We use this parametrization
also in the superfluid state. Note that $\kappa$ is defined as the spin
diffusion constant for ${\bf S}_q$, so that the effective diffusion
constant for the total magnetization ${\bf S}$ is $(1-\lambda)\kappa$.

We continue by writing the equations for variables $\hat{\bf d}$, ${\bf
S}$, and the deviation from local equilibrium
\begin{eqnarray}
\boldsymbol{\eta} = {\bf S}_p - \lambda {\bf S}.
\end{eqnarray}   
Similarly as in Sec.\ \ref{s.hyrdo}, we linearize the equations and
assume harmonic time dependence. We use the fact that 
$l_z=0$ in solitons. As a consequence, the transversal and longitudinal
modes separate at any field $B_0$. 
Differing from Sec.\
\ref{s.hyrdo} we introduce dimensionless quantities by defining
$S^\pm = (\Omega^2/\lambda_{\rm d} \omega_0) (S_x' \pm \imagu S_y')$ and
$\eta^\pm = 
(\Omega^2/\lambda_{\rm d} \omega_0)(\eta_x' \pm \imagu
\eta_y')$. For transversal oscillations we get the equations 
\begin{subequations}\label{e.eigvgen}\begin{eqnarray}
\frac{\omega}{\omega_0}S^{\pm} &=& 
\left[\pm 1+\imagu (1-\lambda)\frac{\kappa}{\omega_0}\nabla^2\right]S^{\pm}
- \imagu\frac{\kappa}{\omega_0}\nabla^2\eta^{\pm}
\mp\frac{\Omega^2}{\omega_0^2}e^{\pm \imagu\theta}
(\mathcal{D}+U_\perp)d_z
\mp \frac{B_\pm}{B_0} \label{e.eigvgena} \\
%%%
\frac{\omega}{\omega_0}\eta^{\pm} &=& 
\left(\pm \frac{1}{1+F_0^a}-\frac{\imagu}{\omega_0\tau} +
\imagu\lambda\frac{\kappa}{\omega_0}\nabla^2\right)\eta^{\pm} 
-\imagu\lambda(1-\lambda)\frac{\kappa}{\omega_0}\nabla^2 S^{\pm}
\label{e.eigvgenb} \\
&&\mp(1-\lambda)\frac{\Omega^2}{\omega_0^2}e^{\pm\imagu\theta}
(\mathcal{D}+U_\perp)d_z
\\
%%%%
\frac{\omega}{\omega_0}d_z &=&
\frac{1}{2}\left(S^{-}e^{\imagu\theta}-S^{+}e^{-\imagu\theta}\right)
+\frac{\chi}{2\lambda\chi_0}\left(\eta^{-}e^{\imagu\theta} -
\eta^{+}e^{-\imagu\theta}\right) 
\nonumber \\ && + \frac{1}{2B_0}
\left(B_{+}e^{-\imagu\theta}-B_{-}e^{\imagu\theta}\right).
\end{eqnarray}\end{subequations}
The full solution of this problem can be written as a five component
vector $\psi = (S^{-}\ \eta^{-}\ d_z\ \eta^{+}\ S^{+})^T =
\sum_i c_i \psi_i $, where the $\psi_i$:s satisfy the homogeneous
equation, where $B_\pm=0$. The eigenvalues $\omega_i$ of
the homogeneous equation are now complex valued, and
the eigenvectors 
$\psi_i$  
are not orthogonal to each other. In order to solve for the absorption
spectrum we find that the following
adjoint eigenvalue problem\cite{MorseF} 
\begin{subequations}\label{e.eigvgen2}
\begin{eqnarray}
\frac{\omega}{\omega_0}\underline{S}^{\pm} &=& 
\left[\pm 1+\imagu (1-\lambda)\frac{\kappa}{\omega_0}\nabla^2\right]
\underline{S}^{\pm}
-\imagu\lambda(1-\lambda)\frac{\kappa}{\omega_0}\nabla^2 
\underline{\eta}^{\pm}
\mp\frac{1}{2}e^{\mp\imagu\theta}\underline{d}_z
 \\
%%%
\frac{\omega}{\omega_0}\underline{\eta}^{\pm} &=& 
\left(\pm \frac{1}{1+F_0^a}-\frac{\imagu}{\omega_0\tau} +
\imagu\lambda\frac{\kappa}{\omega_0}\nabla^2\right)\underline{\eta}^{\pm} 
- \imagu\frac{\kappa}{\omega_0}\nabla^2 \underline{S}^{\pm}
\mp\frac{\chi}{2\lambda\chi_0} e^{\mp\imagu\theta}\underline{d}_z
 \\
%%%%
\frac{\omega}{\omega_0}\underline{d}_z &=&
\frac{\Omega^2}{\omega_0^2}(\mathcal{D}+U_\perp)\left[
e^{-\imagu\theta}\underline{S}^{-} - e^{\imagu\theta}\underline{S}^{+}
+(1-\lambda)(e^{-\imagu\theta}\underline{\eta}^{-} -
e^{\imagu\theta}\underline{\eta}^{+}) \right]
\end{eqnarray}\end{subequations}
has the same eigenvalues $\omega_i$ than the homogeneous version of
problem (\ref{e.eigvgen}) and that the eigenvectors
$\underline{\psi}_i=(\underline{S}_i^{-}\ \underline{\eta}_i^{-}\
\underline{d}_{z,i}\ \underline{\eta}_i^{+}\ \underline{S}_i^{+})^T$
are orthogonal to $\psi_j$:s: 
\begin{eqnarray} 
\int \underline{\psi}_i^T \psi_j d^3r = \delta_{ij}.
\label{e.ortog}
\end{eqnarray} 
When deriving this we have assumed zero derivate for the wave
functions far from the soliton by considering a lattice of solitons as
was discussed in Sec.\ \ref{s.intes}. The power absorption can now
be written as
\begin{eqnarray}
P(\omega) &=& \frac{1}{2}\gamma\omega\int {\rm Im}\left( {\bf
B}'\cdot {\bf S}' \right) d^3r \nonumber \\ 
&=& \frac{\chi B_0 \omega}{4\mu_0} \sum_j {\rm Im}
\left[ c_j \int d^3r \left( B_{-}^{*} S_j^{-}+B_{+}^{*} S_j^{+}
\right) \right]
\label{e.powerabs}
\end{eqnarray}
where the coefficients $c_j$ are given by
\begin{eqnarray}
c_j = \frac{\gamma}{\omega-\omega_j} \int d^3r \left[
 B_{-}\underline{S}_j^{-} -B_{+}\underline{S}_j^{+}
+ \frac{1}{2}(e^{-\imagu\theta}B_{+} -
e^{\imagu\theta}B_{-})\underline{d}_{z,j} \right] . 
\end{eqnarray}
We observe that the eigenfunctions are not symmetric with respect
to the center of the soliton. This apparently is caused by the
spin-diffusion term.

In the hydrodynamic limit where $\omega\tau \ll 1$ one may solve
$\eta^\pm$ from Eq. (\ref{e.eigvgen}) and to linear order in $\tau$
\begin{eqnarray}
\eta^\pm = -\tau\lambda(1-\lambda)\kappa\nabla^2 S^\pm
\pm \imagu\tau(1-\lambda)\frac{\Omega^2}{\omega_0}e^{\pm\imagu\theta}
(\mathcal{D}+U) d_z .
\end{eqnarray}
If we additionally ignore the term $\nabla^2\eta^\pm$, the equations
for the three component vector $(S^{-}\ d_z\ S^{+})^T$ read 
\begin{subequations}\label{e.eigvhydr}
\begin{eqnarray}
\frac{\omega}{\omega_0}S^{\pm} &=& 
\left[\pm 1+\imagu (1-\lambda)\frac{\kappa}{\omega_0}\nabla^2\right]S^{\pm}
\mp\frac{\Omega^2}{\omega_0^2}e^{\pm
\imagu\theta}(\mathcal{D}+U_\perp)d_z
\mp \frac{B_\pm}{B_0} \\
%%%%
\frac{\omega}{\omega_0}d_z &=&
\frac{1}{2}e^{\imagu\theta}
\left[1-\frac{\tau\chi(1-\lambda)\kappa}{\chi_0}
\nabla^2\right] S^{-} 
-\frac{1}{2}e^{-\imagu\theta}
\left[1-\frac{\tau\chi(1-\lambda)\kappa}{\chi_0}
\nabla^2\right] S^{+}  \\
&-&\imagu\frac{\tau\chi\Omega^2(1-\lambda)}{\chi_0\omega_0\lambda} 
(\mathcal{D}+U_\perp)d_z
+ \frac{1}{2B_0}
\left(B_{+}e^{-\imagu\theta}-B_{-}e^{\imagu\theta}\right) .
\end{eqnarray}\end{subequations}
The homogeneous adjoint problem for this is given by 
\begin{subequations}\label{e.eigvhydr2}
\begin{eqnarray}
\frac{\omega}{\omega_0}\underline{S}^{\pm} &=& 
\left[\pm 1+\imagu (1-\lambda)\frac{\kappa}{\omega_0}\nabla^2\right]\underline{S}^{\pm}
\mp \frac{1}{2}\left(1-\frac{\tau\chi(1-\lambda)\kappa}{\chi_0}
\nabla^2\right) \left(e^{\mp\imagu\theta} \underline{d}_z\right)
 \\
%%%%
\frac{\omega}{\omega_0}\underline{d}_z &=&
\frac{\Omega^2}{\omega_0^2}(\mathcal{D}+U_\perp)
\left( e^{-\imagu\theta} \underline{S}^{-} - e^{\imagu\theta}
\underline{S}^{+}\right)  -
\imagu\frac{\tau\chi\Omega^2(1-\lambda)}{\chi_0\omega_0\lambda}
(\mathcal{D}+U_\perp) \underline{d}_z.
\end{eqnarray}\end{subequations}
The equation for the power absorption does not need any
modifications. One must be careful since the validity region of the
hydrodynamic approximation is not very large. For temperatures near
$T_{\rm c}$ one must, for typical values of $\tau$, have $B_0 \lesssim
15$ mT. 
%(tarkista miten suuri virhe rajalla tulee) 
We mainly used the hydrodynamic approximation for checking our
calculations in the limit of $\tau \rightarrow 0$.

For parallel case, where ${\bf B}' = B'\hat{\bf z} \parallel {\bf
B}_0$, we also define dimensionless quantities by writing ${\bf
S}'=\hat{\bf z}\lambda_{\rm d} S_z/\Omega$,
$\boldsymbol{\eta}' = \hat{\bf z}\lambda_{\rm d} \eta_z/\Omega$
and
${\bf d}'= d_\theta \hat{\bf z}\times \hat{\bf d}_0$.  The equations of
motions are
\begin{subequations}\label{e.eigvpargen}
\begin{eqnarray}
\frac{\omega}{\Omega} S_z &=& 
\imagu(1-\lambda)\frac{\kappa}{\Omega} \nabla^2 S_z 
-\imagu\frac{\kappa}{\Omega} \nabla^2 \eta_z
-\imagu(\mathcal{D}+U_\parallel) d_\theta  \\
\frac{\omega}{\Omega} \eta_z &=& 
-\imagu\lambda(1-\lambda)\frac{\kappa}{\Omega}\nabla^2 S_z
-\imagu\left(\frac{1}{\tau\Omega} - \lambda \frac{\kappa}{\Omega}
\nabla^2 \right) \eta_z 
-\imagu (1-\lambda) (\mathcal{D}+U_\parallel) d_\theta  \\
\frac{\omega}{\Omega} d_\theta &=& 
\imagu\left(S_z +\frac{\chi\eta_z}{\lambda\chi_0} \right)
-\imagu\frac{\gamma B'}{\Omega} 
\end{eqnarray}\end{subequations}
and the homogeneous adjoint problem for this is given by
\begin{subequations}\label{e.eigvpargen2}
\begin{eqnarray}
\frac{\omega}{\Omega} \underline{S}_z &=& 
\imagu(1-\lambda)\frac{\kappa}{\Omega} \nabla^2 \underline{S}_z
-\imagu\lambda(1-\lambda)\frac{\kappa}{\Omega}\nabla^2 \underline{\eta}_z 
+ \imagu \underline{d}_\theta  \\
\frac{\omega}{\Omega} \underline{\eta}_z &=&
-\imagu\frac{\kappa}{\Omega}\nabla^2 \underline{S}_z
-\imagu\left(\frac{1}{\tau\Omega} - \lambda \frac{\kappa}{\Omega}
\nabla^2 \right) \underline{\eta}_z
+\imagu \frac{\chi}{\lambda\chi_0} \underline{d}_\theta  \\
\frac{\omega}{\Omega} \underline{d}_\theta &=& 
-\imagu (\mathcal{D}+U_\parallel) \left[ \underline{S}_z +
(1-\lambda)\underline{\eta}_z \right] .
\end{eqnarray}\end{subequations}
Power absorption for parallel case is given by
\begin{eqnarray}
P(\omega) &=& \frac{\chi \Omega \omega}{2\mu_0\gamma}
\sum_j {\rm Im}\left( c_j \int  B'^* S_{z,j} d^3r \right),
\label{e.powerabspar}
\end{eqnarray}
where the $c_j$:s are the coefficients of the full solution 
$\psi = (S_z\ \eta_z \ d_\theta)^T = \sum_j c_j (S_{z,j}\ \eta_{z,j} \ d_{\theta,j})^T $ and given by
\begin{eqnarray}
c_j &=& \frac{\imagu\gamma B'}{\omega_j-\omega}\int 
\underline{d}_{\theta,j} d^3r .
\end{eqnarray}
Similarly to the transversal case one could write the hydrodynamic
equations using only $d_\theta$ and $S_z$.

The numerical solution for these different eigenvalue problems is
obtained by dividing the calculation length $x=-L/2\ldots L/2$, for
example, to 1000 points and approximating the spatial derivatives by
differences. The eigenvalues and vectors of the resulting sparse
5000$\times$5000 (or in the parallel case 3000$\times$3000)
matrix $A$ are then solved by Matlab. We make use of sparse
matrices and calculate normally only $20$ lowest eigenvalues with
${\rm Re}[\omega_k] > 0$ that give the main contribution to the
absorption spectrum. The complex mode frequencies $\omega_k$ are
related to the reduced frequency shifts $\alpha_{k}$ as given by
Eqs.\ (\ref{e.freqll}) and
(\ref{e.freqperpgen}).  

Additional parameters appear in the calculation compared to the
dissipationless case. The most crucial ones are the relaxation times
$\tau$ and $\tau_{\rm D}$. The Leggett-Takagi time $\tau$ can be
extracted from the width of the bulk peak. Measurements of longitudinal
and transverse resonance are in good agreement.\cite{Gully} We have
reanalyzed the data by Gully et.\ al.\cite{Gully} including
strong-coupling corrections as described in Sec.\ \ref{s.princnmr} and
get the fit
$\tau=[1.70+ 6.71(1 - T/T_{\rm c})](1+F_0^a)10^{-7}$ s in the range
of $T/T_{\rm c}=0.78\ldots 0.98$.
In our calculations we have used 
$\tau = 
0.22\times 10^{-6}T^{-2}$ s mK$^2$ (value quoted for the normal state in
Ref.\ \onlinecite{Wheatley}) and
$F_0^a=-0.746$. These give a magnitude of the same order, which is
sufficient for the qualitative conclusions below. The magnitude of spin
diffusion time in the normal state is also of the same
order.\cite{sachrajda}  In the absence of any calculation of spin
diffusion in the A phase, we simply use for
$\tau_{\rm D}$ the same value as for
$\tau$. A few absorption spectra are plotted in
Figs.~\ref{spectrum}, \ref{spectrumHighH0}
and \ref{parallelsplayabs}. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!tb] 
\begin{center} 
\includegraphics[width=8cm]{pabsL50t92t95t97.eps}
\caption{Absorption spectra for transversal twist soliton (dash-dotted
lines) and splay soliton (solid lines) at $T/T_{\rm c} = 0.92$,
0.95 and 0.97, when $L=50\xi_{\rm d}$, $B_0 = 3$ mT, $\tau = \tau_{\rm
D} = 0.22\times 10^{-6}T^{-2}$ s mK$^2$. Other parameters are obtained
using weak coupling, pressure of 3.4 MPa and $F_1^a=-1$.}  
%
%Used 20 lowest eigenvalues with ${\rm Re}[\omega_k] > 0$
%Additionally $v_{\rm F}=32.2267$ m/s
\label{spectrum} 
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!tb]
\begin{center} 
\includegraphics[width=8cm]{pabsL50t75H28tratwist.eps}
\caption{Transversal absorption spectrum for a twist soliton
at
$T=0.75T_{\rm c}$ and $B_0 = 28.4$ mT (solid line). Other parameters are
are the same as in Fig.\ \protect\ref{spectrum}. The effect of the
two dissipation mechanisms is demonstrated by setting $\tau=0$
(dash-dotted line) and $\tau_{\rm D}=0$ (dashed line).}   
%Case tau=0 is calculated using hydrodynamic approximation and
%using 20 lowest eigenvalues with ${\rm Re}[\omega_k] > 0$
\label{spectrumHighH0} 
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!tb] 
\begin{center} 
\includegraphics[width=8cm]{parallelsplayabs.eps}
\caption{Longitudinal absorption spectrum for a splay soliton at
$T=0.95T_{\rm c}$ (solid line). There is no dependence on the magnetic
field, and other parameters are the same as in Fig.\
\protect\ref{spectrum}. The effect of the two dissipation mechanisms is
demonstrated by setting
$\tau=0$ (dash-dotted line) and $\tau_{\rm D}=0$ (dashed line).}   
%Case tau=0 is calculated using hydrodynamic approximation and
%using 20 lowest eigenvalues with ${\rm Re}[\omega_k] > 0$
\label{parallelsplayabs} 
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The effect of the normal-superfluid relaxation seems to be simple
line broadening. In the transversal resonance the linewidth measured on
the
$\alpha_\perp$ scale is inversely proportional to the magnetic field
$B_0$ for $B_0\gg B_{\rm d}$. This is caused by $\imagu/(\omega_0\tau)$
term in Eqs.\ (\ref{e.eigvgenb}).  For the longitudinal resonance there
is no field dependence since the equations (\ref{e.eigvpargen}) do not
contain
$\omega_0$. For longitudinal case the linewidth
on the $\alpha$-scale has no strong temperature dependence and
approaches a finite constant when
$T\rightarrow T_{\rm c}$. In the transverse case the linewidth
on the $\alpha$-scale vanishes as $T\rightarrow T_{\rm c}$. All these
characteristics are the same as predicted for the bulk peaks. 

The effect of spin diffusion is more complicated. In addition to
broadening, it shifts the principal soliton peak to higher frequencies.
Since the spin diffusion is due to inhomogeneous texture, it's strength
in the trans\-ver\-sal case is obtained by comparing the term $\imagu
(1-\lambda)(\kappa/\omega_0)
\nabla^2 S^\pm$ with the term $(\Omega^2/\omega_0^2)
\exp(\pm\imagu\theta) (\mathcal{D}+U_\perp)d_z$ in Eq.\
(\ref{e.eigvgena}). Therefore the effective diffusion constant
$\kappa_{\rm eff}^\perp = (1-\lambda)
\kappa \omega_0/\Omega^2$ is linearly proportional to the magnetic
field. As a consequence also the reduced frequency $\alpha_\perp$ of the
satellite peak is field dependent.   
When $B_0=5$ mT the effect of diffusion is already quite
large at
$T=0.95T_{\rm c}$ and the satellite peak in the absorption spectrum
resulting from the soliton is almost smeared out. For parallel case
there exist no field dependence and the effective diffusion
constant reduces to $\kappa_{\rm eff}^\parallel = (1-\lambda)
\kappa/\Omega$. In both transversal and longitudinal cases the
effective diffusion coefficients $\kappa_{\rm eff}^\parallel$ and 
$\kappa_{\rm eff}^\perp$ diverge when $T\rightarrow T_{\rm c}$, as
$\Omega\rightarrow 0$. 

The relative contributions of spin diffusion and 
normal-superfluid relaxation are illustrated in
Figs.\ \ref{spectrumHighH0} and \ref{parallelsplayabs}, where both are
separately shut off by setting $\tau=0$ or $\tau_{\rm D}=0$. It can be
seen that their contributions to satellite line broadening are
comparable in the longitudinal satellite peak at $T=0.95 T_{\rm c}$. In
the transversal case at high field the spin diffusion becomes the
dominant relaxation mechanism, as well as approaching $T_{\rm c}$. In
uniform order parameter the width of the bulk peak comes solely from
normal-superfluid relaxation. Our soliton lattice has many higher peaks
in the absence of dissipation, but dissipation seems to remove
them. Note that the higher peaks a suppressed by spin diffusion in
the case of Fig.\ \ref{spectrumHighH0} but by normal-superfluid
relaxation in the case of Fig.\ \ref{parallelsplayabs}.

The relatively high field in Fig.\ \ref{spectrumHighH0} has been
chosen to allow comparison to the spectrum of Hakonen et
al.\cite{Hakonen} The shape of the twist-soliton spectra are very
similar even if the temperature is quite low, where our modelling of
the spin diffusion is under suspect. 
  Further spectra at lower field are
shown by Parts et al.\cite{exp} 
The comparison to Ref.\
\onlinecite{GouldBartolacBozler} is a different case. In this
experiment such a dense soliton lattice was created that no bulk
peak was observable. Avenel et al.\cite{Avenel} and Gould and
Lee\cite{GouldLee} show soliton spectra measured by sweeping
temperature.   In all cases a proper comparison to our calculations
would require first estimation of the density of solitons and then
correcting our calculations for that density, which we have not done.  
What we can state that there seems to be no obvious contradictions
between theory and experiment concerning the line width.  

A summary of our results at $B_0=3$ mT is shown Fig.\ \ref{r2spindiff}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!tb]
\begin{center} 
\includegraphics[width=8cm]{alphawith50and75.eps}
\caption{The resonance frequencies and line widths as a function of
temperature at $B_0=3$ mT. The solid lines give the frequency of
maximum  satellite absorption, the dash-dotted lines 75\%  and dotted
lines  50\% of the maximum. The dashed lines give resonance frequency
in the absence of dissipation. Other parameters are
$L = 50\xi_{\rm d}$, $\tau = \tau_{\rm D} = 0.22\times
10^{-6}T^{-2}$ s mK$^2$, pressure 3.4 MPa  ($F_1^s=14.45$ and
$F_1^a=-1$ and weak coupling). The experimental points are same as in
Fig. \ref{R2}.} 
%Used 20 lowest eigenvalues with ${\rm Re}[\omega_k] > 0$
%Additionally $v_{\rm F}=32.2267$ m/s
\label{r2spindiff} 
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We see that the frequency of maximum absorption is shifted towards
higher frequencies, especially at temperatures near $T_{\rm c}$. This
alleviates the disagreement found between dissipationless
theory and the experiment by Gould and
Lee\cite{GouldLee} in the transverse mode of splay soliton.
However, the experimental data does not show any sign of the divergence
at $T_{\rm c}$ predicted by our model. 
The shift of the resonance frequency results from spin diffusion. For
the longitudinal mode it is field independent, but for the transverse
mode it increases with increasing field. In high field this shift can
be substantial even at $T\approx 0.7T_{\rm c}$, see Fig.\
\ref{spectrumHighH0}. This poses a problem since no
field-dependent shift has been observed experimentally. This is the
case particularly for the transverse mode of twist
soliton measured at $B_0=9.9$ mT
($\square$)\cite{exp} and 28 mT ($+$)\cite{Hakonen}.
Similar evidence for transverse mode of splay soliton measured   
at $2.5 \ldots 3.7$ mT ($\circ$)\cite{GouldLee} and at $15\ldots20$
mT
($\triangle$)\cite{GouldBartolacBozler} is not so
clear since the latter data is measured under different conditions, as
discussed above.

\section{CONCLUSIONS}

The frequencies of soliton satellite peaks in the NMR spectrum are
calculated at all temperatures. The agreement of dissipationless
theory with experiments is very good using weak coupling
and $F_1^a\approx -1$ at the melting pressure. However, there is a small
difference in the transverse mode frequency of a splay soliton near
$T_{\rm c}$. This difference is partly explained by taking into
account spin diffusion. Simultaneously the spin diffusion shifts
the reduced frequencies $\alpha$ in high field, which has not
been observed experimentally. We hope that new experiments
could clarify this problem. We also point out
the rather narrow longitudinal line of the splay soliton, that has not
been studied experimentally.

It would be of interest to extend the present calculations to vortex
lines in $^3$He-A. This requires two-dimensional calculation and thus
would be computationally much more demanding than the present
one. 


\section*{ACKNOWLEDGMENTS}
We thank Pete Sivonen for contribution on early stages of this work.
This research is supported by the Academy of Finland and the
V\"ais\"al\"a foundation.

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\end{document}

