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auto-parametric, resonance, whispering, autonomous, gallery, modes, nonabelian, nonlinear, non-pertubative, supratransmission , supraconductivity, breather, nonpertubative, pseudosphere, phase, berry, Gordon, sine-Gordon, Baecklund, Thirring, Rayleigh, fine structure, iteration, iterative
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%keywords: auto-parametric, resonance, whispering, autonomous, gallery, modes, nonabelian, nonlinear, non-pertubative, supratransmission , supraconductivity, breather, nonpertubative, pseudosphere, phase, berry, Gordon, sine-Gordon, Baecklund, Thirring, Lobachevski, Chebyshev, Rayleigh, fine structure, iteration, iterative
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\begin{document}
\title{Soliton Coupling Driven by Phase Fluctuations in Auto-Parametric Resonance}
\author{B{\sc ernd} ~B{\sc inder}}

\date{13-20.11.2002}
\affiliation{\it 88682 Salem, Germany, \href
{mailto:binder@quanics.com}{binder@quanics.com} \copyright 2002}
%#############################################################
\begin{abstract}
In this paper the interaction of sine-Gordon solitons and
mediating linear waves is modelled by a special case of
auto-parametric resonance, the Rayleigh-type self-excited
non-linear autonomous system driven by a statistical phase
gradient related to the soliton energy. Spherical symmetry can
stimulate "whispering gallery modes" (WGM) with integral coupling
number $M=137$.
\end{abstract}
\maketitle
%#############################################################

%#############################################################
\paragraph*{Introduction.}
%#############################################################

For nonlinear field theory models in 1+1--dimensional space--time
the equations of motion admit finite energy and finite width
solutions called solitons \cite{AbS}. The interest in such
low-dimensional sine--Gordon (SG) models as an universal concept
in nonlinear science arises from their integrability, duality
properties, non-perturbative aspects, and electric-magnetic
duality in gauge theories. The solitons retain their identity
after collisions, can annihilate with anti--solitons,
many--soliton solutions obey Pauli's exclusion principle. As
pointed out by Skyrme \cite{Skyrme} this can be interpreted as a
fermion--like behavior. Vertex solitons are localized and
non-singular solutions of a non-linear field theory, whereas the
energy-momentum transfer between solitons can be assigned to
linear bosons. In this paper we will try to determine the coupling
strength of vertex soliton and linear waves. With the focus on
low-dimensional autonomous behavior, the interaction can be
approached by the equation of Lord Rayleigh with coupling and
dissipation driven by phase fluctuations. The second order
equation includes two terms that balance wave dissipation and
regeneration as a special case of self--excited auto--parametric
resonance \cite{trvn}. This system can i.e. model simplified music
instruments (in the original work
of Rayleigh a clarinet reed).\\

%#############################################################
\paragraph*{Preliminary measurements.}
%#############################################################

It is well known, that the Josephson junction qualifies as the
proper nonlinear medium for the experimental study of quantum
interaction between linear waves and vertex solitons. Since linear
waves and vertex solitons oscillate and propagate in the same
medium, the perturbed SG equation will have the same dissipation
and regeneration terms as the linear wave differential equation.
In recent experiments \cite{Ustinov98,Ustinov00} the excitation of
linear electromagnetic modes by a topological SG soliton in a
cylindrically curved Josephson junction waveguides has been
reported. The Rayleigh terms show a small difference to the
dissipation/regeneration terms in \cite{Ustinov98,Ustinov00} that
describe their experimental setup using phenomenological
electrode/junction impedance terms. The excited quantum modes that
mediate circularly and couple to solitons refer to the classical
acoustic phenomenon with name ``whispering gallery modes'' (WGM)
\cite{rayleigh}. The interaction results in the quantum
current--voltage characteristic of the junction. \\
%---------------------------------------------------------------------------------------------------

%#############################################################
\paragraph*{Quantum field theory.}
%#############################################################

In quantum field theory, particles correspond to fluctuations of a
space-time function that minimizes the action for some Lagrangian
$L$ which is itself the space integral of a Lagrangian density
$\cal L$. A low-dimensional (bosonic) SG field of a single
hermitian scalar field $\theta$ in 1+1 dimensions with Lagrangian
density ${\cal L}= {\mu\over 2}
\partial_{\nu}\theta\partial^{\nu}\theta~-V(\theta)$ is a function
of one space dimension and time. The time independent field
equations reads $\mu {\partial_r^2 \theta} = {\partial_\theta V }$
which can also be written as
\begin{equation}
V(\theta) =  {\mu \over 2} (\partial_r \theta)^2 \,
\labeqn{potenergy}.
\end{equation}
In the quantum model, the expectation of the square of the phase
gradient contributes to the soliton self-energy, where the phase
gradient usually scales proportional to the wave number that is
proportional to the frequency. \\

%#############################################################
\paragraph*{Rayleigh--type perturbed SG.}
%#############################################################

The coupling between linear waves and non--linear solitons could
be important for the understanding for scale-dependent
electromagnetic coupling. The Rayleigh-type perturbed SG equation
is given by
\begin{equation}
      \(\partial_{rr} - \partial_{tt}\) \theta
    - \sin\theta =   \[b^2 (\partial_r \theta)^2 - a^2 \]\partial_t\theta
\, \labeqn{josephson0V},
\end{equation}
where $\partial_{rr} - \partial_{tt}$ is the D'Alembert wave
operator. On the left side are the usual SG terms, on the right
side are with \eqn{potenergy} the typical Rayleigh wave
dissipation and regeneration terms. These two terms balance wave
dissipation and regeneration and can be found in self--excited
auto-parametric systems \cite{trvn}. The term $b^2 (\partial_r
\theta)^2 - a^2$ controls $\theta$ by weighting the dissipative
first order term $\partial_t\theta$. The amplitude of
the coupling wave can be assigned to a phase difference between two solitons. \\

%#############################################################
\paragraph*{Rayleigh--type auto--parametric system.}
%#############################################################

Following \cite{trvn} we characterize auto--parametric systems as
a vibrating system which consists of at least two subsystems: an
oscillator that is generally in a vibrating state and the excited
system which is excited indirectly and coupled to the oscillator
in a nonlinear way. A linear wave will be defined as a disturbance
which transports energy from one location to another location
without transporting matter evolving at light speed $c$ with $c
\,\partial_{r} \theta =
\partial_{t}\theta$. In the auto--parametric system the
dissipation and excitement parameter will depend on the phase
statistics. The Rayleigh--type system
\begin{equation}
    \partial_{tt}\theta + \theta = a^2 \partial_{t} \theta -
    b^2 c^{-2} (\partial_{t}\theta)^3
\, \labeqn{josephson2}
\end{equation} is called self--excited auto--parametric, and can be
exactly treated by phase averaging methods providing for the
semi--trivial solution \cite{trvn}. The excitation is here also
balanced by the (soliton vortex) regeneration term $V(\theta)$ of
\eqn{potenergy} with parameter $b^2$ and (linear mode) damping
term $\partial_{t} \theta$ with parameter $a^2$. So $a$, $b$, and
$c$ control the amplitude $\theta$ of intermediating linear waves.\\

%#############################################################
\paragraph*{The coupling mechanism.}
%#############################################################

The vertex solitons are assumed to be identical and not
synchronized in phase, such that the excitement of coupling modes
is based on the (statistical) phase difference. Let the SG
solitons be coupled such, that the amplitude of the perturbed
linear wave will be the phase difference $\theta$ of the perturbed
nonlinear SG solitons controlled by the potential of the phase
gradient $V(\theta)$. Usually the non-trivial equilibrium
fluctuation amplitude $\theta_0$ can be found with given parameter
$a$ and $b$ or ratio $q=b/a$, but in our case the scattering phase
fluctuation amplitude $\theta_0$ is given by the relative phase
fluctuation range of solitons. Regarding the kink-antikink
solution there exists a characteristic time scale or oscillation
frequency that can be related via coupling wave velocity to a
characteristic length scale or wavelength $\lambda_{\mu}$. The
fermion--like behavior pointed out by Skyrme \cite{Skyrme}
suggests to define an additive phase--fluctuation range by $2
\times 2\pi = 4\pi$. Therefore, the coupling amplitude for the
mediating wave will be given by
\begin{equation}
\theta_0 = 4 \pi \lambda_{\mu} \, \labeqn{rayleigh2}.
\end{equation}
Later $\lambda_{\mu}$ will be reintroduced as the Compton
wavelength of wave--soliton coupling.
\\

%#############################################################
\paragraph*{Phase averaging.}
%#############################################################

Statistically, there are phase oscillations or fluctuations
randomly modulated in a special phase interval, so we can assume
in the most simple case a maximum entropy phase modulation and
fluctuation between two vertex solitons and find the most likely
energy flow and coupling ratio from phase averaged solutions
\cite{Sanders85}. The phase--amplitude modulation with random
$\tau$ as a solution to \eqn{josephson2} can be written as
\cite{trvn}
\begin{equation}
      \theta = \theta_0 \cos(\tau + \psi_0)
\, \labeqn{josephson3},
\end{equation}
averaging over $\tau$ yields \cite{Sanders85,trvn} with constant
$\psi_0$
\begin{eqnarray}
      \partial_{t}\theta_0 \propto {1 \over 2} \(a^2 \theta_0
       - {3 b^2 \over 4 \lambda_{\mu}^2} \theta_0^3\)
      , \quad \partial_{t} \psi_0=0
\, \labeqn{josephson4}.
\end{eqnarray}

%#############################################################
\paragraph*{1-dimensional coupling soliton energy.}
%#############################################################

With constant fluctuation strength $\partial_{t}\theta_0 = 0$, the
averaged equations provide with \eqn{rayleigh2} and
\eqn{josephson4} for a balanced non-trivial equilibrium
\begin{equation}
{q } ={b \over a} = {1 \over {4 \pi} \sqrt{3\over 4} }, \quad { 2
\overline{V} \over E_{1d}} = \overline{(\partial_r \theta)^2} =
q^{-2} \, \labeqn{josephson5}.
\end{equation}
Consequently, the 1-dimensional coupling strength $q$ will enter
the energy definitions via mean unit energy $E_{1d}$ of
1-dimensional coupling
\begin{equation}
{E_{1d} } = q^2\overline{(\partial_r \theta)^2}\,=1\mu c^2\,
\labeqn{josephson6},
\end{equation} where $\mu$ is a unit mass.
To compare our theoretical soliton coupling model to real existing
couplings, mass/energy has to be quantified. Let the unit mass be
the 1d coupling mass--energy of a soliton with $E_{1d}=1\mu c^2$.
The mutual 1-d coupling to a photon with amplitude/wavelenght
fluctuation $\lambda_{\mu}$ can be regarded as a permanent Compton
scattering process with mass--energy value related to
$\lambda_{\mu}$ via Compton relation
\begin{equation}
{E_{1d} } =  2 q^2 \overline{V } = {h c \over \lambda_{\mu}} %\overline{\phi_{1d}^2},\quad
%\phi_{1d} = q \,\partial_r \theta
\, \labeqn{couple1}.
\end{equation}\\
%-----------------------------------------------------------------------------------

%#############################################################
\paragraph*{3-dimensional coupling.}
%#############################################################

The Gauss relation can connect the 1-d coupling strength to a 3-d
coupling strength with potential $\phi_{3d}$ such, that
\begin{equation}
%{E_{1d}(\lambda_{\mu})}=
{E_{3d}(\lambda_{\mu})} = { q  \over \epsilon_0}  \,
\phi_{3d}(\lambda_{\mu})
 \, \labeqn{couple3},
\end{equation}
where $\phi_{3d}$ is defined by
\begin{equation}
E_{3d}(r)= - {1 \over \epsilon_0} \int_{\infty}^r \phi_{3d}^2 4\pi
dr' = {q^2 \over 4\pi  \epsilon_0 r}, \quad \phi_{3d}= {q \over
4\pi r} \, \labeqn{josephson9}.
\end{equation}
Let's define the fine structure by
\begin{equation}
\alpha = {q^2 \over  4 \pi \epsilon_0 \hbar c } =
{{E_{3d}(\lambda_{\mu})} \over {E_{1d}}}\, \labeqn{couple3},
\end{equation}
where the relations at the special reference distance
$\lambda_{\mu}$ given by dimensionless Planck units
$h=c=\lambda_{\mu}=1$ must obey the unit condition
\begin{equation} {E_{3d}(\alpha
\lambda_{\mu})} = {E_{1d}}= \phi_{3d}(\alpha \lambda_{\mu}) =
{\phi_{1d}}\equiv 1 \, \labeqn{couple3},
\end{equation}
that provides for
\begin{equation} \epsilon_0=q ,\quad \alpha = {q \over 4\pi} ={ 1 \over 136,75725...}\, \labeqn{couple4}.
\end{equation}
\eqn{josephson5}-\eqn{couple4} correspond to a flux density
reduction given by an 3-dimensional integral flux factor $M$
\begin{equation}
M =\[ 1 \over \alpha \] = 137 \, \labeqn{josephson7},
 \end{equation}
where $[\,]$ means next higher integral value. Why integral? When
the round-trip path fits integer numbers of the wavelength
(single--valuedness), WGM are formed. \\

%#############################################################
\paragraph*{Virtual photons.}
%#############################################################

In \cite{sepp,phasetospace,Mapping} it was possible to describe
coupling vertex solitons in (auto)resonance by a circularly closed
system of B\"acklund transformations. With projective geometry and
symmetry the local soliton coupling can be be assigned via Bohlin
transformation to a non-local Coulomb potential, and leads to an
iterative determination of generalized fine structure constants,
provided the coupling strength is known. It turned out that
$M=137$ with $q=e$ corresponds to the Sommerfeld fine structure
constant $\alpha = e^2/( 4 \pi \epsilon_0 \hbar c )$ according to
the Dirac theory of magnetic monopoles \cite{Dirac-31} (a
generator of the topological Berry phase \cite{berry}). The
meaning behind $M=137$ now becomes clearer: virtual photons (the
mediator of the electromagnetic force) could act similarly to Lord
Rayleighs WGM in a circular quantum billiard-type orbit. The
projective resonance condition that provides for generalized fine
structure constants in \cite{sepp,phasetospace,Mapping} is given
by $\partial_r \theta \propto r$, the energy expectation in
\eqn{josephson5} is a typical spatial variance
$\overline{(r-r_0)^2}$ driven by phase
fluctuations.\\

%#############################################################
\paragraph*{Conclusion.}
%#############################################################


The interaction of linear waves and nonlinear solitons realizes a
self-excited and integrable autonomous system driven by phase
fluctuations defined in \eqn{rayleigh2} and \eqn{josephson3}. The
non-linear regeneration term can be decomposed into the product of
a vertex soliton energy factor $(\partial_{r}\theta)^2$ and a
linear dissipation factor $\partial_{t}\theta$. The balancing
factor $q^{-2}=12\pi^2$ and also $M=137$ indicate, that the
averaged phase fluctuations providing for the soliton energy are
considerably reduced with respect to the linear wave or photon
amplitude of \eqn{rayleigh2}. Regarding the space variance of the
soliton compared to the photon wavelength this would mean, that
the interaction is driven by a
thermodynamic phase space compression of strength $q^{-2}$. \\

\begin{thebibliography}{aaa}

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Scattering Transform}, SIAM, Philadelphia, (1981).

\bibitem{Skyrme} T. H. R. Skyrme, Proc. Roy. Soc. A {\bf 247}, 260 (1958); {\it
ibid.} A {\bf 262}, 237 (1961).

\bibitem{trvn}  A. Tondl, T. Ruijgrok, F. Verhulst, R. Nabergoj, {\it Autoparametric Resonance in Mechanical
Systems} Cambridge University Press, New York, (2000).

\bibitem{Ustinov98} A.V. Ustinov, Physica D {\bf 123},  315
(1998).

\bibitem{Ustinov00} A. Wallraff, A.V. Ustinov, V.V.
Kurin, I.A. Shereshevsky, N.K. Vdovicheva, Phys. Rev. Lett. {\bf
84}, 151 (2000)
\href{http://xxx.lanl.gov/abs/cond-mat/9911260}{cond-mat/9911260}.%

\bibitem{rayleigh} L. Rayleigh, Philos. Mag. {\bf 27}, 100 (1914).

\bibitem{Sanders85} J. A. Sanders, F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems", Appl. Math. Sci. {\bf 59}, Springer--Verlag, New
York (1985).

\bibitem{sepp} B. Binder,{\it Josephson Effect, B\"acklund Transformations, and Fine Structure Coupling};
\href{http://philsci-archive.pitt.edu/documents/disk0/00/00/08/61/index.html}
{PITT-PHIL-SCI00000861}.

\bibitem{phasetospace} B. Binder,{\it Topological Phase Fields, Baecklund Transformations, and Fine Structure};
\href{http://philsci-archive.pitt.edu/documents/disk0/00/00/08/41/index.html}
{PITT-PHIL-SCI00000841}.

\bibitem{Mapping} B. Binder,{\it Charge as the Stereographic Projection of Geometric Precession on
Pseudospheres};\\
\href{http://philsci-archive.pitt.edu/documents/disk0/00/00/08/18/index.html}
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\bibitem{Dirac-31} P. A. M.  Dirac, Proc. Roy. Soc. London A {\bf
133}, 60 (1931).
\bibitem{berry} M. V. Berry, Proc. Roy. Soc. Lond. A {\bf 392},
45 (1984).


\end{thebibliography}
\end{document}
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