Universal criterion and amplitude equation for a nonequilibrium Ising-Bloch transition
D. Michaelis,
1
U. Peschel,
1
F. Lederer,
1
D. V. Skryabin,
2
and W. J. Firth
2
1
Insititut fu ¨r Festko ¨rpertheorie und Theoretische Optik, Friedrich-Schiller Universita ¨t, Jena, D-07743, Germany
2
Department of Physics and Applied Physics, University of Strathclyde, Glasgow, G4 0NG, United Kingdom
Received 22 May 2000; published 15 May 2001
We identify a universal criterion for the onset of a nonequilibrium Ising-Bloch NIB transition, and describe
the behavior near the bifurcation by a generic amplitude equation. We found that a NIB transition is caused by
an antisymmetric eigenvector passing the translational mode of the system at a critical point. In this context we
discuss Hamiltonian and dissipative systems. We report on a NIB in nonlinear optics, manifesting itself in a
transition from static to moving polarization fronts.
DOI: 10.1103/PhysRevE.63.066602 PACS numbers: 42.65.Ky, 42.65.Sf
Nonlinear systems can exhibit spatially homogeneous, pe-
riodic, or localized structures with nontrivial dynamical be-
havior. One of the basic issues in nonlinear physics is to
correctly predict the interaction of such solutions where they
coexist. For example, bistable systems can exhibit fronts
connecting two stable homogeneous states. The properties
and dynamics of such fronts have attracted attention across
many branches of science, including chemistry, biology,
fluid dynamics, and optics 1.
Generally, a front connecting two nonequivalent homoge-
neous states moves in such a way that the more stable state
annihilates the other. Sometimes, as a consequence of a dis-
crete symmetry, a system may possess two equivalent states.
The front between such states is generally at rest due to
symmetry. But such fronts can destabilize via a bifurcation
on changing a system parameter. A prominent example
among gradient systems 1 is the so-called Ising-Bloch tran-
sition 2,3, known from the physics of ferromagnets and
liquid crystals 4. However, many interesting nonlinear sys-
tems are far from equilibrium, and cannot be described by a
free energy functional, i.e., the governing order-parameter
equation is not of gradient type. Nevertheless, the symme-
tries of the system are frequently preserved even far from the
gradient limit. Thus pairs of equivalent solutions exist, but
the net force acting on an interface between them is not
necessarily zero. Examples of a transition from resting to
moving fronts were found in the complex parametrically
driven Ginzburg-Landau equation 2,5, in an activator-
inhibitor reaction-diffusion system 3, and in optical para-
metric oscillators 8. The corresponding bifurcation is often
referred to as the nonequilibrium Ising-Bloch NIB transi-
tion.
The main result of this paper is to obtain a rather simple
but universal criterion for the onset of a NIB transition and a
generic amplitude equation, which can be applied to all
known cases. To this end we identify symmetries, which are
essential for a NIB transition, but more general than those
presented in Refs. 2–5. The investigations are based on the
crucial finding that for each nonlinear nonequlibrium system
a bifurcation of a resting solitary wave to a moving one is
linked to an internal mode which comes into exact coinci-
dence with the translational mode. In this context we com-
pare dissipative gradient and nongradient and Hamiltonian
systems. The formalism and bifurcation scenario described
in this work is rather general. For definiteness, we study an
example from nonlinear optics where the formation and con-
trol of localized structures and fronts recently attracted a
considerable deal of interest 6,7.
Due to the availability of materials with large second or-
der suceptibilities, much attention has been given to paramet-
ric processes 8–17. Here we study intracavity type-II sec-
ond harmonic generation in a planar waveguide resonator,
where two orthogonally polarized pump photons at fre-
quency generate one signal photon at 2 . The normalized
set of equations for the slowly varying envelopes of the two
orthogonally polarized fundamental harmonic fields A
1,2
FH1, FH2 and of the second harmonic field B SH reads,
in the mean field limit, as 16
i
t
+
x
2
+
A
+i A
1,2
+A
2,1
* B =E ,
1
i
t
+
1
2
x
2
+
B
+i
B +A
1
A
2
=0,
where
x
2
describes diffraction, and t is the dimensionless
time. The incident field E is a monochromatic plane wave
with a polarization angle of 45°, thus driving both FH waves
with the same intensity. The FH and SH fields are detuned
by
A
and
B
from a resonator resonance, respectively. is
the ratio of the photon lifetimes at the two frequencies 17.
A typical experimental configuration could consist of a
500- m-thick KTP crystal sandwiched between two mirrors
with 95% reflectivity for both fundamental and second har-
monic waves. If the d
31
coefficient is employed, and phase
matching occurs at a certain tilt at 1.06 m, one obtains the
length and time scales as 30 m and 110 ps. The driving
intensity | E |
2
=1 corresponds to about 50 kW/cm
2
.
Equations 1 exhibit the translational symmetry and two
discrete symmetries:
Z: A
1
, A
2
, B → A
2
, A
1
, B , P: x →-x . 2
Z allows for a pitchfork bifurcation of the stationary (
t
=0), homogeneous (
x
=0), and symmetric ( A
1
=A
2
) solu-
tions 16. There are two different resting fronts, i.e., hetero-
clinic trajectories, which connect equivalent states appearing
after the pitchfork bifurcation. They are transformed into
each other on applying Z see Fig. 1a. On the other side,
PHYSICAL REVIEW E, VOLUME 63, 066602
1063-651X/2001/636/0666024/$20.00 ©2001 The American Physical Society 63 066602-1