Stationary Dissipative Solitons of Model G
20/05/2014 17:41
February 17, 2013 model˙g
Stationary Dissipative Solitons of Model G
Matthew Pulvera† and Paul A. LaVioletteb‡
aBlue Science, Los Angeles, California USA
bStarburst Foundation, Schenectady, New York USA
Model G, the earliest reaction-diffusion system proposed to support the existence of
solitons, is shown to do so under distant steady-state boundary conditions. Subatomic
particle physics phenomenology, including multi-particle bonding, movement in concentration
gradients, and a particle structure matching Kelly’s charge distribution
model of the nucleon, are observed. Lastly, it is shown how a three-variable reversible
Brusselator, a close relative of Model G, can also support solitons.
Keywords: Brusselator; dissipative soliton; Model G; reaction-diffusion system;
subquantum kinetics; Turing instability; Turing wave
1. Introduction
Since the seminal work of Turing (1952), stationary and oscillating patterns have
been studied and observed in a variety of open reaction-diffusion (R-D) systems, one
example being the three-variable Belousov-Zhabotinsky reaction (Winfree, 1974;
Zaikin and Zhabotinsky, 1970). The Brusselator, first proposed in 1968, is the
simplest R-D system known to produce wave patterns of precise wavelength. It is a
two-variable R-D system specified by the following reactions (Lefever, 1968; Nicolis
and Prigogine, 1977):
A → X (1a)
B + X → Y + D (1b)
2X + Y → 3X (1c)
X → E. (1d)
In the Brusselator, only the X and Y reactants are variable while A, B, D, and E are
held constant. The value of either source reactant A or B serves as the bifurcation
parameter determining whether the system is able to spawn a dissipative structure.
But because their concentrations are kept invariant, the system’s criticality remains
uniform throughout the reaction volume and, in the case where the system is supercritical,
gives rise to a nonlocalized dissipative structure. Herschkowitz-Kaufman
†matt@blue-science.org
‡plaviolette@starburstfound.org
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0.0 0.2 0.4 0.6 0.8 1.0
0
5
10
15
20
Space
Concentration
X
A
Y
(a) Box length = 1.
0.0 0.5 1.0 1.5 2.0
0
5
10
15
20
Space
Concentration
X
A
Y
(b) Box length = 2.
Figure 1. (a) The localized steady state dissipative structure of Herschkowitz-Kaufman and Nicolis (1972)
in a 1D box of length 1. (b) Same system parameters with box length = 2, demonstrating the dependence
of structure localization upon system boundaries.
and Nicolis (1972), on the other hand, have been able to create a localized dissipative
structure by allowing A to vary while holding its concentration fixed at the
system boundaries at a level above the critical threshold value Ac. Consumption of
A through reaction (1a) creates the formation of an “A-well” which in the steady
state forms a hypercosine function solution satisfying the relation A = DA∇2A.
The reactions become supercritical wherever A attains a value less than Ac, allowing
X and Y to self-organize a dissipative structure localized in the A-well’s
interior.
While Herschkowitz-Kaufman and Nicolis (1972) note that this version of the
Brusselator produces a dissipative structure that is localized within boundaries
that are distinct from the reaction system boundaries, a dissipative structure generated
in this fashion is not an autonomous entity since its morphology depends
on the particular placement and extent of those boundaries; see Figure 1. If these
boundaries are sufficiently expanded or removed, the localized X-Y structure dissolves.
In the Brusselator, the depth of the A-well determines whether X and Y
will form an ordered pattern, but not vice versa. That is, the X and Y values
forming the soliton do not themselves affect the concentration of A; they do not
determine the character of the A-well which in turn determines whether or not the
X-Y ordered state can exist.
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February 17, 2013 model˙g
-10 -5 0 5 10
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Space
Concentration
a
X
Y
Figure 2. Localized dissipative structure of Koga and Kuramoto (1980). DX = 1, DY = 5, a = 0.254,
b = 0.5, c = 0.5.
Koga and Kuramoto (1980) simulated a localized dissipative structure supported
by a system defined by the equations:
@X
@t
= DX∇2X − X − Y − H(X − a) (2a)
@Y
@t
= DY∇2Y + bX − cY (2b)
where a, b, c are system constants, and H is the Heaviside step function. In the
center of the soliton, there is a finite region in which X > a and thus H = 1, outside
of which H = 0. See Figure 2. With this mechanism the structure maintains its
localized form. It is important to note, however, that an equation that uses the
unit step H in this way is not expressible as a finite set of reaction equations
(e.g. eqs. (1)) and therefore we would not classify this dissipative system as a R-D
system.
However, a minor variation of the Brusselator, a three-variable R-D system
known as Model G, has been theorized in 1980 to support localized, self-stabilizing
Turing patterns within a subcritical environment when the values of its system
parameters are properly chosen (LaViolette, 1980, 1985, 1994, 2008, 2010). It is
able to form a true dissipative soliton, one whose structure is stable, autonomous,
localized, and unaffected by the positions of the system boundaries. Its system of
partial differential equations are derived from a simple set of kinetic reaction equations
with diffusion, and is in fact the earliest R-D system proposed to support
solitons. Others have since reported simulation results of dissipative solitons, such
as Schenk et al. (1998) who investigated a two-variable system of the FitzHugh-
Nagumo (FN) type. Surveys of dissipative solitons produced by dissipative systems
of both the R-D and non-R-D types are given by Purwins et al. (2005), B¨odeker
(2007) and Vanag and Epstein (2007).
Dissipative solitons have generated substantial interest because of the variety of
particle-like properties they have been shown to produce such as scattering, reflection,
particle-particle bonding, orbital motion, particle annihilation, and particle
replication (Bode et al., 2002; B¨odeker, 2007; Liehr et al., 2004; Nishiura et al.,
2005; Purwins et al., 2005; Schenk et al., 1998; Vanag and Epstein, 2007). Previous
publications have demonstrated that Model G can form the basis for a unified field
theory that effectively accounts for a variety of microphysical and macrophysical
phenomena (LaViolette, 1985, 1986, 1992, 2005, 2008, 2010). In this paper, we
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February 17, 2013 model˙g
present computer simulation results carried out for the first time on this promising
reaction system.
2. Model G
Model G is a modification of the Brusselator in which a third intermediary variable
G is inserted into the first reaction step (1a). It is specified by the following five
transformations:
A
k1
⇋
k−1
G (3a)
G
k2
⇋
k−2
X (3b)
B + X
k3
⇋
k−3
Y + Z (3c)
2X + Y
k4
⇋
k−4
3X (3d)
X
k5
⇋
k−5
(3e)
where reaction step (1a) is replaced by the two reactions (3a, 3b). This R-D system
is represented by the following system of partial differential equations:
@G
@t
= DG∇2G − (k−1 + k2)G + k−2X + k1A (4a)
@X
@t
= DX∇2X + k2G − (k−2 + k3B + k5)X
+ k−3ZY − k−4X3 + k4X2Y + k−5
(4b)
@Y
@t
= DY∇2Y + k3BX − k−3ZY + k−4X3 − k4X2Y. (4c)
Since the forward kinetic constants have values much greater than the reverse
kinetic constants, the reactions have the overall tendency to proceed irreversibly to
the right. Nevertheless, the reverse reaction G ← X in eq. (3b) plays an important
role. This allows the concentration of the X variable to influence the concentration
value of the G variable which in turn serves as the system’s bifurcation parameter.
So, because Model G’s bifurcation parameter is able to vary in both time and space,
and be influenced by its X variable which participates in forming a soliton-like
dissipative structure, this reaction system is able to nucleate a structure which, in
the positive Y polarity (negative X), is autonomous and self-stabilizing. All that is
needed is a momentary localized fluctuation in one of the reactants sufficiently large
in amplitude and spatiotemporal breadth to initiate the growth of the soliton. This
technique of allowing a third reactant to serve as a variable bifurcation parameter
for the other two reactants is a recipe general enough for potential applicability
to other dissipative-structure-producing R-D systems, allowing them to support
dissipative solitons as well.
The seed fluctuation may be in the form of an extra term that is added to the
R-D equations, or it may be directly incorporated within the R-D equations by
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February 17, 2013 model˙g
representing the concentration of each reactant with a stochastic term. Such “zeropoint”
fluctuations would be present if each reactant is comprised of a plurality
of constituent units (e.g., X-ons, Y-ons, G-ons) which engage in Markovian birthdeath
transformations.
A positive polarity seed fluctuation (negative X, or positive Y), arising spontaneously
in this fashion, is able to spawn a periodic structure even when the system
is initially in a subcritical homogeneous steady state. The Brusselator, on the other
hand, must always begin from an initially supercritical homogeneous steady state.
As a result, its structures are always destined to fill the entire reaction volume to
its supercritical limits. This autogenic ability, wherein a stable dissipative soliton
can form spontaneously out of system noise, is an important distinctive feature of
Model G. LaViolette (1985, 2010) has shown that this feature makes Model G a
promising candidate for modeling the formation of subatomic particles out of the
zero-point energy continuum vacuum state. How Model G has been employed to
model subatomic particles is further elaborated on in Section 3.1.
2.1 Nondimensionalization
In analyzing eqs. (4), we first pass to dimensionless variables. This reduces the
number of independent system parameters by seven, significantly simplifying the
goal of finding a particular set of system parameters (eqs. (17) below) that support
the formation of a stable soliton. In addition, this conveniently scales the dimensionless
space, time, and concentration potential values to the order of unity in the
vicinity of the soliton formation event.
We assume that the three diffusion constants DG, DX, DY, four concentrations
A, B, Z,
, and the ten kinetic reaction rates k±i are constant in time and space.
The dimensional space, time, and concentration variables x, y, z, t, G, X, Y are
converted to their corresponding dimensionless variables x, y, z, t,G,X, Y via the
following substitutions:
x = Lx, y = Ly, z = Lz, t = Tt
G(x, y, z, t) = CG(x, y, z, t) (5)
X(x, y, z, t) = CX(x, y, z, t)
Y(x, y, z, t) = CY (x, y, z, t)
where the time, space, and concentration constants are defined, respectively, as:
T ≡
1
k−2 + k5
, L ≡
p
DGT, C ≡
1
√k4T
. (6)
These substitutions allow for eqs. (4) to be expressed in terms of dimensionless
variables:
@G
@t
= ∇2G − qG + gX + a (7a)
@X
@t
= dx∇2X + pG − (1 + b)X + uY − sX3 + X2Y + w (7b)
@Y
@t
= dy∇2Y + bX − uY + sX3 − X2Y (7c)
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February 17, 2013 model˙g
where
dx ≡
DX
DG
, dy ≡
DY
DG
,
a ≡
k1√k4
(k−2 + k5)3/2
A, b ≡
k3
k−2 + k5
B, g ≡
k−2
k−2 + k5
, p ≡
k2
k−2 + k5
, (8)
q ≡
k−1 + k2
k−2 + k5
, s ≡
k−4
k4
, u ≡
k−3
k−2 + k5
Z, w ≡
√k4k−5
(k−2 + k5)3/2
.
The vector operator ∇ is taken with respect to the dimensional x, y, z terms in
eqs. (4) and with respect to the dimensionless x, y, z in eqs. (7). Since there will
never be ambiguity in its context, we use the same symbol ∇ for both.
2.2 Redimensionalization
The dimensional diffusion coefficients, kinetic constants, and constant concentrations
A, B, Z,
are given as follows when expressed in terms of their dimensionless
counterparts and T, L, C:
DG =
L2
T
, DX =
L2
T
dx, DY =
L2
T
dy,
k1A =
C
T
a, k2 =
1
T
p, k3B =
1
T
b, k4 =
1
C2T
, k5 =
1
T
(1 − g), (9)
k−1 =
1
T
(q − p), k−2 =
1
T
g, k−3Z =
1
T
u, k−4 =
1
C2T
s, k−5
=
C
T
w.
2.3 Homogeneous Steady State
The homogeneous steady state is one in which
0 =
@G
@t
=
@X
@t
=
@Y
@t
(10a)
0 = ∇G = ∇X = ∇Y. (10b)
Under these conditions the differential equations (7) become algebraic with G, X,
Y having the following respective solutions:
G0 =
a + gw
q − gp
, X0 =
pa + qw
q − gp
, Y0 =
sX2
0 + b
X2
0 + u
X0. (11)
2.4 Concentration Potentials
It proves useful to consider concentration values relative to the homogeneous steady
state (LaViolette, 1985, 1994, 2008). These concentration potentials are defined as:
'G ≡ G − G0, 'X ≡ X − X0, 'Y ≡ Y − Y0. (12)
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February 17, 2013 model˙g
Eqs. (7) are expressed in terms of concentration potentials as:
@'G
@t
= ∇2'G − q'G + g'X (13a)
@'X
@t
= dx∇2'X + p'G − (1 + b)'X + u'Y
− s
¤
('X + X0)3 − X3
0
+
¤
('X + X0)2('Y + Y0) − X2
0Y0
(13b)
@'Y
@t
= dy∇2'Y + b'X − u'Y
+ s
¤
('X + X0)3 − X3
0
−
¤
('X + X0)2('Y + Y0) − X2
0Y0
.
(13c)
This substitution is especially important when calculating numerical solutions for
G, X, and Y that correspond to a dissipative soliton, which, as will be seen below,
consists of small deviations about the homogeneous steady state values G0, X0,
and Y0.
3. Particle Formation and Structure in Model G
We examine the evolution of the system in 1, 2, and 3 dimensions of space. In
the case of 2 and 3 dimensions, circular and spherical symmetry, respectively, are
imposed upon the system.1
Each concentration potential is a function of both position x in 1D (radius r in
the case of 2D and 3D) and time t. The reaction spatial domain and total time is
specified as −50 ≤ x ≤ 50 in 1D (0 ≤ r ≤ 50 in 2D and 3D) and 0 ≤ t ≤ 100.
The boundary conditions for 1D are:2
∀x ∈ [−50, 50] : ∀t ∈[0, 100] :
'G(x, t) ≥ −G0, 'X(x, t) ≥ −X0, 'Y (x, t) ≥ −Y0 (14a)
'G(x, 0) = 0, 'X(x, 0) = 0, 'Y (x, 0) = 0 (14b)
'G(±50, t) = 0, 'X(±50, t) = 0, 'Y (±50, t) = 0. (14c)
For 2D and 3D, the boundary conditions are:
∀r ∈ [0, 50] : ∀t ∈[0, 100] :
'G(r, t) ≥ −G0, 'X(r, t) ≥ −X0, 'Y (r, t) ≥ −Y0 (15a)
'G(r, 0) = 0, 'X(r, 0) = 0, 'Y (r, 0) = 0 (15b)
'G(50, t) = 0, 'X(50, t) = 0, 'Y (50, t) = 0 (15c)
('G)r (", t) = 0, ('X)r (", t) = 0, ('Y )r (", t) = 0. (15d)
The subscript r in eqs. (15d) denote the partial derivative with respect to r.
1The circular and spherical symmetry conditions in 2 and 3 dimensions that are imposed upon the system
are due entirely to the limited availability of computational resources for this research project at the time of
this publication. The mathematics, as well as the simulation software, is general enough for these symmetry
restrictions to be removed—it is only a matter of acquiring sufficient computational time (CPU cycles)
and space (memory) to carry out the simulations.
2“8” is the universal quantifier symbol. The first line of eqs. (14) reads “For all x between −50 and 50,
and for all t from 0 to 100:”
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February 17, 2013 model˙g
Since the soliton shall be centered at the point r = 0, the boundary condition at
r = 0 must not constrain the concentration potential values at this point. Instead,
the boundary condition is placed upon their first order derivatives with respect
to r. Under the imposed constraints of angular symmetry in 2D and 3D, these
derivatives must vanish.
Epsilon, ", is a small positive number to numerically approximate zero. " 6= 0
for computational reasons only, due to the indeterminate form the rotationallysymmetric
Laplacian takes in 2D and 3D at r = 0:
∇2' =
@2'
@r2 +
n − 1
r
@'
@r
(16)
where n ∈ {1, 2, 3} is the number of spatial dimensions. The point at r = 0 is
smoothly approximated since the limit of ∇2' as r → 0 is always finite. The
numerical simulations in this work use " = 10−9.
There are an infinite continuum of parameter values that allow for the creation of
a stationary soliton. The particular set of values that is investigated in this article
are the following:
dx = 1, dy = 12, a = 14, b = 29, g = 1/10,
p = 1, q = 1, s = 0, u = 0, w = 0. (17)
If the system is left to evolve according to eqs. (13–17), the concentrations will
remain at their homogeneous steady state values, i.e. 0 = 'G = 'X = 'Y for all r
and t. In order to initiate the formation of a soliton, we introduce a momentary
seed fluctuation in the X variable specified as :
(r, t) ≡ −e−r2
2 e−(t−10)2
18 . (18)
The time evolution of this fluctuation is graphed in Figure 3 as /10 for four
particular points in time.
For one spatial dimension, r in the above equation is replaced with x. This seed
fluctuation is incorporated into the system by adding to the right-hand side of
eq. (13b). In addition, substituting values (17) into eqs. (13), yields the following
system of nonlinear PDEs:
@'G
@t
= ∇2'G − 'G + 'X/10 (19a)
@'X
@t
= ∇2'X + 'G − 30'X − 4060/9 + ('X + 140/9)2('Y + 261/140) + (19b)
@'Y
@t
= 12∇2'Y + 29'X + 4060/9 − ('X + 140/9)2('Y + 261/140). (19c)
Eqs. (19), along with the boundary conditions (14 or 15), may then be numerically
solved on a computer.
3.1 Particle Structure in 3D
Figure 4 shows the computer-simulated evolution of 'G, 'X, and 'Y under eqs. (19)
in three spatial dimensions, subject to boundary conditions (15) and graphed at
times t = 0, t = 10, t = 13 and t = 20.
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February 17, 2013 model˙g
-10 -5 0 5 10 -2
-1
0
1
2
Space
X Potential Χ10
(a) t = 0
-10 -5 0 5 10 -2
-1
0
1
2
Space
X Potential Χ10
(b) t = 10
-10 -5 0 5 10 -2
-1
0
1
2
Space
X Potential Χ10
(c) t = 13
-10 -5 0 5 10 -2
-1
0
1
2
Space
X Potential Χ10
(d) t = 20
Figure 3. The seed fluctuation /10.
-10 -5 0 5 10 -2
-1
0
1
2
Space
Y, G, X Potentials jY, jG, jX10
(a) t = 0
-10 -5 0 5 10 -2
-1
0
1
2
Space
Y, G, X Potentials jY, jG, jX10
(b) t = 10
-10 -5 0 5 10 -2
-1
0
1
2
Space
Y, G, X Potentials jY, jG, jX10
(c) t = 13
-10 -5 0 5 10 -2
-1
0
1
2
Space
Y, G, X Potentials jY, jG, jX10
(d) t = 20
Figure 4. 3D Particle Formation. In frames with t > 0, the three curves from top to bottom along the
central vertical axis, respectively, are 'Y (yellow), 'G (blue), and 'X/10 (magenta). The stable soliton
pattern that emerges is induced by the seed fluctuation of Figure 3.
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February 17, 2013 model˙g
Y, G and X Potentials
-10 -5 0 5 10
-2
-1
0
1
2
jY
jG
jX
10
Space
(a) Spherically-symmetric 3D stationary particle.
Y, G and X Potentials
-10 -5 0 5 10
-0.02
-0.01
0.00
0.01
0.02
jY
jG
jX
10
Space
(b) Zoomed 100×.
Figure 5. (a) Spherically-symmetric 3D stationary particle formed from eqs. (19) and boundary conditions
(15). (b) Same data zoomed 100× vertically.
The system converges to the stationary structure shown in Figure 5(a). This
stable soliton is what is identified in Model G as an individual particle. These
simulation results show that the reaction variables produce a periodic pattern of
precise wavelength, 0 = 3.08 units, that progressively decreases in amplitude as
it extends outward from a central bell-shaped core. Figure 5(b) exhibits the extent
of the periodicity of the particle’s potential fields by zooming the vertical axis of
the simulation 100×.
The subquantum kinetics physics methodology developed by LaViolette (1985,
2008, 2010, 2012) postulates that subatomic particles are electric and gravity potential
solitons. It identifies the 'G variable with gravity potential and the 'X and
'Y variables with electric potential. Negative 'G values are associated with positive,
matter-attracting gravitational mass and positive 'G values are associated
with negative, matter-repelling gravitational mass. Positive and negative 'Y values
(negative and positive 'X values) are associated respectively with positive and negative
charge, the 'X and 'Y variables having a reciprocal relation to one another.
Since the 'G and 'X potentials are closely coupled in Model G due to the linkage of
species G and X through both forward and reverse reactions, subquantum kinetics
predicts that the electric and gravitational potentials should be closely coupled:
negative gravity potential with positive electric potential. Thus in subquantum kinetics,
subatomic particles are envisioned as local field potential inhomogeneities.
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February 17, 2013 model˙g
Since these particular inhomogeneities share a common Turing wave pattern morphology,
this leads to a natural quantization of these fields into identical particles.
This is reminiscent of the quantization of fermionic matter fields in quantum field
theory which is invoked to explain why those fields’ particles are identical.
The simulation displayed in Figures 4 and 5 would be interpreted in subquantum
kinetics as the nucleation of a neutral subatomic particle, a neutron for example,
being nucleated by a positive charge polarity electric potential fluctuation emerging
from the subquantum zero-point energy field. In this case the particle’s core has a
positive electric potential coinciding with a negative gravity potential and creating
a local stabilizing supercritical domain. In subquantum kinetics, such fluctuations
can trigger particle nucleation even though they themselves have a field energy
potential magnitude much smaller than that of the particle they nucleate. A negatively
charged fluctuation would lead to the formation of an antimatter particle
having a core with a negative electric potential coinciding with a positive gravity
potential. But, as LaViolette (1985) has pointed out, such negative charge polarity
fluctuations emerging from system noise where the reaction system is initially
subcritical, as would be the case for the primordial vacuum state, fail to nucleate
a particle. He notes that Model G’s ability to nucleate solitons with an inherent
polarity bias favoring the matter state is a feature that is advantageous in the application
to cosmology where nature has favored the primordial creation of matter
over antimatter.
These modeling results confirm LaViolette’s 1985 prediction that Model G should
generate a dissipative space structure whose field potential has this same form: a
gaussian core surrounded by an extended asymptotic field periodicity. He proposed
that such a field potential could serve as a useful model of a subatomic particle
provided that the reaction system parameters are chosen such that the wavelength
of the soliton’s oscillatory tail equals the subatomic particle’s Compton wavelength.
He demonstrated that this periodic tail, which he termed the particle’s Turing
wave, accounts for the experimental results of both particle diffraction and orbital
quantization (LaViolette, 1985, 1994, 2010).
Later, LaViolette (2008, 2010) noted that the 'Y (and 'X) wave pattern predicted
for Model G’s Turing wave soliton field pattern has a morphology very similar to
the charge distribution model that Kelly (2002) derived for the core of the nucleon
by analyzing form factor data obtained from high energy electron beam particle
scattering experiments. Kelly obtained a best fit to this scattering data by assuming
that the nucleon’s charge and magnetization distribution had the form of a gaussian
core surrounded by a periodicity having a wavelength approximating the nucleon’s
Compton wavelength. The simulation results presented here further support this
suggestion in that they indicate that the 'X and 'Y field pattern generated by
Model G displays morphological features similar to those of the radial electric
charge distribution in the neutron and proton and should therefore serve as an
appropriate soliton field structure for modeling subatomic particles.
The amplitude of the 'X (or 'Y ) field maxima forming the simulated Model G
soliton are observed to decline as r−3.7 at r ≈ 20, as r−7 at r ≈ 40, and as r−10
at r ≈ 60. This may be compared with r−7 in standard theories for the radial
decline of the nuclear force. Because of this rapid decline, the product of the Turing
pattern field amplitude with incremental shell volume diminishes toward zero for
shells of successively greater radius, and the sum of these shell product increments
converges to a finite value. A finite value for the soliton negentropy makes Model G
attractive as a model of a subatomic particle since subquantum kinetics identifies
the 'X, 'Y field magnitudes forming the Model G soliton, with both the electric
field forming the particle’s core and with the particle’s inertial mass (LaViolette,
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February 17, 2013 model˙g
Boundary Radius
0 5 10 15 20 25
22
23
24
25
26
27
Particle Radius r
(a) Zeroes of 'X with a variable system boundary.
Zeropoint Spacing
æ
æ
æ
æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ
æ
æ
à
à
à
à
à
à
à
à
à à à à à à à à à à à à à à à à à à à à à
à
à
à
ì
ì
ì
ì
ì
ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì
ì
ì
ì
0 10 20 30 40 50
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
æ jY
à jG
ì jX
Midpoint Between Zeropoints
(b) Differences and locations of zeroes for 'Y , 'G, and 'X.
Figure 6. The Turing wavelength is independent of the distance between the particle core and system
boundary, and is the same value for each reactant. Figure (a) shows the periodic locations of the zeroes
of 'X (horizontal axis) for a spherically-symmetric 3D particle centered at r = 0, with a variable system
radius (vertical axis), demonstrating that the Turing wavelength is independent of the size of the enclosing
system volume. Figure (b) shows the successive differences between zeroes for 'Y , 'G, and 'X (vertical
axis) for a 1D particle at the midpoints between each zero-pair (horizontal axis). The central alignment
of zero differences demonstrates that the Turing wavelength is the same for each reactant, equal to 3.08
(twice the distance between zeroes). Similar results were found for circularly and spherically symmetric
2D and 3D particles, respectively.
1985, 2010).
We also report the simulation results of producing ordered patterns with Model
G where the reaction vessel size was progressively increased. The Turing pattern
periodicity was found to persist to the vessel boundaries and to maintain a constant
wavelength as the vessel size was expanded. This is shown in Figure 6(a) which
plots the radial distance locations of the 'X zero values (horizontal axis) against
the radial size of the spherical reaction volume (vertical axis), and shows how the
particle’s fields adjust in the vicinity of the steady-state boundary conditions. The
persistence of this wave pattern and invariance of its wavelength with increasing
vessel size demonstrates that the Turing pattern is independent of the boundary
conditions. These results also confirm the predictions of LaViolette (1994, 2008)
that the Model G Turing pattern should persist outward to large radial distances in
order to properly model the particle diffraction phenomenon. The pattern, though,
would have the inherent radial limit at the radial distance where the noise amplitude
present in the stochastic variation of each reactant concentration exceeds the
Turing wave pattern amplitude.
12
February 17, 2013 model˙g
3.2 Comparison of Particle Structures Simulated in 1D, 2D, and 3D
When Model G is simulated in 1D and 2D, it produces particles whose cores are
narrower and lower in amplitude, although their Turing wavelength remains the
same. Figure 7 shows the 1D and 2D particles formed from eqs. (19) and boundary
conditions (14) and (15), respectively. Compare these to Figure 5(a). Specific values
of the structural characteristics of the 1D, 2D, and 3D particles are shown in
Table 1. The core radius, Table 1, row (b), is defined as the inner-most zero-point
field potential crossing r0 (the least positive value for which '(r0) = 0) for each
of the concentration potentials). Note that the radius in all cases is largest for
'G and smallest for 'X. The core RMS radius, the root mean square radius rRMS
listed in row (c), for each concentration potential in each of the three dimensions
is calculated as:
rRMS =
p
hr2i
=
sR
S '(r)r2dS R
S '(r)dS
.
(20)
The integral in the denominator of the above formula is the core integral, row (d),
which is given by:
Z
S
'(r)dS =
¤¤¤¤¤¤¤¤¤
¤¤¤¤¤¤¤¤
Z r0
−r0
'(r)dr in 1D,
2
Z r0
0
'(r)rdr in 2D,
4
Z r0
0
'(r)r2dr in 3D
(21)
where the domain of integration S is the space within the core radius r0 defined
by each of the concentration potentials. The full integral values, row (e), use the
same formulas as the core integral, but with r0 going out to the system boundary.
The Turing wavelength, row (f), is found by first calculating successive differences
between the zeroes of each of the 3 concentration potentials 'Y , 'G, 'X. Figure 6(b)
illustrates that this is a well-defined value for all 3 concentrations at a sufficient
distance away from both the particle’s central core and the system boundaries.
The mean is calculated from the centrally-aligned data points and doubled (the
distance between zeroes is a half-wavelength) resulting in a value of 3.08. This same
0 value to 3 significant figures was found in dimensions 1D, 2D, and 3D, where
circular and spherical symmetry was imposed in the 2D and 3D cases, respectively.
4. Particle Physics in Model G
The solitons of Model G exhibit a number of properties resembling those commonly
associated with subatomic particles. Earlier we noted that the morphology of the
Model G solitons, their bell-shaped core surrounded by a Turing wave pattern of
precise wavelength, has been found to be a good description of the core field of a
nucleon. Another advantage mentioned earlier is that the full integral of the field
potential converges to a finite value. Other characteristics discussed here include
particle-particle bonding (i.e., equivalent to nuclear bonding), the gravitational
13
February 17, 2013 model˙g
Y, G and X Potentials
-10 -5 0 5 10
-1.0
-0.5
0.0
0.5
1.0
jY
jG
jX
10
Space
(a) 1D stationary particle.
Y, G and X Potentials
-10 -5 0 5 10
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
jY
jG
jX
10
Space
(b) Circularly-symmetric 2D stationary particle.
Figure 7. (a) 1D and (b) 2D stationary particles formed from eqs. (19) and boundary conditions (14, 15).
mass polarity in a particle with spin, and particle movement in a gravitational
gradient.
4.1 Multi-particle States in 1D
The single particle discussed in Section 3 was seeded from a single gaussian fluctuation
defined in eq. (18). In a similar way, multiple particles may be seeded from
multiple gaussian fluctuations, and when seed fluctuations are spaced sufficiently
close together, the resulting particles are able to coexist with one another at specific
distances of separation. We examined cases in which particles are seeded in
1D from two and three gaussian seed fluctuations.
To seed two proximal particles, we define the double-gaussian seed fluctuation:
2(x, t) ≡ (x + d2/2, t) + (x − d2/2, t) (22a)
d2 = 3.303 (22b)
and replace with 2 in eqs. (19) under boundary conditions (14). Figure 8(a)
shows the stationary two-particle solution that the system converges to by t = 100.
The final distance between the particles, defined as the distance between the core
extrema of 'X, is found to be 3.303, or about one Turing wavelength. This final
particle distance is converged to even when d2 in eq. (22a) is varied by small
14
February 17, 2013 model˙g
Table 1. Particle structure values in dimensionless units.
1D 2D 3D 2D
(a) Core amplitude
'Y 0.930 1.50 1.70 1.53
'G -0.161 -0.308 -0.411 -0.320
'X -8.36 -13.6 -14.6 -13.8
(b)
Core radius
(first zero)
'Y 0.724 1.21 1.68 1.67
'G 0.867 1.37 1.85 1.87
'X 0.642 1.13 1.61 1.58
(c) Core RMS radius
'Y 0.302 0.665 1.07 0.929
'G 0.363 0.739 1.14 1.02
'X 0.272 0.634 1.04 0.892
(d) Core integral
'Y 0.806 3.12 13.2 6.31
'G −0.167 −0.710 −3.17 −1.44
'X −6.59 −26.3 −111 −53.6
(e) Full integral
'Y 0.281 0.893 3.30 2.96
'G 8.39 × 10−5 2.88 × 10−3 2.87 × 10−2 −0.480
'X 6.01 × 10−4 0.128 1.93 −13.7
(f) Turing wavelength 3.08 3.08 3.08 3.08
amounts both higher and lower. For example when d2 = 4, the same 3.303 distance
between the particles is converged to. For this reason we may conclude that the
particles coexist in a bonded relationship to one another.
To seed three proximal particles, we define the triple-gaussian seed fluctuation:
3(x, t) ≡ (x + d3, t) + (x, t) + (x − d3, t) (23a)
d3 = 3.314. (23b)
Figure 8(b) shows the stationary three-particle solution the system converges to
by t = 100. The final distance between the particles is 3.314, which again is a
stable value that is converged to even when we make small variations in d3. Future
work will determine if such particle bonding also occurs in 2D and 3D simulations of
Model G. Others working with the FN and FN-type models, who have simulated 2D
dissipative solitons with oscillatory tails, also report particle-particle bonding and
the formation of aggregate structures variously termed “clusters” or “molecules”,
although these are formed when the solitons have initial velocities relative to one
another (Bode et al., 2002; Purwins et al., 2005; Schenk et al., 1998). As is the case
with the 1D simulations of Model G, these bound FN solitons are similarly found
to align their concentration peaks with those of their partner.
When a two-particle simulation is performed with d2 = 6.628, the two fluctuations
being separated by the same distance as between the outer two particles
of the three-particle stable state, two particles are initially created from the 2
seed fluctuations. However, the G potential well and Y potential hill formed in
the space between them by the overlapping soliton tails provide a fertile environment
that allows the spontaneous emergence of a third particle. This three-particle
15
February 17, 2013 model˙g
Y, G and X Potentials
-10 -5 0 5 10
-1.0
-0.5
0.0
0.5
1.0
jY
jG
jX
10
Space
(a) Two 1D stationary bonded particles.
Y, G and X Potentials
-10 -5 0 5 10
-1.0
-0.5
0.0
0.5
1.0
jY
jG
jX
10
Space
(b) Three 1D stationary bonded particles.
Figure 8. (a) Two and (b) three 1D stationary particles formed from eqs. (19) and boundary conditions
(14), where is replaced by (a) 2 and (b) 3.
state then converges to the same three-particle state depicted in Figure 8(b). This
confirms the earlier expectation that in Model G existing particles should produce
favorable conditions facilitating additional particle autogenesis (LaViolette, 1985,
1994, 2010). Again, further work is needed to determine if mother-daughter particle
creation also takes place in 2D and 3D simulations of Model G.
Two dimensional simulations of the FN model carried out by other researchers
have also demonstrated the dissipative soliton particle replication phenomenon
(B¨odeker, 2007; Liehr et al., 2003; Purwins et al., 2005). In one case four particles
move towards one another, collide, and form a stable bound cluster. Then a
fifth soliton nucleates at the cluster’s geometrical center where the concentration
maxima of their innermost shells intersect and thereby produce a combined concentration
maximum that is sufficiently great to induce spontaneous generation of the
fifth particle. So this replication process occurs in the FN model in much the same
fashion, although Model G spawns its progeny particles from initially stationary
parent particles.
4.2 Particle “Mass”
Looking at Figure 5(a), it may appear as if the full integral of 'G should be negative,
due to the large central G-well, compared to the smaller G-hills that surround it.
However, the contribution from these surrounding G-hill shells ultimately outweigh
16
February 17, 2013 model˙g
Y, G and X Potentials
-10 -5 0 5 10
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
jY
jG
jX
10
Space
Figure 9. Circularly-symmetric 2D particle with gaussian diffusion coefficients, centered at the origin. It
is conjectured that 3D particles may naturally form an etheric vortex through its core, producing a similar
2D cross-section perpendicular to the vortex.
the negative contribution of the G-well core and surrounding G-well shells yielding
a positive value for the full integral of 'G; see Table 1, row (e).
Subquantum kinetics interprets the full integral of 'G as modeling a particle’s
gravitational mass which creates its long-range gravity field, positive mass being
associated with a negative 'G potential (LaViolette, 1985, 2010). So, to model a
physically realistic particle of positive gravitational mass, the full integral of 'G
instead needs to be made negative. This could occur if the soliton core became
broadened. To test this possibility, we have artificially broadened the core by introducing
a variable diffusion coefficient. More specifically, we introduce a gaussian
diffusion coefficient (r) with a maximum at the center:
(r) ≡ 1 + e−4r2/9. (24)
The following substitution is made for incorporation into eqs. (19):
∇2 → (r)∇2. (25)
The resulting system of equations is numerically solved in 2D with boundary conditions
(15). The resulting stationary state at t = 100 is shown in Figure 9. The
particle structure values are shown in Table 1 column 2D. Note that in this case
the core RMS radius for the 'G potential increases by 38% and the full integral of
'G potential becomes negative thus yielding a physically realistic positive gravitational
mass.
It is possible that such core broadening might occur with the onset of a rotational
wave mode similar to the spiral waves observed in the B-Z reaction. Mihalache
(2011) describes simulations in which spiral waves form in the dissipative
soliton produced by the complex, cubic-quintic Ginzburg-Landau system. With the
emergence of the rotational wave state they find that the soliton’s core broadens
by 1.5 to 2 fold. We are currently unable to directly simulate Model G particles
with rotational wave modes since, as mentioned earlier, limitations of our available
computational resources required that we impose a symmetry condition when
simulating in 2D and 3D.
17
February 17, 2013 model˙g
4.3 1D Particle Movement in a Gravitational Gradient
The movement of a 1D particle is examined in the presence of a G-gradient. The
G-gradient of slope m is incorporated into the system by adding the function
m(x, t) =
(
0 if t < 100,
mx + 10−4/6 if 100 ≤ t
(26)
to the right-hand side of eq. (13a). This function allows the particle to form from
t = 0 to 100, then “turns on” a G-gradient of slope m from t = 100 to 200, during
which time we examine the particle’s positions and velocities.
Using the parameter substitutions again of eqs. (17) and solving the resulting
system of PDEs under the boundary conditions
∀x ∈ [−50, 50] : ∀t ∈ [0, 200] :
= 3875/4096 (27a)
'G(x, 0) = −0.161e−1
2 ( x
0.363 )2
(27b)
'X(x, 0) = −8.37e−1
2 ( x
0.272 )2
(27c)
'Y (x, 0) = 0.930e−1
2 ( x
0.302 )2
(27d)
'G(±50, t) =
m(±50, t) (27e)
'X(±50, t) = 0 (27f)
'Y (±50, t) = 0 (27g)
yields the same single-particle state of Figure 7(a) as done previously. The difference
here is that the particle grows from gaussian initial conditions eqs. (27b, 27c, 27d)1
rather than the seed fluctuation . The purpose of using these initial conditions is
that eqs. (27) produce the particle more quickly than does. If we were instead
to use the fluctuation, the 1D particle would still be “settling down” at t = 100
with its core zeroes still moving on the order of 10−6 per unit time. This is the
approximate magnitude of the particle velocities due to the induced G-gradients.
In contrast, the 1D particle created from these alternate initial conditions has its
core zeroes effectively stationary at this order of magnitude, allowing us to measure
the particle’s velocity in isolation of this settling effect.
The 11 values for the G-gradients m that are examined are m = −10−5k/3 for
k ∈ {0, 1, 2, ..., 10}. See Figures 10. The position of the particle is defined to be the
midpoint between the two center-most zeroes of 'X flanking its central minimum.
This was found to be a more precise method of determining the particle’s position
than simply calculating the location of the central minimum of 'X, due to the fact
that 'X(x, t) is a numerical solution to the PDEs in this present analysis, and any
specific value is numerically interpolated. Thus points along highly sloping areas of
'X, such as its zeroes, are more accurately interpolated than local extrema, whose
precise locations must generally be extrapolated. The initial hills in Figure 10(b)
are an artifact of this method of determining the particle’s position, as the form
of the particle is slightly altered by the applied G-gradient. Because of this, the
1Note the use of the Core amplitude and Core RMS radius values from Table 1 in these 3 equations. The
constant is an overall factor used to select initial conditions that create the particle in as little time t as
possible.
18
February 17, 2013 model˙g
Particle Position H´10-5L
100 120 140 160 180 200
0
5
10
15
20
Time
(a) 1D particle positions in 11 different G-gradients.
Particle Velocity H´10-7L
100 120 140 160 180 200
0
5
10
15
20
Time
(b) 1D particle velocities in 11 different G-gradients.
Figure 10. Positions (a) and velocities (b) of the 1D particle from t = 100 to 200 for 11 different values
of the G-gradient m = −10−5k/3 for k = 0, 1, 2, ..., 10, corresponding to the data sets displayed from
bottom–up, respectively. After the G-gradient is turned on at t = 100, the particle velocity converges to a
constant value, proportional to the applied G-gradient m.
Particle Velocity H´10-6L
0 5 10 15 20 25 30
0.0
0.5
1.0
1.5
2.0
y = 0.0599x
Applied Negative G Potential Gradient H´10-6L
Figure 11. The 1D particle’s constant velocities vs. the 11 values of the applied G-gradient m form a
linear relationship.
19
February 17, 2013 model˙g
average of the velocities from t = 150 to 200 are used in the particle velocity vs.
G-gradient plot shown in Figure 11. The 11 velocities v were found to be directly
proportional to the applied G-gradient m:
v = −0.0599m. (28)
To realistically represent microphysical phenomena, Model G would need to spawn
solitons that accelerate in a G-gradient field rather than move at a constant velocity.
There are many aspects of Model G that still remain unexplored and we
feel that future work will demonstrate particle acceleration. In particular, future
3D simulations of Model G will investigate whether 3D solitons incorporating rotational
wave modes will be found to accelerate in G-gradients.
5. Conclusion
Features of Model G’s solitons in one, two, and three dimensions of space were
examined, and found to have characteristics resembling those observed for subatomic
particles. These include a structure matching observations of the nucleon’s
stationary wave charge distribution, multi-particle bonding, and movement in field
gradients. Model G was derived from the Brusselator R-D system using a recipe
general enough to potentially generate new soliton-supporting systems from R-D
systems that are unable to. All such systems would be interesting candidates to
study in the context of the subquantum kinetics methodology.
Acknowledgments
MP would like to thank Kerry Cassidy and Bill Ryan of Project Camelot for their
interview of Paul LaViolette in July of 2009.
Appendix A. Brusselator Solitons
Here we describe one other way of modifying the Brusselator in order for it to
support dissipative solitons. This involves keeping the reaction system with just
four reactions, as it was originally specified, but allowing A to vary in addition to
X and Y, and allowing non-zero reverse reaction rates for A ← X and X ← E. We
begin with the reversible Brusselator, which is specified by the following kinetic
reactions (Nicolis and Prigogine, 1977):
A
k1
⇋
k−1
X (A1a)
B + X
k2
⇋
k−2
Y + D (A1b)
2X + Y
k3
⇋
k−3
3X (A1c)
X
k4
⇋
k−4
E. (A1d)
20
February 17, 2013 model˙g
Expressed as PDEs with diffusion:
@A
@t
= DA∇2A − k1A + k−1X (A2a)
@X
@t
= DX∇2X + k1A + k−4E + k−2DY
− (k−1 + k2B + k4)X + k3X2Y − k−3X3
(A2b)
@Y
@t
= DY∇2Y − k−2DY + k2BX − k3X2Y + k−3X3. (A2c)
Passing to a dimensionless system, the units of time, space, and concentration
are identified, respectively, as:
T ≡
1
k−1 + k4
, L ≡
p
DAT, C ≡
1
√k3T
. (A3)
These are used to replace the dimensional parameters with their dimensionless
counterparts, as done in eqs. (5). Together with the dimensionless parameter substitutions
dx ≡
DX
DA
, dy ≡
DY
DA
, b ≡
k2
k−1 + k4
B, g ≡
k−1
k−1 + k4
, (A4)
p ≡
k1
k−1 + k4
, s ≡
k−3
k3
, u ≡
k−2
k−1 + k4
D, w ≡
√k3k−4
(k−1 + k4)3/2
E
eqs. (A2) become
@A
@t
= ∇2A − pA + gX (A5a)
@X
@t
= dx∇2X + pA + w + uY − (1 + b)X + X2Y − sX3 (A5b)
@Y
@t
= dy∇2Y − uY + bX − X2Y + sX3. (A5c)
We again use the same vector operator ∇ in both equation sets (A2,A5) with the
understanding that the prior is taken with respect to dimensional units, and the
latter dimensionless.
The homogeneous steady state values for A, X, and Y are, respectively,
A0 =
gw
p(1 − g)
, X0 =
w
1 − g
, Y0 =
sX2
0 + b
X2
0 + u
X0. (A6)
Defining the concentration potentials
'A ≡ A − A0, 'X ≡ X − X0, 'Y ≡ Y − Y0 (A7)
and additional system constants
c0 ≡ X2
0 + u, c1 ≡ b − 2X0Y0 + 3sX2
0 ,
c2 ≡ 2X0, c3 ≡ Y0 − 3sX0 (A8)
21
February 17, 2013 model˙g
yields
@'A
@t
= ∇2'A − p'A + g'X (A9a)
@'X
@t
= dx∇2'X + p'A − 'X + c0'Y
+ (c2'Y − c1 + ('Y + c3 − s'X)'X)'X
(A9b)
@'Y
@t
= dy∇2'Y − c0'Y
− (c2'Y − c1 + ('Y + c3 − s'X)'X)'X.
(A9c)
Solving this system in 1D with boundary conditions
∀x ∈ [−50, 50] : ∀t ∈ [0, 100] :
'A(x, 0) = −0.161e−1
2 ( x
0.363 )2
'X(x, 0) = −8.37e−1
2 ( x
0.272 )2
'Y (x, 0) = 0.930e−1
2 ( x
0.302 )2
(A10)
'A(±50, t) = 0
'X(±50, t) = 0
'Y (±50, t) = 0
and dimensionless parameter values
dx = 1, dy = 12, b = 29, g = 1/10,
p = 1, s = 0, u = 0, w = 14 (A11)
at t = 100 yields the same stationary soliton configuration shown in Figure 7(a),
with 'G replaced by 'A. Throughout the reaction, all concentrations maintain
a non-negative value, equivalent to the conditions 'A ≥ −A0, 'X ≥ −X0, and
'Y ≥ −Y0.
The same 2D and 3D soliton configurations are found as with Model G, when
circular and spherical symmetry are imposed, respectively, and with the following
modifications:
• The 3 gaussian initial conditions in eqs. (A10) have their heights and widths,
which are the values from rows (a) and (c) from Table 1, modified using the
corresponding 2D or 3D column.
• The Dirichlet boundary conditions at x = −50 for 1D are replaced by the
Neumann boundary conditions at r = 0 for 2D and 3D, as done in equation
sets (14,15).
References
Bode, M., Liehr, A., Schenk, C. and Purwins, H.G., 2002. Interaction of dissipative
solitons: particle-like behaviour of localized structures in a three-component
reaction-diffusion system. Physica D: Nonlinear Phenomena, 161 (1–2), 45–66.
22
February 17, 2013 model˙g
B¨odeker, H.U., 2007. Universal Properties of Self-Organized Localized Structures.
Thesis (PhD). Universit¨at M¨unster.
Herschkowitz-Kaufman, M. and Nicolis, G., 1972. Localized Spatial Structures and
Nonlinear Chemical Waves in Dissipative Systems. J. of Chemical Physics, 56
(5), 1890–1896.
Kelly, J.J., 2002. Nucleon Charge and Magnetization Densities from Sachs Form
Factors. Physical Review C, 66 (6), 065203.
Koga, S. and Kuramoto, Y., 1980. Localized Patterns in Reaction-Diffusion Systems.
Progress of Theoretical Physics, 63 (1), 106–121.
LaViolette, P.A., A Reaction-Diffusion Model of Space-Time. Austin, TX: Paper
presented at the Workshop on Instabilities, Bifurcations, and Fluctuations in
Chemical Systems. [1980].
LaViolette, P.A., 1985. An Introduction To Subquantum Kinetics: Parts I, II, III.
Intern. J. of General Systems, 11 (4), 281–345.
LaViolette, P.A., 1986. Is The Universe Really Expanding? Astrophysical J., 301,
544–553.
LaViolette, P.A., 1992. The Planetary-Stellar Mass-Luminosity Relation: Possible
Evidence of Energy Nonconservation? Physics Essays, 5 (4), 536–544.
LaViolette, P.A., 1994. Subquantum Kinetics: The Alchemy of Creation. 1st ed.
Alexandria, VA: Starlane Publications (out of print).
LaViolette, P.A., 2005. The Pioneer Maser Signal Anomaly: Possible Confirmation
of Spontaneous Photon Blueshifting. Physics Essays, 18 (2), 150–163.
LaViolette, P.A., 2008. The Electric Charge and Magnetisation Distribution of The
Nucleon: Evidence of A Subatomic Turing Wave Pattern. Intern. J. of General
Systems, 37 (6), 649–676.
LaViolette, P.A., 2010. Subquantum Kinetics: A Systems Approach to Physics and
Astronomy. 3rd ed. Niskayuna, NY: Starlane Publications.
LaViolette, P.A., 2012. Subquantum Kinetics: A Systems Approach to Physics and
Astronomy. 4th ebook ed. Niskayuna, NY: Starlane Publications.
Lefever, R., 1968. Dissipative Structures in Chemical Systems. J. of Chemical
Physics, 49 (11), 4977–4978.
Liehr, A.W., et al., 2003. Replication of Dissipative Solitons by Many-Particle
Interaction. In: High Performance Computing in Science and Engineering ’02.,
48–61 Springer.
Liehr, A., et al., 2004. Rotating bound states of dissipative solitons in systems
of reaction-diffusion type. The European Physical J. B: Condensed Matter and
Complex Systems, 37, 199–204.
Mihalache, D., 2011. Spiral solitons in two-dimensional complex cubic-quintic
Ginzburg-Landau models. Romanian Reports in Physics, 63 (2), 325–338.
Nicolis, G. and Prigogine, I., 1977. Self-Organization in Nonequilibrium Systems:
From Dissipative Structures to Order Through Fluctuations. New York: John
Wiley & Sons.
Nishiura, Y., Teramoto, T. and Ueda, K.I., 2005. Scattering of traveling spots in
dissipative systems. Chaos, 15 (4), 047509.
Purwins, H.G., B¨odeker, H. and Liehr, A., 2005. Dissipative Solitons in Reaction-
Diffusion Systems. Lecture Notes in Physics, 661, 267–308.
Schenk, C.P., Sch¨utz, P., Bode, M. and Purwins, H.G., 1998. Interaction of selforganized
quasiparticles in a two-dimensional reaction-diffusion system: The formation
of molecules. Physical Review E, 57, 6480–6486.
Turing, A.M., 1952. The Chemical Basis of Morphogenesis. Philosophical Trans. of
the Royal Society of London B: Biological Sciences, 237 (641), 37–72.
Vanag, V.K. and Epstein, I.R., 2007. Localized Patterns in Reaction-Diffusion Sys-
23
February 17, 2013 model˙g
tems. Chaos, 17 (3), 037110.
Winfree, A.T., 1974. Rotating Chemical Reactions. Scientific American, 230, 82–
95.
Zaikin, A.N. and Zhabotinsky, A.M., 1970. Concentration Wave Propagation in
Two-dimensional Liquid-phase Self-oscillating System. Nature, 225, 535–537.
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