Helix equations, Near Gaussian solitons.
Roger Beresford
identity":-
SUM(i=0 to i=m-1)[a(p) Sin^(2p)[t+2 i Pi/m] = 1 .....(1)
a(p)={2, 2^3/3, 2^4/5, 2^7/(7*5), 2^8/(9*7), 2^10/(11*7*3),
2^11/(13*11*3), 2^15/(13*11*9*5), 2^16/(17*13*11*5),...}
.....(2)
and p is less than m/2 if m is even, p is less than m if m is odd. A
Cos[m*t] ripple occurs if the p limit is exceeded.
My next comments and queries are more relevant to this group.
This result arose from the study of multi-phase sinusoidal functions,
which are solutions to banded sets of differential equations such as
d[x(j)]/d[t] + c(t) - omega*(x(Mod[j+k,m])-x(Mod[j-k,m]))=0, j=1...m,
1<=k<m/2 .....(3)
where j identifies one of the m equi-spaced phases and k is a phase
step. If k is a factor of m there are independent solutions indexed by
s; otherwise there is one solution:-
x(m,j)=C[t]+SUM(s) c(s)Sin[2Sin[2k Pi/m]*
omega*t]+phi(s)-2j*k*Pi/m] .....(4)
with c(s) and phi(s) being amplitude and phase constants of
integration; the frequency factor 2Sin[2k Pi/m] arises from the
identity Sin[x+y]-Sin[x-y]=2Cos[x]Sin[y] that is used in the
derivation from a trial function. (Note that m=12 gives periodicity
4Pi, or half-spin relative to the standard m=4 case.) There is a
related second order equation
d2[y(j)]/d[t]^2 = omega^2 * y(Mod[j+k,m])+y(Mod[j-k,m])-2y(j)
.....(5)
The helices are generalised multi-phase power-sinusoids that can be
defined using a "generalised sign" M(m), which has M as the mth root
of unity, M(m)^m=1, so M(2) is "minus", M(4) is "i", and M(3) is
Cuberoot[1], sometimes called "j" by electrical engineers working with
three-phase systems. Because Mathematica is designed to work with
complex numbers, M(m) has to be written (-1)^(2/m). The Mathematica
definitions have four arguments, an angle x, an optional number of
phases m (with default value 4), a phase number (with default value 0)
and a power(default value 1, but corresponding to 2p in (1)):-
cos[x_, m_:4 ,i_:0, q_:1]:=Re[(-1)^(2(m-i)/m) Exp[i
Im[(-1)^(2/m)]x]]^q
.....(6)
sin[x_, m_:4 ,i_:0, q_:1]:=Im[(-1)^(2(m-i)/m) Exp[i
Im[(-1)^(2/m)]x]]^q
.....(7)
There is an equivalent exponential series definition involving every
m/2th term and alternating signs. The definitions give the correct Cos
and Sin results with the default values.
As the power increases, the sinusoid phases approach the form of
gaussian pulses separated by almost flat sections.
G(q)=Exp[-q/2 k^2(Pi/(2k)-t+phi)^2] .....(8)
They differ from Gausian pulses by reaching zero. The separate phase
are solutions to the non-linear differential equation
d2[x]/d(t)^2=(1-1/q)(d(x)/d(t))^2-omega^2*q*x .....(9)
Numerical integration gives an individual pulse; adjacent pulses are
lost with Mathematica's standard accuracy options because the function
and all its derivatives become zero together.
This pulse seems to stand in a similar relation to Gaussian pulses as
JacobiCN function do to Sech pulses. I have found a general
conservative equation
2 [g]t = SUM(over x,y...)[gxx-gx^2/g]x .....(10)
that has multi-phase gaussian pulses as solutions (of unknown
stability), and
st=SUM(over x,y..)[s^2 sxxx/sx^2-3s sxx/sx+2(sx-s^2kx^2/sx)
.....(11)
that has s=a Sin[2(kx+ky+..)t/q^2+kx x+ky y+..]^p .....(12)
as a solution. The last term in (11) only simplifies if kx=ky=.., so
it is effectively a (1+1) equation.
Queries. (1) Is any of this new? (2) Are there stable integration
procedures for (10) or (11) and do they give solitonic solutions an
any situations?
Finally, I accidentally used the 5-point integration procedure
cnew=c+steplength^3((a-e)c+b^2-d^2), corresponding to
[1/G]t=[Gxx/G+4 Log[G]]x, .....(13)
and found solitonic behaviour. Very narrow (less than 10 mesh points)
solitonic pulses are stable. At first I thought that they were
Gaussian, but they are nearer to a high power sinusoid; these do not
satisfy (13). They travel with speeds proportional to height, and
cross with displacement and longitudinal troughs, unlike the
transverse Sech troughs. Related 2D pulses move at 45 deg, and absorb
smaller, slower pulses without regenerating them. Is this of interest
to anyone?
Roger Beresford.
Michael Pender
"Roger Beresford" <rogerbe...@supanet.com> wrote in message
news:fd882e5e.01060...@posting.google.com...
> Queries. (1) Is any of this new? (2) Are there stable integration
> procedures for (10) or (11) and do they give solitonic solutions an
> any situations?
>
It sounds like the equations are equivalent to polyphase signalling, also
known as MPSK modulation; a direct solution is provided in the book:
Robert M. Gagliardi, Introduction to Communications Enginering: Second
Edition (1988).
See Chapter 9, section 5 - Polyphase (MPSK) signalling.
Michael Pender, Manager
Nanochron, LLC
www.nanochron.com