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Apr 15

Mind, Matter, and Money

Posted by: JackSarfatti
Tagged in: Untagged 
Interesting that it applies to Landau Ginzburg theory so it may be important for the Vitiello-Freeman brain dynamics.

Extreme wave events, also referred to as
freak or rogue waves, are mostly known as an oceanic
phenomenon responsible for a large number of maritime
disasters. These waves, which have height and steepness
much greater than expected from the sea average
state [1], have recently become a topic of intense research.
Freak waves appear both in deep ocean and in shallow
waters [2]. In contrast to tsunamis and storms associated
with typhoons that can be predicted hours (sometimes
days) in advance, the particular danger of oceanic rogue
waves is that they suddenly appear from nowhere only
seconds before they hit a ship. The grim reality, however,
is that although the existence of freak waves has now been
confirmed by multiple observations, uncertainty remains
on their fundamental origins. This hinders systematic
approaches to study their characteristics, including the
predictability of their appearance [3].

In fact, research on rogue waves is in an emerging state
[1, 3, 4]. These waves not only appear in oceans [5] but
also in the atmosphere [7], in optics [8, 9], in plasmas
[11], in superfluids, in Bose-Einstein condensates [12] and
also as capillary waves [13]. The common features and
differences among freak wave manifestations in their different
contexts is a subject of intense discussion [2]. New
studies of rogue waves in any of these disciplines enrich
their concept and lead to progress towards a comprehensive
understanding of a phenomenon which still remains
largely unexplored. A formal mathematical description
of a rogue wave is provided by the so-called Peregrine
soliton [14]. This solitary wave is a solution of the one-dimensional
nonlinear Schr¨odinger equation (NLSE) with
the property of being localized in both the transverse and
the longitudinal coordinate: thus it describes a unique
wave event. This solution is also unique in a mathematical
sense, as it is written in terms of rational functions
of coordinates, in contrast to most of other known solutions
of the NLSE.

In a variety of complex systems such as Bose–Einstein
condensates [22], optical fibers [23], and financial systems
[24, 25], several variables rather than a single wave
amplitude need to be considered. For instance, in the financial
world it is necessary to couple cash to the value of
other assets such as shares, bonds, options, etc., as well
as to consider all correlations between these variables.
The resulting systems of equations may thus describe extreme
waves with higher accuracy than the single NLSE
model. Approaches to rogue wave phenomena involving
multiple coupled waves are the coupled Gross-Pitaevskii
(GP) equations [26] and the Manakov system (or vector
NLSE) [27]. Indeed, vector rogue waves of GP equations
and the Manakov system have been recently presented
[22, 25, 28].

Conclusions. Here we have analytically constructed
and discussed, a multi-parametric vector freak solution
of the vector NLSE. This family of exact solutions includes
known vector Peregrine (rational) solutions, and
novel freak solutions which feature both exponential and
rational dependence on coordinates. Because of the universality
of the vector NLSE model (1), our solutions
contribute to a better control and understanding of rogue
wave phenomena in a variety of complex dynamics, ranging
from optical communications to Bose-Einstein condensates
and financial systems


Begin forwarded message:

From: nick herbert <quanta@cruzio.com>
Subject: deterministic vector freak waves
Date: April 15, 2012 3:07:37 AM GMT+01:00

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deterministic vector freak waves

http://xxx.lanl.gov/abs/1204.1449

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I think you should make it clear, David,
that this paper has absolutely nothing to do with
"The Hippies That Saved Physics"