SelfSimilar Scaling of Solitons and Compactons in Relativistic Jets

1
Self-Similar Scaling of Solitons and Compactons
in Relativistic Jets
HST Image of Quasar Jets
Nonlinear Dispersion Relationship (3,4)
Quasar 3C120
Keith Andrew, Michael Carini, Brett
Bolen Keith.Andrew_at_wku.edu Mike.Carini_at_wku.edu
Brett.Bolen_at_wku.edu Department of Physics and
Astronomy Western Kentucky University Bowling
Green KY
From Dr. Marsher BU websiteFrame from a
conceptual animation of 3C 120 created by
COSMOVISION                         
Large Amplitude Nonlinear Fields-Solitons
Nonlinear Schrödinger Equation
//rst.gsfc.nasa.gov/Sect20/h_accretion_disk_02.jp
gimgrefurlhttp//rst.gsfc.nasa.gov/
Soliton- long lived nondissipative wave form
where nonlinear amplitude growth is balanced by
dissipative losses, need not be topological in
origin Compacton-long lived wave form with well
defined functional relationship between
amplitude, width and speed of propagation, no
exponential envelope (3,7)

AbstractThe jet forming inner region of an
object containing a massive Kerr black hole will
contain a hot turbulent lepton plasma that can be
modeled by a system of relativistic MHD-NPDE. The
nonlinearities in these equations give rise to
long lived localized soliton solutions and
soliton like solutions known as compactons that
exist at all length scales. These objects could
give rise to structure formation at all locations
along the jet that appear as shock bows, vortices
or knots that would cause luminosity variations
along the jet axis. Here we study the scaling
behavior of these solutions in jet environments
by using the dimensionless scaling rules from the
Buckingham Pi Theorem with the self-similar
scaling of nonlinear wavelets in a system of
relativistic NPDE to estimate the resulting
fractional change in jet luminosity.
Buckinghams Pi Theorem Only dimensionless
quantities needed (1)
The field components F are representative of
vector potential components or electric field or
magnetic field components.

• Existence Requirements
• Convection
• Dispersion
• Diffusion
• Nonlinear

G-Newtons constant of gravitation B- magnetic
field l- characteristic length of the
jet -?-density of surrounding medium -c-speed of
light -v-jets ejection velocity M- core
mass dM/dt- mass accretion rate L-jet
luminosity LL(l, v, c, G, M, dM/dt, B, ?)
Wavelet Scaling Rules
Multiscale Wavelet Similarity Analysis
For Nonlinear PDEs (1,5,6)
Waves characterized by 1. Amplitude A 2.
Width w 3. Velocity v-limited by dispersion
relationship Localized soliton and compacton
solutions expanded with Gaussian Family Wavelets
Odd power ?3
MHD Equations
Even power ?4
Conclusions Fractional change in luminosity
Relativistic Hydrodynamic Equations in
terms of the potentials (3)
Luminosity Field Amplitude Squared Width
constrained by jet diameter, velocity constrained
by dispersion
For a given scale, j, the similarity
transformation Maps the NPDE-gtsingle scale
algebraic constraint of the form F(A,w,v)0 for
localized soliton like solutions.
1. Espinosa, M. H., Mendoza, S. Hydrodynamical
scaling laws for astrophysical Jets, arXiv.astro
ph/0503336 v1, (Mar 2005) 2. Tevecchio, F., Jets
at all scales, arXiv.astro-ph/0212254v1, (Dec.
2002) 3. Marklund, M., Tskhakaya, D.D, Shukla,
P.K., Quantum Electrodynamical shocks and
solitons in astrophysical plasmas,
arXiv.astro-ph/0510485 v1 Jan. 2002) 4.
Schwinger, J., On Gauge Invariance and Vacuum
Polarization, Phys. Rev. 82, 664 (1951) 5. P G
Kevrekidis, V V Konotop, A R Bishop and S Takeno
2002 J. Phys. A Math. Gen. 35 L641-L652 6. Ludu,
A, OConnell, R.F., Draayer, J.P., Nonlinear
Equations and Wavelets, Mulit-Scale Analysis,
arXiv.math-ph/0201043 v1(Oct 2005) 7. Tatsumo,
T., Berezhiani, V. I., Mahajn, S. M., Vortex
Solitons-Mass, Energy and Angular momentum
bunching in relativistic electron-positron
plasmas, arXiv.astro-ph/0008212 v1, (Aug 2000)
From BU website http//www.bu.edu/ blazars/resear
ch.html
U-internal energy P-pressure -?-density F-external
gravitational potential V-velocity vector
field B-magnetic field