Heun’s functions and
differential geometry in
Maple15
Plamen Fiziev
The
Goal: To
Open the
Padlocks
of
Nature!
Department of Theoretical Physics
University of Sofia
and
BLTF, JINR, Dubna
Talk at XIV Workshop on Computer Algebra
Dubna, June 03, 2011
The main question:
Where we can find the KEY
?
The
Tool
A GOOD NEWS
After the April 15, 2011
we have
Maple 15
Accordint to Maplesoft:
http://maplesoft.com/products/maple/new_features/
Maple 15 now computes symbolic
solutions to 97% of the 1390 linear
and non-linear ODEs
from the famous text:
Differentialgleichungen by Kamke.
Mathematica® 8 only handles 79%.
or alltogheder (  ) : (a simple Maple calculation)
 97% + 79 %;
 = 176 % ( !!! really a fantastic result !!!)
Maple also solves these ODEs almost 10
times faster than Mathematica.
Heun’s Differential
Equation:
A KEY
for
Huge
amount
Zur Theorie der Riemann'schen
Functionen zweiter Ordnung
mit Vier Verzweigungs-punkten
of
Math. Ann. 31 (1889) 161-179
Physical
Problems
Born in Weisbaden April 3, 1859
found
Died in Karsruhe January 10, 1929
by
The Heun family of equations has been popping
up with surprising frequency in applications
during the last 10 years, for example in general
relativity, quantum, plasma, atomic, molecular,
and nano physics, to mention but a few. This has
been pressing for related mathematical
developments, and from some point of view, it
would not be wrong to think that Heun equations
will represent - in the XXI century - what the
hypergeometric equations represented in the XX
century. That is: a vast source of ideas for linear
differential equations and developments for
special functions.
Edgardo S. Cheb-Terrab,
MITACS and Maplesoft 2004
The General Heun Equation:
Confluent
Heun
Equation:
Mathieu functions, spheroidal wave
functions, and Coulomb spheroidal
Functions are special cases.
The UNIQUE
Frobenius solution
around z = 0 :
Recurrence
relation:
The connection
problem is still
UNSOLVED !
Bi-Confluent Heun Equation:
Exact
solutions
for
Double-Confluent Heun Equation:
Theree-Confluent Heun Equation:
φ4
24 Mobius transformations z -> f(z) of the independent variable z.
These forms of f(z) are:
Examples with
Some Exactly Soluble
in terms of Heun’s functions
physical problems:
1. Hidrogen Molecule
2. Wasserstoffmoleculeon
3. Two-centre problem in QM (Helium).
4. Anharmonic Oscillators in QM and QFT
5. Stark Effect
6. Repulsion and Attraction of Quantum Levels,
7. 3D Hydrodinamical Waves in non-isotermal Atmosphere
8. Quantum Diffusion of Kinks
9. Cristalline Materials
10. In celestial Mechanics: Moon’s motion
11. Cologero-Moser-Sutherland System
12. Bethe ansatz systems
…
At present – more than 200 scientific problems !
Heun’s problems in gravity: perturbations of
1.Schwarzshild metric: PPF, CQG,2006, J Phys C, 2007
2. Kerr metric (for s = 0, 1/2, 1,3/2,2) PPF, gr-qc/0902.1277
3. Reisner-Nortstrom metric (for s = 0, 1/2, 1, 3/2, 2).
4. Kerr-Newman metric (for s = 0, 1/2, 1, 3/2, 2).
5. De Sitter metric (for s = 0, 1/2, 1, 3/2, 2).
6. Reisner-Nortstrom-de Sitter metric (for s = 0, 1/2, 1, 3/2, 2).
7. Interior perturbations of all above solutions of EE.
- for Schwarzschild: PPF gr-qc/0603003.
8. QNM of nonrotating and rotating stars and other compact
objects: naked singularities, superspinars, gravastars,
boson stars, soliton stars, quark stars, fuzz-balls, dark stars…
9. All D-type metrics - Batic D, Schmid H, 2007 JMP 48
10. Relativistic jets: PPF, Staicova, astro-ph:HE/0902.2408
astro-ph:HE/0902.2411
11. Continuous spectrum in TME for s =1/2, 1
Borissov, PPF, gr-qc/0902.3617
An essential GENERALIZATION:
S. Yu. Slavyanov – A Theorem for all Painleve class of classical equations !
Note: All Painleve equations are Euler-Lagrange equations: Slavyanov 1966
Hamilton structure of the Painleve equations : Malmquist, 1922
P.F. , CQG, 2006 (Schwarzschild )
Denitsa Staicova, P.P.F. , Astrophys Space Sci, 2011 (Kerr)
Examples of Relativistic Jets 1
PPF, D. Staicova, astro-ph:HE/0902.2408, BAJ 2010
Discovered by NASA's
Spitzer Space Telescope
``tornado-like``
object Herbig-Haro 49/50,
created from
the shockwaves of powerful
protostellar jet hitting
the circum-stellarmedium.
PPF, D. Staicova,
astro-ph:HE/0902.2411,
BAJ 2010
Confluent Heun’s Functions ???
Cats eye
Some Maple HeunC problems:
HeunC((I)*omega,-(I)*omega+1., (6*I)*omega+1., -((-I+1.*omega))*omega, 20.*omega^2-(1.*I)*omega+.5+omega,z))
Conclusion:
We need a
NEW CODE !
based on new ideas
(tested already)
1. For large |z| = 1..100 :
2. HeunCPrime=fdif(HeunC), but PPF JPA 2011
3. Some values of z are problematic
(for example) :
HeunC(13.7629973824+.199844789*I, -12.7629973824-.199844789*I, -1.0+0.*I, 108.45307688652939865438+2.9503080968932803136*I,
-107.95307688652939865438-2.9503080968932803136*I, 110.988405457376-1.5970801306700*I)
Digits:=10;
-3.216621105*10^(-11)+9.335196121*10^(-12)*I
Digits:=32;
-2.52269564229422256*10^(-12)+5.87236956206153258*10^(-12)*I
Digits:=64;
-1.72317085591748299*10^(-12)+4.00958782709241923*10^(-12)*I
HeunC(-0.1e-1+1.*I, 1.01-1.*I, .94+6.*I, -1.0099+.98*I, -18.4880-1.39*I, 90.03) =.360445353243995
HeunC(-0.1e-1+1.*I, 1.01-1.*I, .94+6.*I, -1.0099+.98*I, -18.4880-1.39*I, 90.04) = Float(infinity)
Another problem:
To find the roots of system of
transcendental equations, written in
terms of Heun’s functions
ArXiv: 1005.5375
We
are
stell
looking
for
the
KEY !
Thank You
for your attention