The Goal: ToOpen thePadlocks of Nature!

Heun’s functions and differential geometry in

Maple15

Plamen FizievDepartment 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

?A GOOD NEWS

After the April 15, 2011we have

Maple 15

The Tool

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.

Born in Weisbaden April 3, 1859Died in Karsruhe January 10, 1929

Heun’s DifferentialEquation:

Zur Theorie der Riemann'schen Functionen zweiter Ordnung mit Vier Verzweigungs-punkten

Math. Ann. 31 (1889) 161-179

A KEYfor

HugeamountofPhysicalProblemsfoundby

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:

The UNIQUEFrobenius solution around z = 0 :

Recurrence relation:

The connectionproblem is still UNSOLVED !

Mathieu functions, spheroidal wavefunctions, and Coulomb spheroidal Functions are special cases.

Bi-Confluent Heun Equation:

Double-Confluent Heun Equation:

Theree-Confluent Heun Equation:

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:

11. Cologero-Moser-Sutherland System

7. 3D Hydrodinamical Waves in non-isotermal Atmosphere

3. Two-centre problem in QM (Helium).

2. Wasserstoffmoleculeon

5. Stark Effect

1. Hidrogen Molecule

9. Cristalline Materials

8. Quantum Diffusion of Kinks

6. Repulsion and Attraction of Quantum Levels,

4. Anharmonic Oscillators in QM and QFT

12. Bethe ansatz systems … At present – more than 200 scientific problems !

10. In celestial Mechanics: Moon’s motion

2. 2. Kerr metricKerr metric (for s = 0, 1/2, 1,3/2,2) (for s = 0, 1/2, 1,3/2,2) PPF, gr-qc/0902.1277PPF, gr-qc/0902.1277

Heun’s problems in gravity: perturbations of

4. Kerr-Newman metric (for s = 0, 1/2, 1, 3/2, 2).3. Reisner-Nortstrom 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).

1.Schwarzshild metric:Schwarzshild metric: PPF, CQG,2006, J Phys C, 2007

5. De Sitter metric (for s = 0, 1/2, 1, 3/2, 2).

7. Interior perturbations of all above solutions of EE. - for Schwarzschild:- 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:Relativistic jets: PPF, Staicova, astro-ph:HE/0902.2408 astro-ph:HE/0902.241111. Continuous spectrum in TMEContinuous spectrum in TME for s =1/2, 1for s =1/2, 1 Borissov, PPF, gr-qc/0902.3617gr-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

PPF, D. Staicova, astro-ph:HE/0902.2411,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.

Cats eye

Confluent Heun’s Functions ???

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))

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)

Conclusion: We need a NEW CODE !based on new ideas (tested already)

Another problem: To find the roots of system of transcendental equations, written in terms of Heun’s functions ArXiv: 1005.5375

Thank You for your attention

Wearestelllookingfor the KEY !