Student: | Michael Alfare |
School: | Rutgers University, Mathematics Dept. |
Email: | alfare@dimax.rutgers.edu |
Research Area(s): | ODEs, Painleve Equations |
Project Name: | Calculation of Stokes (Connection) Constants for the Painleve Equations |
Faculty Advisor: | Dr. Ovidiu Costin, Professor of Mathematics, Rutgers University |
Project Description Stokes constants are more commonly called connection constants. This is because they arise in problems called
connection problems. A connection problem is a situation where we know a lot about a system in the past and we want to use this
information to predict something about the system in the distant future. In other words, given how a system behaves at negative
infinity, can we predict the behavior at positive infinity. One example we can use is our solar system. We are well aware of the
positions of
all the planets for a while now. We know a lot about everything involved with the system. So we can compute, say, the position of Saturn
in 3000 years. This is relatively easy because the system I'm talking about is roughly integrable. Physicists call a system
integrable
if there exist conserved quantities, such as energy, angular momentum, etc. Although our solar system is not exactly integrable, we
can consider it integrable with a very small loss in accuracy. However, this is often not the case. Things
conceptually equivilent to friction can make systems not
integrable. Systems that are not integrable are called chaotic. In integrable systems, if you know all the conserved quantities
you can calculate a lot about the system with relative ease. This is not so in chaotic systems. So we would like to find a way to
calculate information about these systems as well. In recent developments in exponential asymptotics (a field of mathematics that deals
with these connection problems) Prof. Costin has developed a method which can calculate these connection constants regardless of whether
or not the
system in question is integrable. To give a better idea of how this would be applied, consider a diffraction experiment. Say you have
some material that you know, and you expose it to some wave of light, x-ray, uv, something of this sort. Now you want to know what the
diffraction pattern will be after it passes through the material. So we don't really care what happens in the material. We only really
care about what happens on the screen on the other side, which is relatively far away considering the scale of interactions of this
sort. So if we have a good way of pinning down these constants with high accuracy, we can compare our experimental data with our
theoretical expectations to see how good our model is. There is also the inverse problem where we are given the scattering data and we
have to reproduce the material. Please do not confuse these two problems as I will not go into the latter. So we wish to demonstrate
this method with the Painleve equations. They describe completely new functions and they occur in several fields of modern physics such
as quantum mechanics and plasma physics as well as classical mechanics. Also, the first few are relatively simple to work with. All
together they seem like a good choice with which to demonstrate this method. And so, for your enjoyment, here are the first two Painleve
Equations, along with the other four not so nice looking ones:
P_{II}: y'' = 2y^{3} + zy + a P_{III}: y'' = ^{1}/_{y}y'^{2} - ^{1}/_{z}y' + y^{3} + ^{1}/_{z}(ay^{2} + b) - ^{1}/_{y} P_{IV}: y'' = ^{1}/_{2y}y'^{2} + ^{3}/_{2}y^{3} + 4zy^{2} + 2(z^{2} - a)y + b^{1}/_{y} P_{V}: y'' = (^{1}/_{2y} + ^{1}/_{y - 1})y'^{2} - ^{1}/_{z}y' + a^{1}/_{z2}y(y - 1)^{2} + b^{1}/_{z2y}(y - 1)^{2} + c^{1}/_{z}y + d^{1}/_{y - 1}y(y + 1) P_{VI}: y'' = ^{1}/_{2}(^{1}/_{y} + ^{1}/_{y - 1} + ^{1}/_{y - z})y'^{2} - (^{1}/_{z} + ^{1}/_{z - 1} + ^{1}/_{y - z})y' + ^{(y - 1)(y - z)y}/_{z2(z - 1)2}(a + ^{bz}/_{y2} + ^{c(z - 1)}/_{(y - 1)2} + ^{dz(z - 1)}/_{(y - z)2}) |
Sources[1] Costin, O., Costin, R.D., and Kohut, Matthew, "Rigorous Bounds of Stokes Constants for Some Nonlinear ODEs at Rank One Irregular Singularities"Proc. Roy. Soc. Lond., (28 Aug 2003), pg. 1-10. pdf [2] Costin, O., and Costin, R.D., "On the Formation of Singularities of Solutions of Nonlinear Differential Systems in Antistokes Directions" Inventiones Mathematicae, 145, 3, (18 June 2001), pg. 425-485. pdf (Copyright owned by Inventiones) [3] Costin, O., and Kruskal, Martin D., "On Optimal Truncation of Divergent Series Solutions of Nonlinear Differential Systems; Berry Smoothing" Proc. Roy. Soc. Lond., 455, (1999), pg. 1-32. pdf |
Project ResultsAfter P_{I} was transformed and normalized correctly I attempted to find a series solution of the form y = sum(c_{n}*x^{-n}, n = 1..infinity). This substitution leads to a recurrence relation for the c_{n}s. We expect the c_{n}s to behave like a_{n}n!n^{-b}. In this case I found b to be 3/2. So I was left with the task of showing that the a_{n}s converge, thereby showing that this is, in fact, the form of the c_{n}s. The recurrence relation for the a_{n}s is rather ugly, so I won't show it here. However, one may use this to show that {a_{n}} is a decreasing sequence bounded below by a strictly positive number. In fact, I have obtained the following inequality:a_{N}*sqrt(N/(N + 1)) < a_{n} < a_{N}, for n > N This gives us a pretty good bound on our error even with rather crude estimates in the proof. In an attempt to get an idea of what this constant is I ran a script in maple to compute the first 500 terms. It gave an error of roughly .0002. Under suggestions from Prof. Costin I multiplied the 500th term by pi and found the best rational approximation to the remaining term using continued fractions. The method of continued fractions is provably the best rational approximation to a real number. I found that the remaining term was approximately 1/phi. Moreover, the difference of the remaining term and 1/phi was of the same order of magnitude as the error on this constant. I suspect that the lower bound is much tighter than the upper bound because of the estimates I made to show that the sequence was decreasing. Hence I suspect the limit to actually be closer to the lower bound. I then ran the script again to 2000 terms, but decided I wanted my computer back after about 16 hours and I actually only computed 1624 terms. This seemed to give an error of about .00002. So I then again compared the remainder to 1/phi. It was off in the fifth place now instead of the fourth, again agreeing with the error in the constant. 1/(pi*phi) is still in this small interval of possible values of this constant, and it is a lot closer to the lower bound than the upper bound, which agrees with my expectations of the limit being closer to the lower bound. I think it would be quite interesting if phi winds up making yet another appearence in mathematics. |
Upcoming Talk(s) |
When: 22 July 2004, 2:00pm
Where: CoRE 301 What: Complete presentation of results for the first Painleve Equation Last updated 20 July 2004 |