Since I was seventeen my great passion has been the solution of partial differential equations. I used an old book written by Italian mathematicians to face for the first time the technique of variable separation applied to the free Schrödinger equation. The article was written by Paolo Straneo, professor at University of Genova in the first part of the last century and Einstein’s friend, and from it I was exposed to quantum theories in a not too simpler way. At eighteen, some friends of mine, during my vacation in Camdridge, gave to me my first book of mathematics on PDEs: François Treves, Basic Linear Partial Differential Equations. You can find this book at low cost from Dover (see here).
Since then I have never given up with my passion with this fundamental part of mathematics and today I am a professional in this area of research. As a professional in this area, important references come from the work of Terry Tao (see also his blog), the Fields medalist. Terry, together with Jim Colliander at University of Toronto, manage a Wiki, Dispersive Wiki, with the aim to collect all the knowledge about differential equations that are at the foundation of dispersive effects. Most of you have been exposed at their infancy with the wave equation. Well, this represents a very good starting point. On the other side, it would be helpful to add some contributions for Einstein or Yang-Mills equations. Indeed, Dispersive Wiki is open to all people that, like me, is addicted to PDEs and all matter around them.
I have had the chance to write some contributions to Dispersive Wiki. Currently, I am putting down some lines on Yang-Mills equations (I did it before but this was recognized as self-promotion… just look at the discussion there), Dirac-Klein-Gordon equations and other articles. I think it would be important to help Jim and Terry in their endeavor as PDEs are the bread and butter of our profession and to have on-line such a bookkeeping of results would be extremely useful. Just take your time to give a look.
It is a well acquired fact that all the laws of physics are expressed through differential equations and our ability as physicists is to unveil their solutions in a way or another. Indeed, almost all these equations are really difficult to solve in a straightforward way and are very far from the exercises at undergraduate courses. During the centuries people invented several techniques to manage such equations and the most generally known is surely perturbation theory. Perturbation theory applies when a small parameter enters into the equations and a series solution is so allowed. I remember I have seen this method the first time at the third year of my “laurea” course and was Giovanni Jona-Lasinio that showed it to me and other fellow students.
Presently, we see as small perturbation theory has become so pervasive that conclusions derived just at a perturbation level are sometime believed always true. An example of this is the Landau pole or, generally, what implies the renormalization program. It is not generally stated but it is quite common the prejudice that when a large parameter enters into a differential equation we are stuck and nothing can be done than using our physical intuition or numerical computation. This is true despite the fact that the inverse of such a large parameter is indeed a small parameter and most known functions have both a small parameter and a large parameter series as well.
As I said elsewhere this is just a prejudice and I have proved it wrong in a series of papers on Physical Review A (see here, here and here). I have given an overview in a recent paper. With such a great innovation to solve differential equations at hand is really tempting to try to apply it to all fields of physics. Indeed, I have worked for a lot of years in quantum optics testing the approach in a lot of successful ways and I have also found applications to condensed matter physics appeared on Physical Review B and Physica E.
The point is quite clear. How to apply all this to partial differential equations? What is the effect of a large perturbation on such equations? Indeed, I have had this understanding under my nose since the start but I have not been so able to catch it immediately. The reason is that the result is really counterintuitive. When a physical system is strongly perturbed all the terms that imply spatial variation can be neglected. So, a strong perturbation series is a gradient expansion and the converse is true as well. I have proved it numerically in a quite easy way using two or three lines of Maple. These results can be found in my very recent paper on quantum field theory (see here and here). Other results can be found by yourself with similar simple means and are very easy to verify.
As strange as may seem this conclusion, it has obtained a striking confirmation through numerical computations in general relativity. Indeed, I have applied this method also to general relativity (see here and here). Indeed this paper gives a sensible proof of the Belinski-Khalatnikov-Lifshitz or BKL conjecture on the behavior of space-time approaching a singularity. Indeed, BKL conjecture has been analyzed numerically by David Garfinkle with a very beatiful paper published on Physical Review Letters (see here and here). It is seen in a striking way how all the gradient contributions from Einstein equations become increasingly irrelevant as the singularity is approached. This is a clear proof of BKL conjecture and our approach of strong perturbations at work. Since then Prof. Garfinkle has done a lot of other very good work on general relativity (see here).
We hope to show in future posts how this machinery works for pdes. In case of odes we have already posted about (see here).