A few years ago a masters student, Harry, came to me and wanted to write his thesis on solitons (for a more mathematical introduction to solitons, see here, and for a picture of a soliton, see here, and for a history of the observation of the first soliton, see here) in space plasmas, and wanted me to be his advisor. I told him that while I did know a lot about space plasmas, I didn't know much about solitons, and therefore I couldn't guide him in the proper direction, but that I'd be happy to be his advisor and we'd learn about the subject together. This is unusual in that in addition to technical guidance, and advisor usually knows the particular field, and can suggest certain fruitful directions to the student. However, in my experience I've found that my most successful masters students are those that come to me with their own definitive idea, and in addition to learning the subject with them (sometimes from them) I simply give them technical guidance in plasma physics in general. This way they already have motivation, because it's a topic of their choice. I've heard that Feynman was of a similar mind, and didn't want students who came to him and asked what topic they should work on.
In any case, we started, as usual, by reading the literature, and one of the papers that Harry sent me was written by someone who I had shared an office with while a grad student at UCLA. He was an expert on solitons! So I contacted Bob, and after some email exchanges, where he gave Harry and me some advice, I invited him out to Daytona for a visit. He was on a sabbatical at the time, and was able to visit us for a week, during which he gave a departmental colloquium, and also suggested some very good directions for Harry's thesis. Harry duly finished his thesis, graduated, and is now working for Lockheed out in California while we finished polishing up his thesis for publication.
So, in addition to writing a great thesis, I have renewed a friendship that was on a 20-year hiatus, and started a new collaboration. Several of the ideas in Harry's thesis have suggested other problems to work on, and Harry and Bob and I will be pursuing these ideas in the future.
So, what's this neat idea? It's the fact that you can use the inverse scattering transform (IST) to determine whether or not a soliton (or solitons) is embedded in a particular magnetic field profile that has been observed (in our case, by the Ulysses satellite that is in the solar wind). In the first figure below, there's a plot of the magnetic field observed by Ulysses, along with a plot of the soliton that we found using the IST analysis. You can see that they don't match exactly, because there are other waves in the observed magnetic field - not just the soliton - and those waves show up as a difference in the two curves, solid versus dashed.
To see the soliton and the other waves distinctly, we did a numerical simulation and propagated the magnetic field forward in time, and compared that to the soliton moved forward in time. This comparison is shown in the second figure below. Notice that the two curves match much better in the region of the single bump - that's the soliton - but you can also see the waves propagating away from this region. Those are the waves (not part of the soliton) that were initially part of the magnetic field profile.
Normally, given a particular magnetic field, it is in general difficult to tell if there is a soliton embedded in that profile simply by looking. It may resemble a soliton, but that resemblance may simply be masking the fact that there are lots of waves interfering in such a way as to look like a soliton. The IST method, however, gives a definitive answer, although you must run a numerical simulation to see what other waves are embedded in the profile. You can clearly see below that there are some other waves, but that they propagate away at a different speed, and leave the soliton behind.
In any case, we started, as usual, by reading the literature, and one of the papers that Harry sent me was written by someone who I had shared an office with while a grad student at UCLA. He was an expert on solitons! So I contacted Bob, and after some email exchanges, where he gave Harry and me some advice, I invited him out to Daytona for a visit. He was on a sabbatical at the time, and was able to visit us for a week, during which he gave a departmental colloquium, and also suggested some very good directions for Harry's thesis. Harry duly finished his thesis, graduated, and is now working for Lockheed out in California while we finished polishing up his thesis for publication.
So, in addition to writing a great thesis, I have renewed a friendship that was on a 20-year hiatus, and started a new collaboration. Several of the ideas in Harry's thesis have suggested other problems to work on, and Harry and Bob and I will be pursuing these ideas in the future.
So, what's this neat idea? It's the fact that you can use the inverse scattering transform (IST) to determine whether or not a soliton (or solitons) is embedded in a particular magnetic field profile that has been observed (in our case, by the Ulysses satellite that is in the solar wind). In the first figure below, there's a plot of the magnetic field observed by Ulysses, along with a plot of the soliton that we found using the IST analysis. You can see that they don't match exactly, because there are other waves in the observed magnetic field - not just the soliton - and those waves show up as a difference in the two curves, solid versus dashed.
Observed magnetic field components (solid lines) and predicted magnetic field components of a single soliton (dashed lines) deduced from the IST. At time t = 0.
To see the soliton and the other waves distinctly, we did a numerical simulation and propagated the magnetic field forward in time, and compared that to the soliton moved forward in time. This comparison is shown in the second figure below. Notice that the two curves match much better in the region of the single bump - that's the soliton - but you can also see the waves propagating away from this region. Those are the waves (not part of the soliton) that were initially part of the magnetic field profile.
Normally, given a particular magnetic field, it is in general difficult to tell if there is a soliton embedded in that profile simply by looking. It may resemble a soliton, but that resemblance may simply be masking the fact that there are lots of waves interfering in such a way as to look like a soliton. The IST method, however, gives a definitive answer, although you must run a numerical simulation to see what other waves are embedded in the profile. You can clearly see below that there are some other waves, but that they propagate away at a different speed, and leave the soliton behind.
Observed magnetic field components (solid lines) and predicted magnetic field components of a single soliton (dashed lines) deduced from the IST. At time t = 55.2.
If you are interested in reading the entire paper, it's posted on the arXiv, here. The arXiv is a repository of "preprints," papers in draft form before they get published in order to disseminate more quickly than waiting for publication. It's a good place to watch to keep abreast of the latest developments in a particular field.
No comments:
Post a Comment