Dina's Blog
Week 1
Our first meeting started vaguely with professor Rybkin talking about research in general. He solidified the notion that research is 99% learning and only 1% discovery. Self-education and motivations are key components to gaining knowledge.
Our subject of study is soliton waves. Their behavior is fascinating compared to other waves. Not only are they non-linear, but fully preserve their shape and speed after a collision. Examples of soliton waves in nature include tsunamis, freak waves (the ones that form suddenly in the middle of the ocean and disappear without trace leaving a lot of damage behind) and bore waves. Because of the incredible stability of solitons, they are widely used in telecommunications: what a convenient and clever way to pass on information!
Nonlinear partial differential equations come into play when modeling soliton waves. In fact there is a soliton solution to the KdV equation (named after scientists Korteweg and de Vries) describing the propagation of waves. However, this represents a very ideal situation with a soliton wave of perfect shape. We still don’t know how to model a soliton wave ‘tipping over’ when it hits the shore. After the wave’s top ‘slides over’ the object is no longer a function, which poses a number of difficulties for mathematical modeling.
Our group members looked puzzled after the information has been presented. Who wouldn’t be?? The mathematical knowledge required to even approach the problem of modeling soliton waves seemed far beyond the college math curriculum. Our main coordinator, Odile Bastille, reassured us that we will gain the necessary “machinery” to understand solitons. A time has come to brush up on our differential equations knowledge.
One of the very useful tools introduced to us during the first week was the mathematical software designed to write expressions involving complex mathematic symbols. Typing a mathematics paper in Microsoft Word or preparing a math presentation in Power Point is not my idea of fun, since it takes ages to type the simplest possible equation. This is when TeXnicCenter comes for a rescue. This cool program lets you create LaTeX documents with formulas and text. Moreover, you can compile them in PDF and create PDF presentation slides. LaTeX language is quite simple indeed. All is required to understand it is a minimal amount of practice and a couple of sample documents.
The next stop in the ‘modern marvels’ series was the program called MATLAB. It lets you build the most unimaginable graphs ranging from simple 2D sine curves to 3D plots of the unnormalized sinc functions. To help one visualize the function better, MATLAB offers animated graphing. This is not a computer graphing calculator, but a whole engineering workshop! Curve fitting, optimization, PDE solving, vector fields - this is only a short list of what MATLAB is capable of. I am definitely happy to be a part of undergraduate math research and have such a great piece of software available to me.
After brushing up on our theoretical knowledge, the group members gave presentations on useful functions that in some way are related to partial differential equations.
- Deven talked about the Fourier trasforms that can represent any piecewise continuous functions, making them easy to work with when solving PDE’s.
- Jennifer’s subject was the Burgers’ equation and the method of characteristics to solve it. Interestingly enough, shock waves can also be solutions to a specific type of Burgers’ equation.
- The Airy function was presented by Jessica - it is a solution to a simple linear ODE and has a variety of applications: from optics to quantum mechanics. It also appears in the study of oscillating integrals, which hints that it will emerge in our soliton waves research.
- The dirac delta function is more of a limiting procedure than a function, since it is zero everywhere except for the origin. Its unique quality is what it does to other functions: when you have a product of dirac delta of (x-a) times f(x) under the integral you get out f(a). This property is widely used in solving PDE’s especially when there is a product of a Green’s function and the dirac delta.
Besides the research, our group is actively engaged in putting together an REU 2010 website. We are looking for interesting images and creative content to get people interested in our program. The upcoming trip to Girdwood will be the culmination of our research: we will get to see real soliton bore waves.
Week 2
This week at the REU started excitingly. I was looking forward to Monday all weekend because our group accompanied by coordinator, Odile, and professor Rybkin had plans to go out for dinner. Professor Rybkin chose his favorite Thai restaurant called Lemongrass. What a good way to spice up a Monday evening! After a productive workday we indulged ourselves in good food and interesting company. Shrimp with eggplant, scallops with green beans, halibut, chicken, fried banana and mango dishes were going around the table and settling on our plates.
Presentation of the new material marked our Tuesday afternoon. To study the process of how solitons transform themselves back into shockwaves, we need to understand the conservation laws that the KdV equation possesses. When I think of this concept, I recall conservation of mass or energy, but here the situation is somewhat different. In a mathematical sense, conservation laws are differential equations. Interestingly enough, the KdV equation has an infinite number of conservation laws! As shown by mathematicians Kruskal, Zabusky and Miura, they can be found by a recurrence relation (discovered in the 1960s).
Then our topic suddenly switched to quantum physics and the Schrodinger equation. First, I had a hard time understanding how this is tied to our subject of study, but soon my confusion went away. It turns out that KdV equations can be solved by the method of Inverse Scattering Transform and finding the Schrodinger operator is one of the crucial steps. This neat way of solution was only discovered in the second half of the 20th century and it also can be applied to some other non-linear partial differential equations; they all share common properties such as presence of solitons and infinite number of conservation laws.
This ocean of new information seemed incomprehensible from the first glance. Fortunately, reviewing ordinary differential equations got us thinking in the right direction. Here is why: the reduction of order method (introduced to us long ago!) proved to be useful in simplifying the Schrodinger equation.
After cranking the math, Jessica and I immersed in our creative assignment: putting together the REU website. I am very glad that professor Rybkin selected us in charge of this exciting task. Together we have finalized the outline of the layout and step by step are trying to come up with original content. Working on different designs means taking my imagination to new levels and I truly enjoy this. I am glad that REU offers such a diverse range of activities allowing everyone to contribute, while doing what he/she likes best.
Next week we are heading to Odile’s place for dinner and to observe her cute, newborn kittens. I will be sure to take plenty of pictures.
Week 3
It is becoming a tradition that our week at REU starts with a nice evening dinner. This time we got to visit Odile’s house and observe the newly born kittens nurtured by the adopted cat. Our menu consisted of the food from the Lemongrass restaurant and appetizers prepared by Odile. During the meal, we discussed Alaska adventures involving trips to the Circle hot springs, unpaved roads and abandoned hotels. We agreed that Alaska is an astounding place full of mysteries and got more excited about our trip to Girdwood.
On Tuesday we faced our biggest difficulty – trying to graph the wave equation with the finite difference method and plotting piecewise continuous functions. Dmitry from the UAF Geophysical Institute gave us an entire tutorial on MATLAB, making our lives a whole lot easier! The key element is using proper format when inputting functions, their solutions, vectors and matrices. Punctuation (especially putting the semicolon at the end of each line) matters a lot: one little mistake and the program stops cooperating. Graphing in MATLAB became not only doable but fun, since it is possible to pick a color for your graph, “time” it properly and produce an animation. I watched Deven play around with the program; she was very open to the experiment, trying different functions, time intervals and vectors. What started as a series of broken lines gradually turned into an elaborate animation with “dancing” loops. Deven definitely succeeded at understanding the program’s language.
The presentation on Wednesday talked about reflection less potentials and how they are tied to the Inverse Scattering Transform that we discussed last week. It all goes back to the idea that solitons are the physical manifestation of the Schrodinger operator’s discrete spectrum. Any short range potential eventually evolves as a train of solitons for a large value of time t.
Intern presentations put this information into perspective. Jennifer introduced the cubic Nonlinear Schrödinger equation that guides deep sea events in 1+1 dimension. She also talked about the circumstances under which rogue waves form. Modulational instability and dispersive focusing mechanism were main subjects of interest. The first one involves nonlinear wave trains suffering from the Benjamin-Feir instability that causes rapidly growing modulations capable of developing into a short group of steep waves. These waves may or may not become rogue waves.
Jessica continued the topic by giving us insight into the shallow water waves. She examined the momentum-balance equation in the integral form, the mass balance law and the mass flux. Jump conditions were especially emphasized since they are of key importance in the formation of the bore.
Jessica and I continue working on the website. At this point we have a clear layout and solid content. We have agreed on the navigation links, but are still in the process of developing interesting graphics. I am quite disappointed that Jessica is leaving next week; her input into this project and the program in general has been invaluable. You rarely see a person with such good work ethic and an eye for detail. I will continue on this path of excellence to reach our goal!
Week 4
We have been discussing our trip to Girdwood. It is unclear yet, when professor Rybkin will be available, but we are going to shoot for the first week of August.
Deven and Jennifer are occupied with reading different research papers on bore waves and such. They are still in the process of tackling MATLAB and getting it to graph what they need. Jennifer is studying shallow water and jump conditions. For the water to be considered shallow its length must be much greater than its depth. Shallow water wave equations model one-dimensional waves on a horizontal bottom.
Meanwhile Odile installed Dreamweaver on my computer and I am discovering what it is capable of. Last time I used it was during my junior year of high school, so I have very vague memories. The hardest thing so far is to make sure that different browsers display the same thing… my navigation bar appears to be crooked in Internet Explorer. I get a little frustrated because html is hard to read and the formatting feature in Dreamweaver is not always cooperating. I suppose practice is the key. I already found many useful tutorials online that explain how to make navigation bars, headers, etc.
Week 5
We are all consumed with our projects. I hear Jennifer and Deven discussing the method of characteristics. Characteristics are curves in spacetime on which certain properties, or expressions, are propagated. Partial differential equations reduce to ordinary differential equations along these curves. I remember Jennifer giving a presentation about this at the beginning of our internship; it indeed proved to be useful. The important thing is to find such curves and that requires transforming the system of shallow water equations into a new system which involves derivatives in only one direction. The girls are actively engaging in discussions with each other – I like this spirit of teamwork. Too bad Jessica is not here to share her website ideas with me.
While, Deven and Jennifer are consumed with David Logan’s book An Introduction to Nonlinear Partial Differential Equations, I am battling Dreamweaver. For some reason, my horizontal navigation bar is not centered at all. To top it off, the images are overly large and the text is not wrapping around them properly. I go online and learn about the V-space and the H-space: they allow a decent amount of area between the picture and the text. Next step is straightening the navigation bar. Through trial and error, I find the right width percentage. Finally, it looks better!
Week 6
We now have more or less concrete information on our Girdwood trip. I am happy to know that we will also go to Seward for a cruise around glaciers. This exciting research/entertainment journey will take place on August 6-11. I can’t wait!
Moving away from hyperbolic equations, Jennifer and Deven will concentrate on tidal bores in the Turnagain Arm, which is located next to Cook Inlet. Large tides are attributed to the natural resonance of Inlet. Turnagain arm is narrow, shallow and gently sloping, but the topography of its mudflats does not exist. This is a good place to observe large tidal bores; they can reach 3 meters in height and travel as fast as 5 m/s. How and where the bore forms is impacted by the wind speed and the channels of the mudflats.
Jennifer sent me gif images for a soliton wave animation that needs to be put together using Photoshop. For some reason, the frames arrange themselves in the opposite order, so I have the wave traveling from the farthest point to the origin and not vice versa. After finding the ‘reverse’ button on the animation tab, everything starts to work! Jennifer was kind enough to tweak the images in MATLAB for me, changing the colors. I am glad things are going good with the website.
Week 7
This week I found out we have another REU intern that is working at the UAF Geophysical Institute. Enzo Wendler is an undergraduate student in math and physics and this summer he is being mentored not by Prof. Rybkin, but by a Geophysical Institute staff member, Dmitry Nicolsky. Dmitry’s correct job title is a tsunami numerical modeler and his research is focused on development and implementation of various geophysical models. The spotlight of the day, however, went to Enzo who gave a presentation on forced long waves and the runup of solitary waves on a canonical beach. He too was using the shallow water equations but in a different context.
I took a picture of Dmitry, Enzo and Prof. Rybkin just outside of the Chapman building after the presentation. It will go on the website, which is by the way on its last stages of completion. I am astonished with my ability to do web design. This just proves the point that nothing is impossible.
Also, this week I went to the library and reserved the media equipment for us to use on the trip. I was tutored on how to video tape things and use a professional picture camera. It is quite sophisticated indeed. Turns out we will need a Mac Book laptop to hook up our video camera to, so the video is streamed live onto the computer. We can later edit the movie file any way we want, which is always a plus. A 400X lens will be available to me in case we want some high quality pictures (and we do). I am definitely going to practice using the equipment before we head out to Girdwood.
Week 8
Our trip to Girdwood has been moved to August 8-12. We are still in accordance with the tidal bore schedule.
Jennifer and Deven are polishing up their tidal bore presentation. They have gathered some highly technical and deep material. Good job girls! Starting from the general and going to the specific, the girls first introduce the formation of the bore. Next, they move on to a distinction between turbulent and undular bores (the latter ones are weakly dissipative). The geographic location of the Turnagain arm and its significance are discussed after that. The mathematical context comes in the form of the shallow water wave equations, the conservation laws and the method of characteristics to solve PDE’s. There is still more to come, but even at this point there is an astounding amount of information.
Sadly, this is the last week of the REU. I am trying to rap things up. The website will need to get updated after our trip, since we’ll have a ton of pictures.