Presented by Professor Lev Ostrovsky, Universities of Colorado and North Carolina, USA
The beauty of theoretical science is that quite different physical, biological, etc. phenomena can often be described as similar mathematical objects by similar differential (or other) equations. In most physical and engineering applications, the oscillatory process is characterized by a slow (as compared with the period) modulation of its parameters, such as the amplitude and frequency. The same is true of the wave processes. For example, the electromagnetic waves radiated from the radio transmitting antenna, impulses from radars and lasers, and many other kinds of natural and man-made signals contain many periods of “carrier” oscillations; same for sound and ultrasonic waves. Moreover, the notion of modulation can be extended to non-harmonic oscillators and waves, such as solitary impulses with slowly varying amplitude and duration.
Such processes are commonly studied with different perturbation methods. Using approximations to find solutions to differential equations has a long history, especially in the areas of applied sciences and engineering where the relatively simple ways to obtain quantitative results are often crucial. However, in many cases, a small perturbation can strongly change the solution even if the basic equations are only slightly disturbed. An adequate tool for solving such problems is the asymptotic perturbation theory, which constructs a series in which the main term of the expansion differs from the unperturbed solution in that its parameters are slowly varying. Their variation can be found from the “compatibility,” or “orthogonality,” conditions that secure the smallness of the higher-order perturbations.
In the lecture, I will consider examples of this approach for oscillations and waves, beginning from a simple linear oscillator with damping, and including practically important cases of van der Pol oscillator with a limit cycle and parametric resonance. Then the modulated wave processes such as modulation instability and nonlinear solitary waves (solitons) in a variable environment will be considered with electromagnetic and oceanic examples. The presentation will be given in simple language avoiding as much formal mathematics as possible and focusing on physical phenomena.
Professor Lev Ostrovsky is a world renowned expert in the wave theory, acoustics, laser optics, biophysics, fluid mechanics, applied mathematics, and physical oceanography. He has published more than 350 high-quality papers and reviews. He is the author of five books and and many book chapters. For his contribution to science, he was awarded an International Lagrange prize (2010), the Mandelstam award of the Russian Academy of Sciences (2009); he obtained a Grant of the London Mathematical Society (2004), Orson Andersen Distinguished Scholarship of the Institute of Geophysics and Planetary Physics (1998), the USSR State Prize (1985), Discovery Certificate for the “A phenomenon of self-compression of wave packet in nonlinear dispersive media with formation of envelope solitons and envelope shocks” (1982). His name is included into various editions of "Who's Who in the World,” “Who’s Who in America,” “Who’s Who in Science and Engineering,” Strathmore’s “Who’s Who,” Reference Book “Renown Russians,” and different biographical volumes of American Biographical Institute and International Biographical Centre (Cambridge, England). One of the basic equations in the physical oceanography is named after him the Ostrovsky equation. His H-index is 37 in Google Scholar and 29 in Scopus.
For more information, please contact the Research Training Officer.