The Fourier transform of a soliton скачать в хорошем качестве

The Fourier transform of a soliton

Here's a lovely contour integral calculation that applies to a problem in soliton theory: it's an integral that appears in a calculation of wobbles around a kink in phi^4 theory, but this integral is actually the Fourier transform of the sech-squared soliton of the Korteweg-de Vries (KdV) equation. KdV is a model equation for shallow water waves, and it has many remarkable properties. After telling of the story of how John Scott Russell discovered the soliton (or "wave of translation") in 1834, showing a Maple animation, and giving the exact formula for it, I go on to briefly mention a paper by Oxtoby & Barashenkov on phi^4 kinks, which includes the integral in question. The rest of the video is about the details of calculating the definite integral by integrating over a suitable contour in the complex plane and applying Cauchy's residue theorem. I actually calculate a slightly more general integral, which I found listed in the Digital Library of Mathematical Functions (DLMF). The notes I wrote during the talk are here: drive.google.com/file/d/1oouGIhXI4xQY2mOSHJ8glfKB0WKb7uQQ/view?usp=sharing For more details see: M. Buchanan, Wave of translation. Nature Phys 2, 575 (2006). doi.org/10.1038/nphys395 Wobbling kinks in phi^4: O.F. Oxtoby and I.V. Barashenkov, Asymptotic Expansion of the Wobbling Kink, Theoretical and Mathematical Physics, 159(3): 863–869 (2009). DLMF: Go to dlmf.nist.gov/ ( I made this video on holiday over Christmas, and must apologize for the poor sound quality in some parts, exacerbated by downloading problems with my editing software.)