Title of the talk: The near integrability of the water wave equations.
We derive a form of the two-dimensional water wave equation found by Levi-Civita in 1907. This is functional differential equation that was overlooked for almost a century. We then analyze the equation and show that its solution can be written as an infinite series where the first term satisfies the Korteweg-De Vries equation for shallow water waves. Surprisingly, all the remaining terms satisfy a linear equation, driven by the lower order terms that can be solved explicitly. This situation is called near-integrability.
Then we solve the equations when the leading term is the KDV soliton. All the higher order terms turn out to be traveling waves. The leading term of the solution of the Levi-Civita equation is actually the solution of the complex KDV equation, that was integrated in 1981. The special case of the solution of the real KDV equation gives the waves on the surface of the water. The complex solutions give the motion of the whole water column below the surface.
Finally, we discuss the situation when the solution of the complex KDV equations blows-up in finite time as it was shown that it could do in 1987.
Short bio: Björn Birnir is the Director of the Center for Nonlinear, Complex and Data Science (CNLS) at the University of California, Santa Barbara. In 1981 he obtained his PhD (Mathematics) from the Courant Institute New York University, and held positions at the University of Arizona, Tucson and UC Berkeley. Since 1993 Björn has been Professor in Mathematics at the University of California in Santa Barbara. In 1998, he became Director, Center for Complex, Nonlinear and Data Science (UCSB). He established an exchange program in 2000, that developed into an exchange between UCEAP (UC Education Abroad Program) and the University of Iceland. He was also the Study Center Director for the UC Education Abroad Program in Madrid, Spain, from 2009-2011, overseeing the student exchange between the UC systems and Spanish Universities. In 2010, Björn was elected Fellow of the American Association for the Advancement of Science. In 2013, he became Editor in Chief of the International Journal of Nonlinear, Complex and Data.
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