Asymptotics in Fourier space of self-similar solutions to the modified Korteweg-de Vries equation

摘要

We give the asymptotics of the Fourier transform of self-similar solutions for the modified Korteweg-de Vries equation. In the defocussingcase, the self-similar profiles are solutions to the Painleve II equation; although they were extensively studied in physical space, no result to our knowledge describe their behavior in Fourier space. These Fourier asymptotics are crucial in the study of stability properties of the self-similar solutions for the modified Korteweg-de Vries flow.