Asymptotics in Fourier space of self-similar solutions to the modified Korteweg-de Vries equation
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We give the asymptotics of the Fourier transform of self-similar solutions for the modified Korteweg-de Vries equation. In the defocussing case, the self-similar profiles are solutions to the Painlevé 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. Our result is obtained through a fixed point argument in a weighted W1,∞ space around a carefully chosen, two term ansatz, and we are able to relate the constants involved in the description in Fourier space with those of the description in physical space.