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

Abstract

We give the asymptotics of the Fourier transform of self-similar solutions to the modified Korteweg-de Vries equation, through a fixed point argument in weighted W 1,8 around a carefully chosen, two term ansatz. Such knowledge is crucial in the study of stability properties of the self-similar solutions for the modified Korteweg-de Vries flow. In the defocusing 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. We are able to relate the constants involved in the description in Fourier space with those involved in the description in physical space. 7

Self-Similar Dynamics for the Modified Korteweg–de Vries Equation

“…We will also use the Landau notation a = o n (b) when a and b are two complex quantities (depending in particular of n) such that a/b → 0 as n → +∞; and mutatis mutandis a = o ǫ (b) when a/b → 0 as ǫ → 0. We use often the japanese bracket 〈 y〉 = 1 + | y| 2 , and the (complex valued) Airy-Fock function (3) Ai(z) := 1 π +∞ 0 e ipz+ip 3 d p.…”

“…Our goal in this paper is to continue our work initiated in [3], and to study the (mKdV) flow in spaces in which self-similar solutions naturally live. As we will see, the number of technical problems increases dramatically with respect to the case of (NLS).…”

Self-similar solutions and critical spaces for the modified Korteweg-de Vries equation

“…In what follows, we will give some insight of the main steps and of the main ideas of the proofs of Theorems 1, 3 and 4. Proposition 2, Corollary 5, Proposition 6 and Proposition 7 follow easily from the analysis developed for the theorems: we will not give further details on these matters and rather refer directly to [3,4].…”

“…where V : → is the self-similar profile, so that S λ = S. After an integration we see that the profile V solves a Painlevé II-type equation 3 + α. (1) for some α ∈ .…”

“…Motivation. In this review paper, we give an account of our work in collaboration with Simão Correia and Luis Vega [3,4]. We are interested in the dynamics near self-similar solutions of the modified Korteweg-de Vries equation:…”