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
We focus on the study of ground-states for the system of M coupled semilinear Schrödinger equations with power-type nonlinearities and couplings. General results regarding existence and characterization are derived using a variational approach. We show the usefulness of such a characterization in several particular cases, including those for which uniqueness of ground-states is already known. Finally, we apply the results to find the optimal constant for the vector-valued Gagliardo-Nirenberg inequality and we study global existence, L 2 -concentration phenomena and blowup profile for the evolution system in the L 2 -critical power case.
Abstract. In this paper we consider the nonlinear Schrödinger equation iut + ∆u + κ|u| α u = 0. We prove that if α < 2 N and ℑκ < 0, then every nontrivial H 1 -solution blows up in finite or infinite time. In the case α > 2 N and κ ∈ C, we improve the existing low energy scattering results in dimensions N ≥ 7. More precisely, we prove that if, then small data give rise to global, scattering solutions in H 1 .
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