Around self-similar solutions to the modified Korteweg-de Vries equation

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

For smooth and decaying data, this problem (the so-called soliton resolution conjecture) can be studied via the inverse transform method ((mKdV) is integrable): for generic such

As a consequence, we can study the flow of (mKdV) around self-similar solutions: in particular, we give an as- ymptotic description of small solutions as t → +∞ and construct

Obviously, we shrank considerably the critical space by taking smooth perturbations of self- similar solutions, but the above result still shows some kind of stability of

To prove the orbital stability result stated in Theorem 1.3, we first need to prove the uniqueness of the minimizer under equimeasurability and symmetric constraints which are

We apply the variational method and the blow-up analysis to the self-dual Chern–Simons–Higgs vortex equation on a flat torus to obtain two solutions for certain values of

Local existence and uniqueness theory for the inverted profile equation The purpose of this section is to prove an existence and uniqueness theorem for solutions of (1.12) on

These generalise the famous result for the mean curvature flow by Huisken [20], although in that case and for some other flows no curvature pinching condition on the