Construction and characterization of solutions converging to solitons for supercritical gKdV equations

We consider the generalized Korteweg-de Vries equation in the supercritical case, and we are interested in solutions which converge to a soliton in large time in H^1. In the subcritical case, such solutions are forced to be exactly solitons by variational characterization, but no such result exists in the supercritical case. In this paper, we first construct a "special solution" in this case by a compactness argument, i.e. a solution which converges to a soliton without being a soliton. Secondly, using a description of the spectrum of the linearized operator around a soliton due to Pego and Weinstein, we construct a one parameter family of special solutions which characterizes all such special solutions.Comment: 38 pages ; submitted ; v2: margins modifie

Multi-soliton solutions for the supercritical gKdV equations

For the L^2 subcritical and critical (gKdV) equations, Martel proved the existence and uniqueness of multi-solitons. Recall that for any N given solitons, we call multi-soliton a solution of (gKdV) which behaves as the sum of these N solitons asymptotically as time goes to infinity. More recently, for the L^2 supercritical case, Cote, Martel and Merle proved the existence of at least one multi-soliton. In the present paper, as suggested by a previous work concerning the one soliton case, we first construct an N-parameter family of multi-solitons for the supercritical (gKdV) equation, for N arbitrarily given solitons, and then prove that any multi-soliton belongs to this family. In other words, we obtain a complete classification of multi-solitons for (gKdV).Comment: 32 pages, submitted, v2: hyperref links adde

Construction of multi-bubble solutions for the critical gKdV equation

We prove the existence of solutions of the mass critical generalized Korteweg-de Vries equation $\partial_t u + \partial_x(\partial_{xx} u + u^5) = 0$ containing an arbitrary number $K\geq 2$ of blow up bubbles, for any choice of sign and scaling parameters: for any $\ell_1>\ell_2>\cdots>\ell_K>0$ and $\epsilon_1,\ldots,\epsilon_K\in\{\pm1\}$, there exists an $H^1$ solution $u$ of the equation such that u(t) - \sum_{k=1}^K \frac {\epsilon_k}{\lambda_k^\frac12(t)} Q\left( \frac {\cdot - x_k(t)}{\lambda_k(t)} \right) \longrightarrow 0 \quad\mbox{ in }\ H^1 \mbox{ as }\ t\downarrow 0, with $\lambda_k(t)\sim \ell_k t$ and $x_k(t)\sim -\ell_k^{-2}t^{-1}$ as $t\downarrow 0$. The construction uses and extends techniques developed mainly by Martel, Merle and Rapha\"el. Due to strong interactions between the bubbles, it also relies decisively on the sharp properties of the minimal mass blow up solution (single bubble case) proved by the authors in arXiv:1602.03519.Comment: 70 page

Micronutrient deficiencies, vitamin pills and nutritional supplements

In the 21st century, it is hard to reconcile the concepts of the Western diet and overconsumption with the risk of micronutrient deficiencies. However, deficiencies can arise from poor dietary intake, alone or combined with physiological or metabolic injury. Micronutrients are essential to fulfil a broad range of biochemical and physiological functions, and are tightly regulated by homeostatic processes. Diagnosis of deficiency is complex and requires the use of separate investigations (dietary, functional, biochemical). While the role of micronutrients in the prevention or treatment of diseases (including cancer, type 2 diabetes) is of interest, a key driver for the vitamins and supplement market is their advertised potential to optimize health and performance in healthy individuals. The evidence so far indicates that multivitamins supplements offer no health protection, increase all-cause mortality, and risk of cancers in some subgroups. A nutritionally balanced diet is a safer way to achieve sufficiency

The radial structure of protostellar accretion disks: influence of jets

The radial structure of accretion disks is a fundamental issue regarding star and planet formation. Many theoretical studies, focussing on different aspects such as e.g. disk emissivity or ionization, have been conducted in the context of the Standard Accretion Disk (SAD) model, where no jet is present. We wish to calculate the structure of YSO accretion disks in an approach that takes into account the presence of the protostellar jets. The radial structure of these Jet Emitting Disks (JED) should then be compared to that of standard accretion disks. The analytical treatment used in this work is very similar to that of standard accretion disks but is using the parameter space of Magnetised Accretion-Ejection Structures that include the jet torque on the underlying disk. In this framework, the analytical expressions of key quantities, such as mid-plane temperatures, surface densities or disk aspect ratio are derived. It is found that JEDs present a structure very different from the SADs and that can be observationally tested. The implications on planet formation in the inner regions of accretion disks are briefly discussed. We also supply sets of analytical formulae, valid in different opacity regimes, for the disk quantities. These expressions can be readily used for any work where the disk structure is needed as an input for the model.Comment: 11 pages, 4 figures. Accepted for publication in A&

Transport of exotic anti-nuclei: I- Fast formulae for antiproton fluxes

The Galactic secondary cosmic ray anti-proton flux calculated with different propagation models is fairly consistent with data, and the associated propagation uncertainty is small. This is not the case for any anti-proton exotic component of the dark matter halo. Detailed propagation models are mandatory if the ultimate goal is to explain an excess. However, simpler and faster approximate formulae for anti-protons are an attractive alternative to quickly check that a given dark matter model is not inconsistent with the anti-proton observed flux. This paper provides such formulae. In addition, they could be used to put constraints on new physics in this channel, where an extensive scan of a large parameter space could otherwise be quite expensive in computer ressources.Comment: 8 pages, 3 figures (submitted). Stand-alone code for exotic anti-proton propagation can be downloaded at http://wwwlapp.in2p3.fr/~taillet/mtc/mtc_code.tar . Paper re-organized (results unchanged