Invariance of the white noise for KdV

We prove the invariance of the mean 0 white noise for the periodic KdV. First, we show that the Besov-type space \hat{b}^s_{p, \infty}, sp <-1, contains the support of the white noise. Then, we prove local well-posedness in \hat{b}^s_{p, \infty} for p= 2+, s = -{1/2}+ such that sp <-1. In establishing the local well-posedness, we use a variant of the Bourgain spaces with a weight. This provides an analytical proof of the invariance of the white noise under the flow of KdV obtained in Quastel-Valko.Comment: 18 pages. To appear in Comm. Math. Phy

Global well-posedness for the KP-I equation on the background of a non localized solution

We prove that the Cauchy problem for the KP-I equation is globally well-posed for initial data which are localized perturbations (of arbitrary size) of a non-localized (i.e. not decaying in all directions) traveling wave solution (e.g. the KdV line solitary wave or the Zaitsev solitary waves which are localized in $x$ and $y$ periodic or conversely)

Random data wave equations

Nowadays we have many methods allowing to exploit the regularising properties of the linear part of a nonlinear dispersive equation (such as the KdV equation, the nonlinear wave or the nonlinear Schroedinger equations) in order to prove well-posedness in low regularity Sobolev spaces. By well-posedness in low regularity Sobolev spaces we mean that less regularity than the one imposed by the energy methods is required (the energy methods do not exploit the dispersive properties of the linear part of the equation). In many cases these methods to prove well-posedness in low regularity Sobolev spaces lead to optimal results in terms of the regularity of the initial data. By optimal we mean that if one requires slightly less regularity then the corresponding Cauchy problem becomes ill-posed in the Hadamard sense. We call the Sobolev spaces in which these ill-posedness results hold spaces of supercritical regularity. More recently, methods to prove probabilistic well-posedness in Sobolev spaces of supercritical regularity were developed. More precisely, by probabilistic well-posedness we mean that one endows the corresponding Sobolev space of supercritical regularity with a non degenerate probability measure and then one shows that almost surely with respect to this measure one can define a (unique) global flow. However, in most of the cases when the methods to prove probabilistic well-posedness apply, there is no information about the measure transported by the flow. Very recently, a method to prove that the transported measure is absolutely continuous with respect to the initial measure was developed. In such a situation, we have a measure which is quasi-invariant under the corresponding flow. The aim of these lectures is to present all of the above described developments in the context of the nonlinear wave equation.Comment: Lecture notes based on a course given at a CIME summer school in August 201

The phase shift of line solitons for the KP-II equation

The KP-II equation was derived by [B. B. Kadomtsev and V. I. Petviashvili,Sov. Phys. Dokl. vol.15 (1970), 539-541] to explain stability of line solitary waves of shallow water. Stability of line solitons has been proved by [T. Mizumachi, Mem. of vol. 238 (2015), no.1125] and [T. Mizumachi, Proc. Roy. Soc. Edinburgh Sect. A. vol.148 (2018), 149--198]. It turns out the local phase shift of modulating line solitons are not uniform in the transverse direction. In this paper, we obtain the $L^\infty$-bound for the local phase shift of modulating line solitons for polynomially localized perturbations

A pedestrian approach to the invariant Gibbs measures for the 2-d defocusing nonlinear Schrödinger equations

42 pagesInternational audienceWe consider the defocusing nonlinear Schr\"odinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in $\R^2$. Our aim is to give a pedagogic and self-contained presentation on the Wick renormalization in terms of the Hermite polynomials and the Laguerre polynomials and construct the Gibbs measures corresponding to the Wick ordered Hamiltonian. Then, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure

Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation

We consider the cubic fourth order nonlinear Schr\"odinger equation on the circle. In particular, we prove that the mean-zero Gaussian measures on Sobolev spaces $H^s(\mathbb{T})$, $s > \frac34$, are quasi-invariant under the flow.Comment: 41 pages. To appear in Probab. Theory Related Field

Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation

The aim of this paper is the accurate numerical study of the KP equation. In particular we are concerned with the small dispersion limit of this model, where no comprehensive analytical description exists so far. To this end we first study a similar highly oscillatory regime for asymptotically small solutions, which can be described via the Davey-Stewartson system. In a second step we investigate numerically the small dispersion limit of the KP model in the case of large amplitudes. Similarities and differences to the much better studied Korteweg-de Vries situation are discussed as well as the dependence of the limit on the additional transverse coordinate.Comment: 39 pages, 36 figures (high resolution figures at http://www.mis.mpg.de/preprints/index.html

Neurotensin(8–13) analogs as dual NTS1 and NTS2 receptor ligands with enhanced effects on a mouse model of Parkinson's disease

: The modulatory interactions between neurotensin (NT) and the dopaminergic neurotransmitter system in the brain suggest that NT may be associated with the progression of Parkinson's disease (PD). NT exerts its neurophysiological effects by interactions with the human NT receptors type 1 (hNTS1) and 2 (hNTS2). Therefore, both receptor subtypes are promising targets for the development of novel NT-based analogs for the treatment of PD. In this study, we used a virtually guided molecular modeling approach to predict the activity of NT(8-13) analogs by investigating the docking models of ligands designed for binding to the human NTS1 and NTS2 receptors. The importance of the residues at positions 8 and/or 9 for hNTS1 and hNTS2 receptor binding affinity was experimentally confirmed by radioligand binding assays. Further in vitro ADME profiling and in vivo studies revealed that, compared to the parent peptide NT(8-13), compound 10 exhibited improved stability and BBB permeability combined with a significant enhancement of the motor function and memory in a mouse model of PD. The herein reported NTS1/NTS2 dual-specific NT(8-13) analogs represent an attractive tool for the development of therapeutic strategies against PD and potentially other CNS disorders

Apple polyphenols in human and animal health*

Apples contain substantial amounts of polyphenols, and diverse phenolics mainly flavonoids and phenolic acids, have been identified in their flesh and skins. This work aimed to analyze the overall landscape of the research literature published to date on apple phenolic compounds in the context of human and animal health. The Web of Science Core Collection electronic database was queried with (apple* polyphenol*) AND (health* OR illness* OR disease* OR medic* OR pharma*) to identify relevant papers covering these words and their derivatives in the titles, abstracts, and keywords. The resulted 890 papers were bibliometrically analyzed. The VOSviewer software was utilized to produce term maps that illustrate how the frequent phrases fared in terms of publication and citation data. The apple polyphenol papers received global contributions, particularly from China, Italy, the United States, Spain, and Germany. Examples of frequently mentioned chemicals/chemical classes are quercetin, anthocyanin, catechin, epicatechin, and flavonol, while examples of frequently mentioned medical conditions are cardiovascular disease, atherosclerosis, diabetes, Alzheimers disease, and obesity. The potential health benefits of apple polyphenols on humans and animals are diverse and warrant further study.Authors acknowledge the support from The National Centre for Research and Development (NCBR) of Poland (project number POIR.01.01.01-00-0593/18).info:eu-repo/semantics/publishedVersio

Impact of whole genome amplification on analysis of copy number variants

Large-scale copy number variants (CNVs) have recently been recognized to play a role in human genome variation and disease. Approaches for analysis of CNVs in small samples such as microdissected tissues can be confounded by limited amounts of material. To facilitate analyses of such samples, whole genome amplification (WGA) techniques were developed. In this study, we explored the impact of Phi29 multiple-strand displacement amplification on detection of CNVs using oligonucleotide arrays. We extracted DNA from fresh frozen lymph node samples and used this for amplification and analysis on the Affymetrix Mapping 500k SNP array platform. We demonstrated that the WGA procedure introduces hundreds of potentially confounding CNV artifacts that can obscure detection of bona fide variants. Our analysis indicates that many artifacts are reproducible, and may correlate with proximity to chromosome ends and GC content. Pair-wise comparison of amplified products considerably reduced the number of apparent artifacts and partially restored the ability to detect real CNVs. Our results suggest WGA material may be appropriate for copy number analysis when amplified samples are compared to similarly amplified samples and that only the CNVs with the greatest significance values detected by such comparisons are likely to be representative of the unamplified samples