On the dispersionless Kadomtsev-Petviashvili equation in n+1 dimensions: exact solutions, the Cauchy problem for small initial data and wave breaking

We study the (n+1)-dimensional generalization of the dispersionless Kadomtsev-Petviashvili (dKP) equation, a universal equation describing the propagation of weakly nonlinear, quasi one dimensional waves in n+1 dimensions, and arising in several physical contexts, like acoustics, plasma physics and hydrodynamics. For n=2, this equation is integrable, and it has been recently shown to be a prototype model equation in the description of the two dimensional wave breaking of localized initial data. We construct an exact solution of the n+1 dimensional model containing an arbitrary function of one variable, corresponding to its parabolic invariance, describing waves, constant on their paraboloidal wave front, breaking simultaneously in all points of it. Then we use such solution to build a uniform approximation of the solution of the Cauchy problem, for small and localized initial data, showing that such a small and localized initial data evolving according to the (n+1)-dimensional dKP equation break, in the long time regime, if and only if n=1,2,3; i.e., in physical space. Such a wave breaking takes place, generically, in a point of the paraboloidal wave front, and the analytic aspects of it are given explicitly in terms of the small initial data.Comment: 20 pages, 10 figures, few formulas adde

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Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking

We have recently solved the inverse spectral problem for integrable PDEs in arbitrary dimensions arising as commutation of multidimensional vector fields depending on a spectral parameter $\lambda$. The associated inverse problem, in particular, can be formulated as a non linear Riemann Hilbert (NRH) problem on a given contour of the complex $\lambda$ plane. The most distinguished examples of integrable PDEs of this type, like the dispersionless Kadomtsev-Petviashivili (dKP), the heavenly and the 2 dimensional dispersionless Toda equations, are real PDEs associated with Hamiltonian vector fields. The corresponding NRH data satisfy suitable reality and symplectic constraints. In this paper, generalizing the examples of solvable NRH problems illustrated in \cite{MS4,MS5,MS6}, we present a general procedure to construct solvable NRH problems for integrable real PDEs associated with Hamiltonian vector fields, allowing one to construct implicit solutions of such PDEs parametrized by an arbitrary number of real functions of a single variable. Then we illustrate this theory on few distinguished examples for the dKP and heavenly equations. For the dKP case, we characterize a class of similarity solutions, a class of solutions constant on their parabolic wave front and breaking simultaneously on it, and a class of localized solutions breaking in a point of the $(x,y)$ plane. For the heavenly equation, we characterize two classes of symmetry reductions.Comment: 29 page

Use of monomeric and oligomeric flavanols in the dietary management of patients with type 2 diabetes mellitus and microalb

__Background:__ Patients with type 2 diabetes mellitus (T2D) are prone to micro- and macro-vascular complications. Monomeric and oligomeric flavanols (MOF) isolated from grape seeds (Vitis vinifera) have been linked to improved endothelial function and vascular health. The aim of this study is to determine the effect of a daily supplementation of 200 mg MOF on renal endothelial function of patients with T2D and microalbuminuria. __Methods/design:__ For this double-blind, placebo-controlled, randomized, multicenter trial 96 individuals (ages 40-85 years) with T2D and microalbuminuria will be recruited. Participants will be randomly assigned to the intervention group, receiving 200 mg of MOF daily for 3 months, or to the control group, receiving a placebo. The primary endpoint is the evolution over time in albumin excretion rate (AER) until 3 months of intervention as compared with placebo. Secondary endpoints are the evolution over time in established plasma markers of renal endothelial function-asymmetric dimethylarginine (ADMA), soluble vascular cell adhesion molecule-1 (sVCAM-1), soluble intercellular cell adhesion molecule-1 (sICAM-1), interleukin-6 (IL-6), and von Willebrand Factor (vWF)-until 3 months of intervention as compared with placebo. Mixed modeling will be applied for the statistical analysis of the data. __Discussion:__ We hypothesize that T2D patients with microalbuminuria have a medically determined requirement for MOF and that fulfilling this requirement will result in a decrease in AER and related endothelial biomarkers. If confirmed, this may lead to new insights in the dietary management of patients with T2D

Implementation of the kidney team at home intervention:Evaluating generalizability, implementation process, and effects

Research has shown that a home-based educational intervention for patients with chronic kidney disease results in better knowledge and communication, and more living donor kidney transplantations (LDKT). Implementation research in the field of renal care is almost nonexistent. The aims of this study were (1) to demonstrate generalizability, (2) evaluate the implementation process, and (3) to assess the relationship of intervention effects on LDKT-activity. Eight hospitals participated in the project. Patients eligible for all kidney replacement therapies (KRT) were invited to participate. Effect outcomes were KRT-knowledge and KRT-communication, and treatment choice. Feasibility, fidelity, and intervention costs were assessed as part of the process evaluation. Three hundred and thirty-two patients completed the intervention. There was a significant increase in KRT-knowledge and KRT-communication among participants. One hundred and twenty-nine out of 332 patients (39%) had LDKT-activity, which was in line with the results of the clinical trials. Protocol adherence, knowledge, and age were correlated with LDKT-activity. This unique implementation study shows that the results in practice are comparable to the previous trials, and show that the intervention can be implemented, while maintaining quality. Results from the project resulted in the uptake of the intervention in standard care. We urge other countries to investigate the uptake of the intervention

Multidimensional Conservation Laws: Overview, Problems, and Perspective

Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of multidimensional hyperbolic conservation laws are revealed, and some samples of multidimensional systems/models and related important problems are presented and analyzed with emphasis on the prototypes that have been solved or may be expected to be solved rigorously at least for some cases. In particular, multidimensional steady supersonic problems and transonic problems, shock reflection-diffraction problems, and related effective nonlinear approaches are analyzed. A theory of divergence-measure vector fields and related analytical frameworks for the analysis of entropy solutions are discussed.Comment: 43 pages, 3 figure

On the solutions of the second heavenly and Pavlov equations

We have recently solved the inverse scattering problem for one parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields, like the heavenly equation of Plebanski, the dispersionless Kadomtsev - Petviashvili (dKP) equation and the two-dimensional dispersionless Toda (2ddT) equation, as well as with the commutation of one dimensional vector fields, like the Pavlov equation. We also showed that the associated Riemann-Hilbert inverse problems are powerfull tools to establish if the solutions of the Cauchy problem break at finite time,to construct their longtime behaviour and characterize classes of implicit solutions. In this paper, using the above theory, we concentrate on the heavenly and Pavlov equations, i) establishing that their localized solutions evolve without breaking, unlike the cases of dKP and 2ddT; ii) constructing the longtime behaviour of the solutions of their Cauchy problems; iii) characterizing a distinguished class of implicit solutions of the heavenly equation.Comment: 16 pages. Submitted to the: Special issue on nonlinearity and geometry: connections with integrability of J. Phys. A: Math. and Theor., for the conference: Second Workshop on Nonlinearity and Geometry. Darboux day