Initial value problem of the Whitham equations for the Camassa-Holm equation

We study the Whitham equations for the Camassa-Holm equation. The equations are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the initial value problem of the Whitham equations. When the initial values are given by a step function, the Whitham solution is self-similar. When the initial values are given by a smooth function, the Whitham solution exists within a cusp in the x-t plane. On the boundary of the cusp, the Whitham solution matches the Burgers solution, which exists outside the cusp.     

From the Solution of the Tsarev System to the Solution of the Whitham Equations

We study the Cauchy problem for the Whitham modulation equations for increasing smooth initial data. The Whitham equations are a collection of one-dimensional quasi-linear hyperbolic systems. This collection of systems is enumerated by the genus g=0,1,2, ... of the corresponding hyperelliptic Riemann surface. Each of these systems can be integrated by the so-called hodograph transformation introduced by Tsarev. A key step in the integration process is the solution of the Tsarev linear overdetermined system. For each g>0, we construct the unique solution of the Tsarev system, which matches the genus g+1 and g–1 solutions on the transition boundaries.     

Existence of a global solution of the Whitham equations

The Cauchy problem for Whitham equations with monotonic analytic initial data is studied. If the initial data f(u) satisfies the condition f^(2N+1)(u)<0 for all u∈ℝ except a number of isolated points, then the genus of the solution of the Whitham equations is at most equal to N, where 1≤N∈ℕ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 1, pp. 58–71, January, 1999.     

Synthesis and structural characterization of potato starch sponges

Materials with hierarchical porous structures have attracted great attention over the past years since they have been widely used in many industrial applications. Starch stands out among the materials commonly used as sacrificial templates in environmentally friendly applications. In addition, starch is inexpensive, easy to process, and readily available. In this work we present a structural characterization of sponges prepared by freezing and thawing of a starch gel. Samples obtained in this study were examined by scanning electron microscopy (SEM) and X-ray microtomography (μ-CT). SEM tests revealed that these samples show an open macroporous framework with continuous walls. μ-CT tests revealed that it is possible to tailor the pore structure of these materials by changing the starch concentration in gels used in their processing. We noticed that increasing starch concentration from 15 wt.% to 60 wt.% leads to sponges with porosities ranging from about 28% to over 65%. It was also observed that the higher the starch concentration, the greater the mean pore size of the prepared sponge.     

Universality of the Break-up Profile for the KdV Equation in the Small Dispersion Limit Using the Riemann-Hilbert Approach

We obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation

Thermal characterization of Flemion® membranes substituted by alkali metal cations

The thermal behavior of perfluorosulfonated membranes of three equivalent mass (EW=910, 1000 and 1100 g eq⁻¹) has been studied for membranes in acid form and in the alkali metal countercations substituted samples. The water contents of the membranes decrease progressively with increasing EW and the countercations charge density. The monovalent cations substitutions increase the membranes thermal stability. DSC curves show a single endothermic peak around 120°C that give low peak temperature for low EW and high peak temperature for large cations size. The membrane mechanical properties changed for different EW and temperatures of membranes. Stress-strain analysis showed that K⁺ substituted membranes at both temperatures present a highest YM compared to the other alkali cation substitutions. The thermal properties of perfluorosulfonated membranes depend on the water contents, cation size, temperature and also on EW value.     

Soliton versus the gas: Fredholm determinants,analysis, and the rapid oscillations behind the kinetic equation

We analyse the case of a dense modified Korteweg–de Vries (mKdV) soliton gas and its large time behaviour in the presence of a single trial soliton. We show that the solution can be expressed in terms of Fredholm determinants as well as in terms of a Riemann–Hilbert problem. We then show that the solution can be decomposed as the sum of the background gas solution (a modulated elliptic wave), plus a soliton solution: the individual expressions are however quite convoluted due to the interaction dynamics. Additionally, we are able to derive the local phase shift of the gas after the passage of the soliton, and we can trace the location of the soliton peak as the dynamics evolves. Finally, we show that the soliton peak, while interacting with the soliton gas, has an oscillatory velocity whose leading order average value satisfies the kinetic velocity equation analogous to the one posited by V. Zakharov and G. El.     

Bow shocks formed by plasma collisions in laser irradiated semi-cylindrical cavities

The formation of shocks in plasmas created by short pulse laser irradiation (λ = 800 nm, I ≈ 1 × 10¹² W cm^?2) of semi-cylindrical cavities of different materials was studied combining visible and soft X-ray laser interferometry with simulations. The plasma rapidly converges near the axis to form a dense bright plasma focus. Later in time a long lasting bow shock is observed to develop outside the cavity, that is shown to arise from the collision of plasmas originating from within the cavity and the surrounding flat walls of the target. The shock is sustained for tens of nanoseconds by the continuous arrival of plasma ablated from the target walls. The plasmas created from the heavier target materials evolve more slowly, resulting in increased shock lifetimes.