Wigner instability analysis of the damped Hirota equation

We address the modulation instability of the Hirota equation in the presence of stochastic spatial incoherence and linear time-dependent amplification/attenuation processes via the Wigner function approach. We show that the modulation instability remains baseband type, though the damping mechanisms substantially reduce the unstable spectrum independent of the higher-order contributions (e.g. the higher-order nonlinear interaction and the third-order dispersion). Additionally, we find out that the unstable structure due to the Kerr interaction exhibits a significant resilience to the third-order-dispersion stabilizing effects in comparison with the higher-order nonlinearity, as well as a moderate Lorentzian spectrum damping may assist the rising of instability. Finally, we also discuss the relevance of our results in the context of current experiments exploring extreme wave events driven by the modulation instability (e.g. the generation of the so-called rogue waves).Comment: 7+4 pages. 3 figures. Comments are welcome. To appear in Physica

Riesz transforms, Cauchy-Riemann systems and amalgam Hardy spaces

In this paper we study Hardy spaces $\mathcal{H}^{p,q}(\mathbb{R}^d)$, $0<p,q<\infty$, modeled over amalgam spaces $(L^p,\ell^q)(\mathbb{R}^d)$. We characterize $\mathcal{H}^{p,q}(\mathbb{R}^d)$ by using first order classical Riesz transforms and compositions of first order Riesz transforms depending on the values of the exponents $p$ and $q$. Also, we describe the distributions in $\mathcal{H}^{p,q}(\mathbb{R}^d)$ as the boundary values of solutions of harmonic and caloric Cauchy-Riemann systems. We remark that caloric Cauchy-Riemann systems involve fractional derivative in the time variable. Finally we characterize the functions in $L^2(\mathbb{R}^d) \cap \mathcal{H}^{p,q}(\mathbb{R}^d)$ by means of Fourier multipliers $m_\theta$ with symbol $\theta(\cdot/|\cdot|)$, where $\theta \in C^\infty(\mathbb{S}^{d-1})$ and $\mathbb{S}^{d-1}$ denotes the unit sphere in $\mathbb{R}^d$.Comment: 24 page

MODELING AND NUMERICAL ANALYSIS OF DEAD-CORE PHENOMENA IN CHEMICAL REACTOR ENGINEERING

Diffusion-reaction processes in chemical reactors are often modelled by differential equations of diffusion-reaction type that describe the change in time and space of concentrations of chemical species. In this work, dead-core phenomena, i.e. depleting of chemical species due to the strong catalytic reactions, are studied analytically and numerically for single reactions with power-law kinetics of fractional order. In the first part of this work, dead-core phenomena are presented for 1-D diffusion-reaction problems for catalytic pellets. The point-wise convergence of the classical solution of non-stationary problems to the solution of the steady-state limit is shown analyti cally which constitutes the basis for the construction of an appropriate time-marching scheme to solve numerically stationary diffusion-reaction problems. In the second part of this work, 2-D reactor problems are studied. The spatial discretization is based on Finite Element Method (FEM) where the modified Crank-Nicolson method is used for the time-marching approach. The developed numerical scheme is implemented in MATLAB using Partial Differential Toolbox (PDE Toolbox). The simulation re sults confirm the theoretical predictions. Also, the phenomenon of dead-cores at the boundary is studied numerically for the model of chemical reactor with a catalytic membrane

PENROSE INSTABILITY ANALYSIS IN THE HIROTA EQUATION

In this research, we study the Penrose instability analysis in the Hirota equation, which is a higher-order version of Nonlinear Schrödinger equation. We apply the Wigner function to Hirota equation in order to obtain Wigner-Hirota equation