arXiv subjects
Publications and source records attributed to Md Shah Alam.
This paper studies the global existence of component-wise nonnegative solutions of the Gray-Scott model in $Ω\subset \mathbb{R}^n$, $n \ge 1$, with a mixture of both local and nonlocal diffusion operators. We use semigroup theory with duality arguments to establish the global existence and boundedness of solutions of our model. We also present the patterns of both local and mixed Gray-Scott models.
We study a class of semilinear reaction-diffusion systems with nonlocal diffusion on a bounded domain $Ω$ in $\mathbb{R}^n$ with smooth boundary. The initial data is assumed to be component-wise nonnegative and bounded, and the reaction vector field is assumed to be quasi-positive and satisfy a generalized mass control condition. We obtain global existence and uniqueness of component-wise nonnegative solutions, and when the reaction vector field satisfies a linear intermediate sum condition, we establish the uniform boundedness of solutions in $L^p(Ω)$ for all $2 \le p<\infty$ on bounded time intervals independent of the kernel of the nonlocal diffusion operator. This allows us to generalize a recent diffusive limit result of Laurencot and Walker \cite{laurencot2023nonlocal}. We also analyze a class of $m$-component reaction-diffusion systems in which some of the components diffuse nonlocally and the other components diffuse locally, and establish both global existence and a diffusive limit.
We analyze the Gray-Scott reaction--diffusion system on $Ω\subset\mathbb{R}^n$ ($n\ge 1$) with mixed diffusion combining local and nonlocal operators. Using semigroup methods and duality estimates, we prove global existence of component-wise nonnegative solutions and establish uniform-in-time bounds. Numerical simulations illustrate pattern formation and highlight qualitative differences between the purely local and mixed-diffusion models.
We study the global existence of solutions to a class of nonlocal Geirer-Meinhardt system. This is a two component reaction-diffusion model on a bounded domain in $\mathbb{R}^n$, $n \ge 1$, with nonlocal diffusion given by a nonlocal convolution operator. We have used semigroup theory and derive estimate to guarantee global existence. Then we build an $L^b$ functional to bound our solution independent of the nonlocal convolution kernel for $2 \le b < \infty$. Next, we have used this result to obtain a diffusive limit similar to \cite{laurenccot2023nonlocal} for our model. We also numerically simulate our model to show the formation of patterns by this model and compare the results with the patterns with the traditional local/classical Geirer-Meinhardt model.
This work studies a nonlocal extension of the Klausmeier vegetation model in $\mathbb{R}^N$ $(N \ge 1)$ that incorporates both local and nonlocal diffusion. The biomass dynamics are driven by a nonlocal convolution operator, representing anomalous and faster dispersal than the standard Laplacian acting on the water component. Using semigroup theory combined with a duality argument, we establish global well-posedness and uniform boundedness of classical solutions. Numerical simulations based on the Finite Difference Method with Forward Euler integration illustrate the qualitative effects of nonlocal diffusion and kernel size on vegetation patterns. The results demonstrate that nonlocal interactions significantly influence the spatial organization of vegetation, producing richer and more coherent structures than those arising in the classical local model.
Chiral nanostructures offer the ability to respond to the vector nature of a light beam at the nanoscale. While naturally chiral materials offer a path towards scalability, engineered structures offer a path to wavelength tunability through geometric manipulation. Neither approach, however, allows for temporal control of chirality. Therefore, in the best of all worlds, it is crucial to realize chiral materials that possess the quality of scalability, tailored wavelength response, and dynamic control at high speeds. Here, a new class of intrinsically chiral phase change nanomaterials (PCNMs) is proposed and explored, based on a scalable bottom-up fabrication technique with a high degree of control in three dimensions. Angular resolved Mueller Matrix and spectroscopic ellipsometry are performed to characterize the optical birefringence and dichroism, and a numerical model is provided to explain the origin of optical activity. This work achieves the critical goal of demonstrating high-speed dynamic switching of chirality over 50,000 cycles via the underlying PCNM.
Magnon spintronics is a prosperous field that promises beyond-CMOS technology based on elementary excitations of the magnetic order that act as information carriers for future computational architectures. Unidirectional propagation of spin waves is key to the realization of magnonic logic devices. However, previous efforts to enhance the Damon-Eshbach-type nonreciprocity did not realize (let alone control) purely unidirectional propagation. Here we experimentally demonstrate excitations of unidirectional exchange spin waves by a nanoscale magnetic grating consisting of Co nanowires fabricated on an ultrathin yttrium iron garnet film. We explain and model the nearly perfect unidirectional excitation by the chirality of the magneto-dipolar interactions between the Kittel mode of the nanowires and the exchange spin waves of the film. Reversal of the magnetic configurations of film and nanowire array from parallel to antiparallel changes the direction of the excited spin waves. Our results raise the prospect of a chiral magnonic logic without the need for fragile surface states.