Studies in Applied Mathematics最新文献

We study orthogonal polynomials on a fully symmetric planar domain $�\Omega$ that is generated by a certain triangle in the first quadrant. For a family of weight functions on $�\Omega$, we show that orthogonal polynomials that are even in the second variable on $�\Omega$ can be identified with orthogonal polynomials on the unit disk composed with a quadratic map, and the same phenomenon can be extended to the domain generated by the rotation of $�\Omega$ in higher dimensions. The connection allows an immediate deduction of results for approximation and Fourier orthogonal expansions on these fully symmetric domains. It applies, for example, to analysis on a sector of a disk in two variables and the conic domain derived from the rotation of a sector in high dimensions.

This paper deals with the Sturm–Liouville operators with distribution potentials of the space $�W_2^{�-�1�}$ on a metric tree. We study an inverse spectral problem that consists in the recovery of the potentials from the characteristic functions related to various boundary conditions. We prove the uniform stability of this inverse problem for potentials in a ball of any fixed radius, as well as the local stability under small perturbations of the spectral data. Our approach is based on a stable algorithm for the unique reconstruction of the potentials relying on the ideas of the method of spectral mappings.

We investigate a reaction–diffusion predator–prey model incorporating indirect prey-taxis, where predators respond to chemical cues released by prey rather than directly tracking prey density. The model incorporates two biologically relevant boundary conditions: (i) homogeneous Dirichlet conditions, representing lethal habitat edges, and (ii) nonlinear boundary fluxes accounting for density-dependent emigration. We establish the global existence and boundedness of solutions under biologically reasonable assumptions, proving that indirect prey-taxis prevents finite-time blowup, even in higher-dimensional settings. Using spectral theory and bifurcation analysis, we characterize the existence and stability of semitrivial and coexistence steady states, identifying critical thresholds for predator conversion efficiency and prey growth rate that determine long-term dynamics. Finally, we classify the long-term dynamics into distinct ecological regimes (extinction, prey dominance, and coexistence) and quantify how boundary conditions modify these regimes through their influence on stability thresholds. These results rigorously demonstrate the stabilizing role of indirect taxis in spatial ecological systems while advancing the mathematical theory of nonlinear boundary value problems in biological contexts.

The study investigates solute dispersion and mass transport phenomena in a composite channel partially filled with a fluid layer with a variable viscosity, sandwiched between two anisotropic porous layers. The motivation for this work comes from the study conducted by Ling, Tartakovsky, and Battiato, who investigated the dispersion phenomena in a fluid layer with a constant viscosity confined between two isotropic porous layers. We obtain an analytical expression for the dispersion coefficient within the fluid layer in connection with the channel matrix properties such as anisotropic permeability ratio, angle, stress jump coefficient, and viscosity ratio. We have compared the one-dimensional upscaled model accuracy with the two-dimensional numerical solution. We present the asymptotic behavior of the dispersion coefficient of the fluid layer for the corresponding clear flow and pure Darcy situations. We see that the dispersion coefficient for the coupled channel matrix situation when the porous nature of the matrix prevails is always less than that of the clear flow situation, that is, when the porous matrix is highly permeable (equivalent to a homogeneous fluid layer). The impact of the anisotropic properties of the porous medium, stress jump coefficient, and viscosity ratio is more prominent on the dispersion coefficient in the convective zone. This model provides valuable insights into the solute transport process within blood arteries, where factors such as the variable viscosity of blood and the anisotropic properties of the endothelium glycocalyx play a crucial role in solute movement.

A new integrable nonlinear evolution equation is proposed, which is a positive flow of the modified short pulse equations. Based on spectral analysis of the Lax pair and the inverse scattering technique, combined with specific spectral function and variable transformations, the initial value problem of the new positive flow modified short pulse equation is converted into a $��2�\times �2�$ matrix oscillatory Riemann–Hilbert problem. The various contour deformations and the principles behind them are elaborated in detail. Then, through appropriate transformations and strict error estimations, the original Riemann–Hilbert problem is simplified into a model problem, and the solution to this model problem can be explicitly expressed using the parabolic cylinder functions. Finally, the long-time asymptotic behavior of the initial value problem of the solution for the new positive flow modified short pulse equation is obtained by employing the nonlinear steepest descent method.

This paper presents a tropical geometry-based edge detection framework that reformulates convolution and gradient operations using min-plus and max-plus algebra. By emphasizing dominant intensity variations, the tropical formulation yields sharper and more continuous edge representations. Three core variants are explored: an adaptive threshold-based method, a multi-kernel min-plus approach, and a max-plus formulation that promotes structural continuity. The framework integrates multi-scale filtering, Hessian-based refinement, and wavelet shrinkage to enhance edge transitions while maintaining computational efficiency. Experiments conducted on MATLAB built-in grayscale and color images demonstrate that tropical formulations, when integrated with classical operators such as Canny and LoG, improve boundary localization in low-contrast and textured regions. Quantitative evaluation using standard edge quality metrics confirms favorable clarity, noise resilience, and structural coherence. These findings underscore the potential of tropical algebra as a scalable, robust, and mathematically grounded alternative for practical edge detection tasks.

We investigate rogue wave formation and spectral downshifting in the higher-order nonlinear Schrödinger (HONLS) equation and its dissipative extensions: the nonlinear mean-flow damping model (NLD-HONLS) and the viscous damping model (V-HONLS). Using Floquet spectral analysis, we characterize (i) the organization of the underlying dynamical background and (ii) the structure of the emergent rogue waves, distinguishing soliton-like rogue waves (SRWs)—which are sharply localized and spectrally coherent—from more diffuse rogue-wave events that lack coherent localization. In the conservative HONLS, SRWs arise only for sufficiently steep initial data, with the evolution intermittently switching between SRW formation and disordered multi-mode dynamics, while moderately steep initial data produce only broader, less coherent rogue waves.

Nonlinear mean-flow damping in the NLD-HONLS model suppresses disorder and promotes a sustained, well-organized Floquet spectrum that supports persistent soliton-like states from which SRWs emerge. In contrast, viscous damping in the V-HONLS model generates a disordered Floquet spectral evolution, broader rogue wave events, and enhanced phase variability. Furthermore, the NLD-HONLS exhibits a close link between rogue wave events and the time of permanent spectral downshift, whereas these phenomena appear decoupled in the V-HONLS model.

These results clarify how dissipation type and wave steepness interact to regulate coherence, rogue wave formation, and spectral downshifting in near-integrable wave systems. Moreover, the persistent coherent states identified in the NLD-HONLS align with the focused, highly organized wave groups observed immediately prior to wave breaking in laboratory experiments, providing a Floquet spectral framework that is conceptually relevant to pre-breaking wave dynamics.

This paper presents a detailed examination of the propagation of shock waves in a medium that exhibits mixed-type weak nonlinearity. The governing equations are the one-dimensional Euler equations, supplemented by the van der Waals equation of state. We consider an undisturbed state in which weak nonlinearity causes the fundamental derivative to change sign twice, leading to the appearance of quartic nonlinearity. This behavior leads to the formation of both expansion and compression waves in a single pulse. We conduct a comprehensive analytical study of the interactions between these waves that reveals several interesting and unconventional results. These include the formation of multiple precursor waves that can extinguish in finite time, the occurrence of a centered wave fan during the interaction, and the splitting of a single shock wave into two distinct shock waves, which are not seen in cases where quartic nonlinearity is absent. Additionally, we investigate how the parameters of non-ideal fluids affect various interaction times and the overall propagation of waves.

In this paper, we consider a gravity-driven unsaturated two-phase flow with nonequilibrium formalism proposed by Hassanizadeh and coworkers. The nonequilibrium effect stems from the dynamic capillary pressure in porous media flow. Consequently, the resulting model is a nonlinear, degenerate pseudo-parabolic diffusion–convection equation. We delve into illustrating and analyzing how the dynamic capillary effect and gravity introduce qualitative differences in the local behavior of wetting saturation fronts in the vicinity of the interface. The underlying idea is to analyze the traveling wave solutions and subsequently perform an asymptotic analysis on solutions with initial support in a half line.