Studies in Applied Mathematics最新文献

This paper investigates the formation of singularities in Bernoulli-type free boundary problems for steady axially symmetric inviscid incompressible flows with general vorticity and swirl. We focus on the asymptotic behavior near degenerate points and provide a complete characterization of singular wave profiles on the free boundary. Degenerate points are classified into two types: Type 1, which occur away from the axis of symmetry, and Type 2, which occur on the axis.

By developing a unified framework that combines variational methods for semilinear elliptic equations with novel analytical techniques for axially symmetric Euler systems, we show that the singular profiles at Type 1 degenerate points are fundamentally limited to three canonical forms: Stokes corners, cusps, or horizontal flat profiles. Our analysis employs monotonicity formulas to construct blow-up limits at degenerate points and utilizes concentration-compactness arguments. A key contribution is the derivation of a frequency formula that rigorously excludes the possibility of horizontal flat singularities. For Type 2 degenerate points, we prove that the singular profiles must be cusps.

This paper introduces two new mathematical functions, the sine Krätzel function and the cosine Krätzel function, which extend the classical Krätzel function into the trigonometric domain. These functions are rigorously defined through integral representations and their fundamental properties, such as absolute convergence, generating functions, and derivative formulas, are investigated in detail. The integral transforms of these functions, including Mellin, Fourier, and Laplace transforms, are derived, highlighting their analytical flexibility. Furthermore, the study explores the applications of these functions in wave optics, specifically in the context of Fresnel and Fraunhofer diffraction patterns, demonstrating their utility in understanding light diffraction phenomena. Numerical examples and graphical visualizations are provided to illustrate the influence of key parameters on diffraction patterns, emphasizing the potential of these functions in applied mathematical and physical research.

In this paper, we propose a novel variational model for color–texture image segmentation by embedding the molecular beam epitaxy (MBE) equation into a multi-cue segmentation (MCS) framework. The MBE equation incorporates a fourth-order diffusion term to smooth high-frequency noise while preserving curvature variations, along with a non-equilibrium term to ensure mass conservation and suppress oscillations, thereby eliminating the need for frequent re-initialization. Inspired by the physical principles of crystal film growth, this approach regulates the level set evolution by controlling thin-film growth dynamics, improving both stability and accuracy. We derive the gradient flow equation of the proposed model and prove the existence of a weak solution using the Galerkin approximation method. To solve the model efficiently, we design an implicit–explicit (IMEX) scheme, and employ an additive operator splitting (AOS) method to obtain the diffusion tensor. Extensive experiments demonstrate that the MBE-MCS model achieves more stable level set evolutions, better preserves fine structural details, and delivers superior segmentation accuracy, even for images with noise, sharp corners, and complex backgrounds.

We study a first-order hyperbolic approximation of the nonlinear Schrödinger (NLS) equation. We show that the system is strictly hyperbolic and possesses a modified Hamiltonian structure, along with at least three conserved quantities that approximate those of NLS. We provide families of explicit standing-wave solutions to the hyperbolic system, which are shown to converge uniformly to ground-state solutions of NLS in the relaxation limit. The system is formally equivalent to NLS in the relaxation limit, and we develop asymptotic preserving discretizations that tend to a consistent discretization of NLS in that limit, while also conserving mass. Examples for both the focusing and defocusing regimes demonstrate that the numerical discretization provides an accurate approximation of the NLS solution.

We investigate a diffusive Lotka–Volterra competition model with temporally distributed memory and Dirichlet boundary conditions, focusing on the interaction between a species with memory and one without. The memory-capable species exhibits both self-memory and cross-memory, while the memoryless species relies solely on random diffusion. We analyze the existence and stability of steady-state solutions, including semi-trivial and positive steady states, under two distinct memory kernel cases. In the weak kernel case, where memory fades over time after immediate acquisition, the positive steady-state solution remains locally asymptotically stable for all non-negative delays. In the strong kernel case, where memory follows both an acquisition and decay phase, Hopf bifurcations arise as delay increases, leading to instability and the emergence of nonhomogeneous periodic solutions. Our findings reveal that species with self-memory gain a competitive advantage, increasing their likelihood of survival, while those relying solely on cross-memory face a higher risk of extinction. This contrast underscores the crucial role of different memory types in shaping competitive outcomes.

This paper investigates the nonlinear Schrödinger equation with a singular convolution potential. It demonstrates the local well-posedness of this equation in a modified Sobolev space linked to the energy. Additionally, we derive conditions under which the solutions are uniformly bounded in the energy space. This finding is closely linked to the existence of standing waves for this equation.

A reduction from the Miura transformation of the self-dual Yang–Mills (SDYM) equations to the unreduced Fokas–Lenells (FL) system is described in this paper. It has been known that the SDYM equation can be formulated from the Cauchy matrix schemes of the matrix Kadomtsev–Petviashvili (KP) hierarchy and of the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy. We show that the reduction can be realized in these two Cauchy matrix schemes, respectively. Each scheme allows us to construct solutions for the unreduced FL system. We prove that these solutions obtained from different schemes are equivalent under a certain reflection transformation of coordinates. Using conjugate reduction, we obtain solutions of the FL equation. The paper adds an important example to Ward's conjecture on the reductions of the SDYM equation. It also reveals the Cauchy matrix structures of the Kaup–Newell hierarchy.

We consider long-range variants of Fermi–Pasta–Ulam–Tsingou lattice and, in particular, allow for particles to interact over arbitrarily long distances. We develop sufficient conditions that allow for the construction of solitary wave solutions.

In this paper, we study the stationary solutions of the inflow problem for a compressible model of the viscous ions motion, which is given by the one-dimensional isentropic compressible Navier–Stokes–Poisson equations. The unique existence of the stationary solutions to the one-dimensional isentropic compressible Navier–Stokes–Poisson equations in the half line is shown provided that the boundary data satisfy some smallness conditions. Moreover, the spatial decay rates of the stationary solutions are presented. By taking the accurate analysis of the cubic characteristic equation for the linearized stationary system, the sign of the real part corresponding to the eigenvalues can be sure. Then our results can be proven by the manifold theory and the center manifold theorem.