Studies in Applied Mathematics最新文献

The Novikov equation is a peakon equation with cubic nonlinearity, which, like the Camassa–Holm and the Degasperis–Procesi, is completely integrable. In this paper, we study the spectral and linear stability of peakon solutions of the Novikov equation. We prove spectral instability of the peakons in $��L^2��\left(�R�\right)��$. To do so, we start with a linearized operator defined on $��H^1��\left(�R�\right)��$ and extend it to a linearized operator defined on weaker functions in $��L^2��\left(�R�\right)��$. The spectrum of the linearized operator in $��L^2��\left(�R�\right)��$ is proven to cover a closed vertical strip of the complex plane. Furthermore, we prove that the peakons are spectrally unstable on $��W^{�1�,�\infty �}��(�R�$

This paper is concerned with the numerical approximation of initial-boundary-value problems of a three-parameter family of Bona–Smith systems, derived as a model for the propagation of surface waves under a physical Boussinesq regime. The work proposed here is focused on the corresponding problem with Dirichlet boundary conditions and its approximation in space with spectral methods based on Jacobi polynomials, which are defined from the orthogonality with respect to some weighted $�L^2$ inner product. Well-posedness of the problem on the corresponding weighted Sobolev spaces is first analyzed and existence and uniqueness of solution, locally in time, are proved. Then, the spectral Galerkin semidiscrete scheme and some detailed comments on its implementation are introduced. The existence of numerical solution and error estimates on those weighted Sobolev spaces are established. Finally, the choice of the time integrator to complete the full discretization takes care of different stability issues that may be relevant when approximating the semidiscrete system. Some numerical experiments illustrate the results.

This paper is concerned with a spatially heterogeneous vector-borne disease model that follows the Fokker–Planck-type diffusion law. One of the significant features in our model is that Fokker–Planck-type diffusion is used to characterize individual movement, which poses new challenges to theoretical analysis. We derive for the first time the variational characterization of basic reproduction ratio $�R_0$ for the model under certain conditions and investigate its asymptotic profiles with respect to the diffusion rates. Furthermore, via overcoming the difficulty of the associated elliptic eigenvalue problem, the asymptotic behaviors of endemic equilibrium for the model are discussed. Our results imply that whether rapid or slow movement of susceptible and infected individuals are conducive to disease control depends on the degree of disease risk in the habitat. Numerically, we verify the theoretical results and detect that Fokker–Planck-type diffusion may amplify the scale of disease infection, which in turn increases the complexity of disease transmission by comparing the impacts of distinct dispersal types on disease dynamics.

In this paper, we study a predator–prey mite model of Leslie type with generalized Holling IV functional response. The model is shown to have very rich bifurcation dynamics, including subcritical and supercritical Hopf bifurcations, degenerate Hopf bifurcation, focus-type and cusp-type degenerate Bogdanov–Takens bifurcations of codimension 3, originating from a nilpotent focus or cusp of codimension 3 that acts as the organizing center for the bifurcation set. Coexistence of multiple steady states, multiple limit cycles, and homoclinic cycles is also found. Interestingly, the coexistence of two limit cycles is guaranteed by investigating generalized Hopf bifurcation and degenerate homoclinic bifurcation, and we also find that two generalized Hopf bifurcation points are connected by a saddle-node bifurcation curve of limit cycles, which indicates the existence of global regime for two limit cycles. Our work extends some results in the literature.

In this paper, we propose a new integrable fractional Fokas–Lenells equation by using the completeness of the squared eigenfunctions, dispersion relation, and inverse scattering transform. To solve this equation, we employ the Riemann–Hilbert approach. Specifically, we focus on the case of the reflectionless potential with a simple pole for the zero boundary condition. And we provide the fractional $�N$-soliton solution in determinant form. In addition, we prove the fractional one-soliton solution rigorously. Notably, we demonstrate that as $��\left|�t�\right|�\to �\infty �$, the fractional $�N$-soliton solution can be expressed as a linear combination of $�N$ fractional single-soliton solutions.

This paper examines an epidemic patch model with mass-action transmission mechanism and asymmetric connectivity matrix. Results on the global dynamics of solutions and the spatial structures of endemic equilibrium (EE) solutions are obtained. In particular, we show that when the basic reproduction number $�R_0$ is less than one and the dispersal rate of the susceptible population $�d_S$ is large, the population would eventually stabilize at the disease-free equilibrium. However, the disease may persist if $�d_S$ is small, even if $��R_0�<�1�$. In such a scenario, explicit conditions on the model parameters that lead to the existence of multiple EE are identified. These results provide new insights into the dynamics of infectious diseases in multipatch environments. Moreover, results in Li and Peng (Stud Appl Math. 2023;150(3):650-704), which is for the same model but with symmetric connectivity matrix, are generalized and improved.

This paper is concerned with a cross-diffusion tumor invasion model with double-taxis effect. We first investigate the global existence of classical solutions of this model in two-dimensional space. The essential difficulty lies in the second-level taxis effect of immune cells on tumor cells, where chemotactic factor (tumor cells) exhibit their own taxis behavior, the double-taxis effect makes us have to use more detailed analysis and calculation, and some new estimation techniques are used. Subsequently, we also investigate the stability of some equilibria. For small proliferation coefficient, we prove the global asymptotic stability or local asymptotic stability of a semitrivial equilibrium point. While, for the other equilibria, the stability analysis is complicated even for some special cases, and both the double chemotactic coefficients will affect the stability of the solution.

We propose a method to numerically compute fractional derivatives (or the fractional Laplacian) on the whole real line via Riesz fractional integrals. The compactified real line is divided into a number of intervals, thus amounting to a multidomain approach; after transformations in accordance with the underlying $�Z_q$ curve ensuring analyticity of the respective integrands, the integrals over the different domains are computed with a Clenshaw–Curtis algorithm. As an example, we consider solitary waves for fractional Korteweg-de Vries equations and compare these to results obtained with a discrete Fourier transform.

In this paper, taking into account the inevitable impact of environmental perturbations on disease transmission, we primarily investigate a stochastic model for measles infection with nonlinear incidence. The transmission rate in this model follows a logarithmic normal distribution influenced by an Ornstein–Uhlenbeck (OU) process. To analyze the dynamic properties of the stochastic model, our first step is to establish the existence and uniqueness of a global solution for the stochastic equations. Next, by constructing appropriate Lyapunov functions and utilizing the ergodicity of the OU process, we establish sufficient conditions for the existence of a stationary distribution, indicating the prevalence of the disease. Furthermore, we provide sufficient conditions for disease elimination. These conditions are derived by considering the interplay between the model parameters and the stochastic dynamics. Finally, we validate the theoretical conclusions through numerical simulations, which allow us to assess the practical implications of the established conditions and observe the dynamics of the stochastic model in action. By combining theoretical analysis and numerical simulations, we gain a comprehensive understanding of the stochastic model's behavior, contributing to the broader understanding of measles transmission dynamics and the development of effective control strategies.