Studies in Applied Mathematics最新文献

In this paper, we consider the instability of a constant equilibrium solution in a general activator–inhibitor (depletion) model with passive diffusion, cross-diffusion, and nonlocal terms. It is shown that nonlocal terms produce linear stability or instability, and the system may generate spatial patterns under the effect of passive diffusion and cross-diffusion. Moreover, we analyze the existence of bifurcating solutions to the general model using the bifurcation theory. At last, the theoretical results are applied to the spatial water–biomass system combined with cross-diffusion and nonlocal grazing and Holling–Tanner predator–prey model with nonlocal prey competition.

We obtain asymptotics of polynomials satisfying the orthogonality relations

We consider a model initial- and Dirichlet boundary–value problem for a nonlinear Schrödinger equation in two and three space dimensions. The solution to the problem is approximated by a conservative numerical method consisting of a standard conforming finite element space discretization and a second-order, linearly implicit time stepping, yielding approximations at the nodes and at the midpoints of a nonuniform partition of the time interval. We investigate the convergence of the method by deriving optimal-order error estimates in the $�L^2$ and the $�H^1$ norm, under certain assumptions on the partition of the time interval and avoiding the enforcement of a Courant-Friedrichs-Lewy (CFL) condition between the space mesh size and the time step sizes.

We focus on the Ablowitz–Ladik equation on the zero background, specifically considering the scenario of $�N$ pairs of multiple poles. Our first goal was to establish a mapping between the initial data and the scattering data, which allowed us to introduce a direct problem by analyzing the discrete spectrum associated with $�N$ pairs of higher-order zeros. Next, we constructed another mapping from the scattering data to a $��2�\times �2�$ matrix Riemann–Hilbert (RH) problem equipped with several residue conditions set at $�N$ pairs of multiple poles. By characterizing the inverse problem on the basis of this RH problem, we are able to derive higher-order soliton solutions in the reflectionless case.

This paper considers a two-species chemotaxis system with chemical signaling loop and Lotka–Volterra competition kinetics under the homogeneous Newman boundary condition in smooth bounded domains. The global existence and boundedness of solutions for the parabolic–elliptic/parabolic–parabolic system are established. In the strong competition case, the global stability of the semitrivial constant steady state is obtained under certain parameter conditions. Linear analyzes and numerical simulations demonstrate that chemical signaling loop can significantly impact population dynamics, and admit the coexistence in the exclusion competitive case, including nonconstant steady states, chaos, and spatially inhomogeneous time-periodic types.

In this paper, we propose and analyze a reaction–diffusion vector–host disease model with advection effect in an one-dimensional domain. We introduce the basic reproduction number (BRN) $�{\Re }_0$ and establish the threshold dynamics of the model in terms of $�{\Re }_0$. When there are no advection terms, we revisit the asymptotic behavior of $�{\Re }_0$ w.r.t. diffusion rate and the monotonicity of $�{\Re }_0$ under certain conditions. Furthermore, we obtain the asymptotic behavior of $�{\Re }_0$ under the influence of advection effects. Our results indicate that when the advection rate is large enough relative to the diffusion rate, $�{\Re }_0$ tends to be the value of local basic reproduction number (LBRN) at the downstream end, which enriches the asymptotic behavior results of the BRN in nonadvection heterogeneous environments. In addition, we explore the level set classification of $�{\Re }_0$, that is, there exists a unique critical surface indicating that the disease-free equilibrium is globally asymptotically stable on one side of the surface, while it is unstable on the other side. Our results also reveal that the aggregation phenomenon will occur, namely, when the ratio of advection rate to diffusion rate is large enough, infected individuals will gather at the downstream end.

We consider the Painlevé asymptotics for a solution of the integrable coupled Hirota equations with a $��3�\times �3�$ Lax pair whose initial data decay rapidly at infinity. Using the Riemann–Hilbert (RH) techniques and Deift–Zhou nonlinear steepest descent arguments, in a transition zone defined by $���|�x�/�t�-�1�/��\left(�12�\alpha �\right)��|��t^{�2�/�3�}�\le �C�$, where $��C�>�0�$ is a constant, it turns out that the leading-order term to the solution can be expressed in terms of the solution of a coupled Painlevé II equations, which are associated with a $��3�\times �3�$ matrix RH problem and appear in a variety of random matrix models.

Steady solutions to the Navier–Stokes equations with internal temperature forcing are considered. The equations are solved in two dimensions using the Boussinesq approximation to couple temperature and density fluctuations. A perturbative Stokes expansion is used to prove that that steady flow variables are parametrically analytic in the size of the forcing. The Stokes expansion is complemented with analytic continuation, via functional Padé approximation. The zeros of the denominator polynomials in the Padé approximants are observed to agree with a numerical prediction for the location of singularities of the steady flow solutions. The Padé representations not only prove to be good approximations to the true flow solutions for moderate intensity forcing, but are also used to initialize a Newton solver to compute large amplitude solutions. The composite procedure is used to compute steady flow solutions with forcing several orders of magnitude larger than the fixed-point method developed in previous work.

A wide class of two-component evolution systems is constructed admitting an infinite-dimensional Lie algebra. Some examples of such systems that are relevant to reaction–diffusion systems with cross-diffusion are highlighted. It is shown that a nonlinear evolution system related to the Ricci flow on warped product manifold, which has been extensively studied by several authors, follows from the above-mentioned class as a very particular case. The Lie symmetry properties of this system and its natural generalization are identified and a wide range of exact solutions is constructed using the Lie symmetry obtained. Moreover, a special case is identified when the system in question is reducible to the fast diffusion equation in one space dimension. Finally, another class of two-component evolution systems with an infinite-dimensional Lie symmetry that possess essentially different structures is presented.