Studies in Applied Mathematics最新文献

Large-time patterns of general higher-order lump solutions in the Kadomtsev–Petviashvili I (KP-I) equation are investigated. It is shown that when the index vector of the general lump solution is a sequence of consecutive odd integers starting from one, the large-time pattern in the spatial $��(�x�,�y�)�$-plane generically would comprise fundamental lumps uniformly distributed on concentric rings. For other index vectors, the large-time pattern would comprise fundamental lumps in the outer region as described analytically by the nonzero-root structure of the associated Wronskian–Hermite polynomial, together with possible fundamental lumps in the inner region that are uniformly distributed on concentric rings generically. Leading-order predictions of fundamental lumps in these solution patterns are also derived. The predicted patterns at large times are compared to true solutions, and good agreement is observed.

We consider in an infinite horizontal layer, the equations of the viscous compressible magnetohydrodynamic flows subject to the gravitational force. On the upper and lower planes of the layer, we consider homogeneous Dirichlet conditions on the velocity while a large constant vector field is prescribed on the magnetic field. The existence of the global strong solution with small initial data and its asymptotic behavior as time goes to infinity are established.

Differential equations with right-hand side functions that are not everywhere differentiable are referred to as nondifferentiable systems. This paper introduces three novel methods to address stability issues in nondifferentiable systems. The first method extends the linearization method as it fails when the equilibrium is in a nondifferentiable region. We find that the stability of a piecewise differentiable system aligns with the behavior of its subsystems as long as the “distance” between these subsystems is sufficiently small. The second method is to examine the eigenvalues of the symmetric part of the Jacobian matrix in the vicinity of the equilibrium. This method applies to functions with even weaker regularity conditions, and does not require the eigenvalues to have a negative upper bound (or positive lower bound) over the domain. The third method establishes a connection between nondifferentiable systems and their approximate counterparts, revealing that their stability can be consistent under certain conditions. Additionally, we reaffirm the first two results via the approximation method. Examples are provided to illustrate the applications of our main results, including piecewise differentiable systems, general nondifferentiable systems, and realistic scenarios.

We discuss applications of the inverse scattering transform, also known as the nonlinear Fourier transform (NFT) in telecommunications, both for nonlinear optical fiber communication channel equalization and time-domain signal processing techniques. Our main focus is on the challenges and recent progress in the development of efficient numerical algorithms and approaches to NFT implementation.

We consider a family of structured population models from adaptive dynamics in which cells transition through a number of growth states, or classes, before division. We prove the existence and global asymptotic stability of invariant (‘resident') manifolds in that family; furthermore, we re-derive conditions under which scarce mutants can invade established resident populations, and we show the existence of corresponding ‘invasion’ manifolds that are obtained as critical manifolds under the additional assumption that resident has attained quasi-steady state, which induces a separation of scales. Our analysis is based on standard phase space techniques for ordinary differential equations, in combination with the geometric singular perturbation theory due to Fenichel.

A bidisperse porous medium is one with two porosity scales. There are the usual pores known as macropores but also cracks or fissures in the skeleton which give rise to micropores. In this article, we develop and analyze a model for thermal convection where a layer of viscous incompressible fluid overlies a layer of bidisperse porous medium. Care has to be taken with the boundary conditions at the interface of the fluid and the porous material, and this aspect is investigated. We propose two Beavers–Joseph conditions at the interface and we argue that the parameters in these relations should be different since they depend on the macro or micro permeability, and these parameters are estimated from the original experiments of Beavers and Joseph. The situation is one in a layer which is heated from below and under appropriate conditions bimodal neutral curves are found. These can depend on the relative permeability between the macro and micropores, the Beavers–Joseph conditions appropriate to the macro or micropores, the ratio $�\hat{d}$ of the depth $�d$ of the fluid layer to the depth $�d_m$ of the porous layer, or generally the nature of the bidisperse medium.