Studies in Applied Mathematics最新文献

The Taylor–Green vortex represents an exact solution of the Navier–Stokes equations in $�R^2$. In this work, an approximation of this solution in two spatial dimensions is proposed for weakly compressible flows. These flows are characterized by small compressibility (or, equivalently, by a small Mach number) and are often employed in computational fluid dynamics to approximate the behaviour of incompressible Newtonian fluids. In this framework, the proposed solution is expected to be a useful benchmark for numerical solvers that implement the weakly compressibility approximation. To this end, some numerical examples are reported in the final section of this work.

In this paper, we address the system made by two coupled one-dimensional Moore–Gibson–Thompson equations

A new method is developed for finding exact solitary wave solutions of a generalized Korteweg–de Vries equation with $�p$-power nonlinearity coupled to a linear Schrödinger equation arising in many different physical applications. This method yields 22 solution families, with $��p�=�1�,�2�,�3�,�4�$. No solutions for $��p�>�1�$ were known previously in the literature. For $��p�=�1�$, four of the solution families contain bright/dark Davydov solitons of the 1st and 2nd kind, obtained in recent literature by basic ansatz applied to the ordinary differential equation (ODE) system for traveling waves. All of the new solution families have interesting features, including bright/dark peaks with (up to) $�p$ symmetric pairs of side peaks in the amplitude and a kink profile for the nonlinear part in the phase. The present method is fully systematic and involves several novel steps that reduce the traveling wave ODE system to a single nonlinear base ODE for which all polynomial solutions are found by symbolic computation. It is applicable more generally to other coupled nonlinear dispersive wave equations  as well as to nonlinear ODE systems of generalized Hénon–Heiles form.

In this work, we analyze the asymptotic behaviors of high-order rogue wave solutions with multiple large parameters and discover novel rogue wave patterns, including modified claw-like, one triple root (OTR)-type, modified OTR-type, two triple roots (TTR)-type, semimodified TTR-type, and modified TTR-type patterns. A correlation is established between these rogue wave patterns and the root structures of the Adler–Moser polynomials with multiple roots. At the positions in the $��(�x�,�t�)�$-plane corresponding to simple roots of the Adler–Moser polynomials, these high-order rogue wave patterns asymptotically approach first-order rogue waves. At the positions in the $��(�x�,�t�)�$-plane corresponding to multiple roots of the Adler–Moser polynomials, these rogue wave patterns asymptotically tend toward lower-order fundamental rogue waves, dispersed first-order rogue waves, or mixed structures of these rogue waves. These structures are related to the root structures of special Adler–Moser polynomials with new free parameters, such as the Yablonskii–Vorob'ev polynomial hierarchy, among others. Notably, the positions of the fundamental lower-order rogue waves or mixed structures in these rogue wave patterns can be controlled freely under specific conditions.

In this paper, we study a dynamic model of hepatitis C virus (HCV) infection with density-dependent proliferation of uninfected and infected hepatocytes and two time delays, which is derived from a three-dimensional model by the quasi-steady-state approximation. The model can exhibit forward bifurcation or backward bifurcation, and an explicit control threshold parameter $�R_c$ is obtained for the case of backward bifurcation. It is shown that if the proliferation rate of infected hepatocytes is greater than the proliferation rate of uninfected hepatocytes by a certain amount, it becomes more difficult for the virus to be removed. The model has rich dynamical properties: (i) In some parameter regions, bistability can occur; (ii) both time delays $�{\tau }_1$ (virus-to-cell delay) and $�{\tau }_2$ (cell-to-cell delay) can lead to Hopf bifurcations; (iii) same length of time delays $�{\tau }_1$ and $�{\tau }_2$ can lead to at most one stability switch, but different time delays can lead to multiple stability switches. Several sufficient conditions for the global stability of the disease-free equilibrium and the endemic equilibrium are obtained for both forward and backward bifurcation scenarios. In particular, several sharp results on global stability are obtained. Theoretical and numerical results portray the complexity of viral evolutionary dynamics in chronic HCV-infected patients.