Studies in Applied Mathematics最新文献

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.

The exact solutions for the Navier–Stokes–Fourier equations in the case of plane shock waves for dilute monatomic gases with constant transport coefficients were found by Becker in 1922. The solutions obtained are limited for a Prandtl's number given by $��P�r�=�3�/�4�$. Besides the solutions for the speed and temperature, profiles were given in an implicit way. In this paper, we consider Becker's model to find some exact explicit solutions for dilute polyatomic gases.

We consider a nonconservative nonlinear Schrödinger equation (NCNLS) with time-dependent coefficients, inspired by a water waves problem. This problem does not have mass or energy conservation, but instead mass and energy change in time under explicit balance laws. In this paper, we extend to the particular NCNLS two numerical schemes which are known to conserve energy and mass in the discrete level for the cubic nonlinear Schrödinger equation. Both schemes are second-order accurate in time, and we prove that their extensions satisfy discrete versions of the mass and energy balance laws for the NCNLS. The first scheme is a relaxation scheme that is linearly implicit. The other scheme is a modified Delfour–Fortin–Payre scheme, and it is fully implicit. Numerical results show that both schemes capture robustly the correct values of mass and energy, even in strongly nonconservative problems. We finally compare the two numerical schemes and discuss their performance.

We perform the so-called rigid lid limit on different shallow water models such as the abcd Bousssinesq systems or the Green–Naghdi equations. To do so, we consider an appropriate nondimensionalization of these models where two small parameters are involved: the shallowness parameter $�\mu$ and a parameter $�\epsilon$ which can be interpreted as a Froude number. When the parameter $�\epsilon$ tends to zero, the surface deformation formally goes to the rest state, hence the name rigid lid limit. We carefully study this limit for different topologies. We also provide rates of convergence with respect to $�\epsilon$ and careful attention is given to the dependence on the shallowness parameter $�\mu$.

In this paper, we derive a higher-order Korteweg–de Vries (HKdV) equation as a model to describe the unidirectional propagation of waves on an internal interface separating two fluid layers of varying densities. Our model incorporates underlying currents by permitting a sheared current in both fluid layers, and also accommodates the effect of the Earth's rotation by including Coriolis forces (restricted to the Equatorial region). The resulting governing equations describing the water wave problem in two fluid layers under a “flat-surface” assumption are expressed in a general form as a system of two coupled equations through Dirichlet–Neumann (DN) operators. The DN operators also facilitate a convenient Hamiltonian formulation of the problem. We then derive the HKdV equation from this Hamiltonian formulation, in the long-wave, and small-amplitude, asymptotic regimes. Finally, it is demonstrated that there is an explicit transformation connecting the HKdV we derive with the following integrable equations of a similar type: KdV5, Kaup–Kuperschmidt equation, Sawada–Kotera equation, and Camassa–Holm and Degasperis–Procesi equations.

In this work, we explore the Riemann problem of the Fokas–Lenells (FL) equation given initial data in the form of a step discontinuity by employing the Whitham modulation theory. The periodic wave solutions of the FL equation are characterized by elliptic functions along with the Whitham modulation equations. Moreover, we find that the $�\pm$ signs for the velocities of the periodic wave solutions remain unchanged during propagation. Thus, when analyzing the propagation behavior of solutions, it is necessary to separately consider the clockwise (negative velocity) and counterclockwise (positive velocity) cases. In this regard, we present the classification of the solutions to the Riemann problem of the FL equation in both clockwise and counterclockwise cases for the first time.

The Riemann problem for the discrete conservation law $��2�{\dot{u}}_n�+�u_{�n�+�1�}^2�-�u_{�n�-�1�}^2�=�0�$ is classified using Whitham modulation theory, a quasi-continuum approximation, and numerical simulations. A surprisingly elaborate set of solutions to this simple discrete regularization of the inviscid Burgers' equation is obtained. In addition to discrete analogs of well-known dispersive hydrodynamic solutions—rarefaction waves (RWs) and dispersive shock waves (DSWs)—additional unsteady solution families and finite-time blowup are observed. Two solution types exhibit no known conservative continuum correlates: (i) a counterpropagating DSW and RW solution separated by a symmetric, stationary shock and (ii) an unsteady shock emitting two counterpropagating periodic wavetrains with the same frequency connected to a partial DSW or an RW. Another class of solutions called traveling DSWs, (iii), consists of a partial DSW connected to a traveling wave comprised of a periodic wavetrain with a rapid transition to a constant. Portions of solutions (ii) and (iii) are interpreted as shock solutions of the Whitham modulation equations.

We consider the stability of bound-state solutions of a family of regularized nonlinear Schrödinger equations which were introduced by Dumas et al. as models for the propagation of laser beams. Among these bound-state solutions are ground states, which are defined as solutions of a variational problem. We give a sufficient condition for existence and orbital stability of ground states, and use it to verify that ground states exist and are stable over a wider range of nonlinearities than for the nonregularized nonlinear Schrödinger equation. We also give another sufficient and almost necessary condition for stability of general bound states, and show that some stable bound states exist which are not ground states.

A weak formulation is devised for the $��K�(�m�,�n�)�$ equation, which is a nonlinearly dispersive generalization of the gKdV equation  having compacton solutions. With this formulation, explicit weak compacton solutions are derived, including ones that do not exist as classical (strong) solutions. Similar results are obtained for a nonlinearly dispersive generalization of the gKP equation in two dimensions, which possesses line compacton solutions.