Studies in Applied Mathematics最新文献

The Cauchy problem of the modified Camassa–Holm (CH) positive flow with the Schwartz initial data is studied by using the Riemann–Hilbert (RH) approach and nonlinear steepest descent method. Due to the existence of energy-dependent potential, the spectral analysis of Lax pairs is extremely difficult. We have to go out of our way and introduce some spectral function transformations, from which a basic RH problem is constructed with the aid of the inverse scattering transformation. Then we convert the basic RH problem into a regular RH problem and take into account the reorientation, local parametrices, and main contributions. Finally, we obtain the long-time asymptotic behavior of solution of the modified CH positive flow.

In this article, an improved fast two-grid mixed element algorithm is developed for a nonlinear time fractional pseudo-hyperbolic wave equation model. By introducing two auxiliary variables, the original problem is transformed into a lower-order coupled system with three equations, and then a fully discrete nonlinear mixed element system is formulated, where the spatial direction is approximated by the provided mixed element method, and the temporal direction is discretized using the $��L�2�-�1_{\sigma }�$ approximation. To further address the computational time issue caused by nonlinear iterations, a fast two-grid mixed finite element algorithm in space is constructed. The stability and error estimates for the fully discrete improved fast two-grid mixed element algorithm are derived. Finally, by the comparison with the computing results of the nonlinear mixed element algorithm, it is evident that the proposed two-grid mixed finite element algorithm significantly improves computational efficiency.

We derive equations for the slow changes of parameters of the soliton of the 1D nonlinear Dirac, or Gross–Neveu, equation under the action of a small perturbation. Our perturbation theory uses the neutral modes of the linearized operator of the nonlinear Dirac equation. In addition to the conventional soliton parameters such as its frequency (related to the amplitude and width), velocity, and shifts of the center and phase, we also account for an additional parameter, related the so-called Bogoliubov symmetry of the Dirac equation, which was first pointed out almost half a century ago and rediscovered in the last decade. This aspect of our theory allows one to explain both asymmetric changes of the soliton profile and large growth of the soliton amplitude, which was observed in previous studies via numerical simulations.

We study the long-time dynamics of the defocusing nonlinear Schrödinger (NLS) equation. Compared with previous literature, we revisit the direct and inverse scattering map to obtain asymptotics in some weighted energy space that requires less restrictive decay and regularity assumptions. The main result is derived from an application of uniform resolvent bound and an approximation argument in the spirit of Riemann–Lebesgue lemma. As a consequence, our result demonstrates that zeros of the solution to the defocusing NLS equation cannot lie in bounded sets as $��t�\to �\infty �$.

In this paper, we investigate the global dynamics of a two-species competition reaction–diffusion model in a time-varying domain under the homogeneous Dirichlet and Neumann boundary conditions. Under appropriate conditions, we establish the competitive exclusion principle for asymptotically bounded and periodic domains, respectively. By the method of upper and lower solutions and comparison arguments, we prove that one species will exclude the other in an asymptotically unbounded domain. We further apply the analytic results to a Lotka–Volterra competition model for its global dynamics and conduct numerical simulations to illustrate our findings.

In this paper, we first investigate the mathematical aspects of supersonic flow of a Chaplygin gas past a conical wing with diamond-shaped cross sections in the case of a shock wave detached from the leading edges. The flow under consideration is governed by the three-dimensional steady compressible Euler equations. For the Chaplygin gas, all characteristics are linearly degenerate, and shocks are reversible and characteristic. Using these properties, we can determine the location of the shock in advance and reformulate our problem as an oblique derivative problem for a nonlinear degenerate elliptic equation in conical coordinates. By establishing a Lipschitz estimate, we show that the equation is uniformly elliptic in any subdomain strictly away from the degenerate boundary, and then further prove the existence of a solution to the problem via the continuity method and vanishing viscosity method.