Applied Mathematics Letters最新文献

This paper provides an improved exponential growth estimate, surpassing the growth rate given in the previous work. This finding elucidates the impact of the power index $k$ in the logarithmic nonlinearity $uln{\left|u\right|}^k$ on the growth behavior of the solution to the initial boundary value problem for the one-dimensional sixth-order nonlinear Boussinesq equation with logarithmic nonlinearity.

In this paper, we study the uniqueness of the stationary sonic–subsonic solution to the isentropic hydrodynamic model of semiconductors with sonic boundary. We provide a new method to improve the proof of the uniqueness of the steady-state sonic–subsonic solution, even for the general isentropic case. In detail, we apply the exponential variation method combining a series of modifications with respect to the degeneracy of electrons at the boundary. The proposed method in the present paper is much simpler and perfecter than the existing methods.

In this letter we report a new type of multi-soliton solutions for the modified Korteweg–de Vries (mKdV) equation. These solutions contain $\tau$ functions of the trigonometric solitons and classical solitons simultaneously. A new bilinear form of the mKdV equation is introduced to derive these solutions. The obtained solutions display as solitons living on a periodic background, which are analyzed and illustrated.

In this paper, an extended (3+1)-dimensional nonlinear Schrödinger equation is studied. By using similarity transformation, some exact solutions of this equation are obtained, which include soliton solutions and periodic function solutions, its nonlinear spatial modulation and external potential are affected by time and space. Based on the solution obtained previously, some figures are displayed.

In this paper, we investigate a generalized (2+1)-dimensional nonlinear wave equation characterizing nonlinear waves in a fluid or solid. We use the Painlevé analysis to test the integrability of that equation. In order to research the modulation instability (MI) of that equation, we obtain the one-soliton solutions of that equation through the Hirota bilinear method. The propagation velocity formula and characteristic line of the one soliton are derived. Then, we perform the MI to that equation through the standard linear stability. We study the distribution of the MI gain under the parameters $K$, $\Omega$ and $R$, which are the perturbation wave numbers in $x$, $t$ directions and the initial amplitude, respectively. We find that the amplitude, bandwidth and distance to the line $\Omega =0$ or $R=0$ of the MI gain vary as the $K$ or $R$ value changes. Finally, we analyze the bifurcation behavior of the system using direction field maps and investigate its chaotic behavior under the influence of a periodic external force using phase portraits.

Markov Chain Monte Carlo algorithms, the method of choice to sample from generic high-dimensional distributions, are rarely used for continuous one-dimensional distributions, for which more effective approaches are usually available (e.g. rejection sampling). In this work we present a counter-example to this conventional wisdom for the von Mises distribution, a maximum-entropy distribution over the circle. We show that Hamiltonian Monte Carlo with Laplacian momentum has exactly solvable equations of motion and, with an appropriate travel time, the Markov chain has negative autocorrelation at odd lags for odd observables and yields a relative effective sample size bigger than one.

In this short paper, we are concerned with a mathematical model, which is the variant of non-isentropic system of polytropic gas. The main contribution is to prove the a-priori $L^{\infty }$ estimates for the viscosity approximation solutions and to obtain a new application, on a large hyperbolic system of three equations, of the global existence framework of weak solutions in the paper “Existence of Global Entropy Solutions to a Nonstrictly Hyperbolic System” (Lu, 2005).

We reconsider an SIS epidemic model studied by Cui et al. [1]. The model is shown that saturation recovery leads to backward bifurcation, Hopf bifurcation and codimension 2 Bogdanov–Takens bifurcation. However, for the case when the Bogdanov–Takens bifurcation of codimension 2 is degenerate, the types and codimensions of Bogdanov–Takens bifurcation have not been investigated. In this paper we prove that this same model can undergo cusp type Bogdanov–Takens bifurcations of codimensions 3. Hence, more complex new phenomena, including degenerate Hopf bifurcation, degenerate homoclinic bifurcation and saddle–node bifurcation of limit cycles, exhibit. Furthermore, we get the bifurcation diagram of codimension 3 Bogdanov–Takens bifurcation with cusp type of the SIS epidemic model.

In this letter, we propose and rigorously analyze a fully implicit difference scheme for the derivative nonlinear Schrödinger equation. We show that the numerical scheme at least preserves two discrete conserved quantities. Next, to facilitate error estimate, the numerical scheme is converted into an equivalent system, which can be regarded as one-stage Gaussian–Legendre Runge–Kutta method in time. Furthermore, with the help of the cut-off function technique, we prove the convergence of the equivalent system for the first time with the convergence order $O({\tau }^2+h^2)$ under discrete $L^{\infty }$-norm without any restriction on step ratio. Finally, the numerical results confirm theoretical findings and capacity in long-time simulations.

The usual definition of scaled monomials found in polytopal finite elements literature leads to elemental matrices with an unnecessarily high condition number. A trivial but apparently overlooked rescaling significantly improves the situation. The extent of the improvement is demonstrated numerically.