Applied Mathematics Letters最新文献

Sasa–Satsuma equation is proposed to model the propagation and interaction of the sub-picosecond or femtosecond pulses in a monomode optical fiber. Different from several integrable equations in the Ablowitz–Kaup–Newell–Segur system, the higher-order zeros of Riemann–Hilbert problem for the Sasa–Satsuma appear in quadruples. A new approach to study the multi-geometric discrete spectral problem with several pairs of zeros for the Sasa–Satsuma equation is proposed. Thus, the complete soliton solutions corresponding to the higher-order zeros with arbitrary geometric and algebraic multiplicities are derived. Moreover, the inelastic interactions between or among the solitons corresponding to the higher-order non-elementary zeros exhibit the shape-changing phenomena.

This study explores multiple soliton and breather solutions on periodic backgrounds in the complex modified Korteweg–de Vries equation. The compact determinant formulas and their detailed derivation process for these solutions are provided via the bilinear method. We confirm that on periodic backgrounds, soliton amplitudes exhibit regular periodic behaviors, while breather amplitudes display quasi-periodic behaviors, as is expected for a breather with one period propagating over a periodic wave with another period. The asymptotic expressions for the solitons and breathers, which establish the high accuracy of the derived solutions, are provided to reveal the soliton and breather dynamics on the periodic backgrounds.

A semilinear initial–boundary value problem with a Caputo time derivative of fractional order $\alpha \in (0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For an L2-type discretization of order $3-\alpha$, we give sharp pointwise-in-time error bounds on graded temporal meshes with arbitrary degree of grading.

In the recent work ([1], 2024) by Y. Li et al., the existence and uniqueness of the two-dimensional compressible subsonic jet flow with general vorticity were established. As a follow-up research, we will investigate the geometry shape of the free boundary for the compressible subsonic rotational jet flow. It is proved that if the nozzle is concave to the fluid, then the free boundary for the jet flow is strictly convex to the fluid. Furthermore, the positivity of the horizontal velocity in the fluid region is also obtained.

We propose an elementary proof based on a penalization technique to show the existence and uniqueness of the solution to a system of variational inequalities modeling the friction-based motion of a two-body crawling system. Here for each body, the static and dynamic friction coefficients are equal.

In this paper we establish the existence of one bounded periodic weak solution for a nonlinear parametric differential problem via variational methods.

The influence of algal cell size on nutrient absorption, and the time delay in reproduction is paramount in biological terms. Furthermore, the control is necessary during algal flocculation harvesting and sewage treatment. Therefore, we investigated the properties of a single cell algal flocculation model with time delay, size structure and control. First, we introduced a dimensionless model. Then, we analyzed the existence of positive equilibrium states and studied the equilibrium states of the model. Finally, the numerical simulation revealed that the model exhibits backward bifurcation.

Using the $\bar{\partial }$-dressing method, we study the general reverse-space nonlocal nonlinear Schrödinger (nNLS) equation. Beginning with a 3 × 3 matrix $\bar{\partial }$-problem, the associated spatial and time spectral problems are obtained through two linear constraint equations. Furthermore, the gauge equivalence between the Heisenberg chain equation and the general reverse-space nNLS equation is established. By employing a recursive operator, a hierarchy for the general reverse-space nNLS equation is proposed. Moreover, by selecting a suitable spectral transformation matrix, the $N$-soliton solutions of the general reverse-space nNLS equation are calculated, yielding the explicit one-soliton and two-soliton solutions.

In this paper, we introduce a novel Gauss–Jacobi quadrature rule designed for infinite intervals, which is specifically applied to the velocity discretization in multi-scale Boltzmann solvers. Our method utilizes a newly formulated bell-shaped weight function for numerical integration. We establish the relationship between this new quadrature and the classical Gauss–Jacobi, as well as the Gauss–Hermite quadrature rules, and we compare the resulting discrete velocity distributions with several commonly used methods. Additionally, we validate the performance of our method through numerical simulations of flows with various Knudsen numbers. The proposed quadrature provides fresh insights into velocity space discretization.

We study the bifurcation behavior of nodal solutions for the Kirchhoff type beam equation $\left(\begin{array}{cc}& u^{\prime \prime \prime \prime }-M\left({\int }_0^1{\left|u^{\prime }\right|}^{2}dx\right)u^{\prime \prime }=\lambda f(x,u),x\in (0,1),\\ & u\left(0\right)=u\left(1\right)=u^{\prime \prime }\left(0\right)=u^{\prime \prime }\left(1\right)=0,\end{array}\right)$ where $\lambda \in R$ is a parameter, $M:[0,\infty )\to [0,\infty )$ and $f:[0,1]\times R\to R$ are smooth functions. We obtain the existence of nodal solutions under some suitable conditions. The proof of our main result is based upon bifurcation techniques.