In this paper, the gravitational deflection of a relativistic massive neutral particle in the Schwarzschild-de Sitter spacetime is studied via the Rindler–Ishak method in the weak-field limit. When the initial velocity $$v_0$$ v 0 of the particle tends to the speed of light, the result is consistent with that obtained in the previous work for the light-bending case. Our result is reduced to the Schwarzschild deflection angle of massive particles up to the second order, if the contributions from the cosmological constant $$\varLambda $$ Λ are dropped. The observable correctional effects due to the deviation of $$v_0$$ v 0 from light speed on the $$\varLambda $$ Λ -induced contributions to the deflection angle of light are also analyzed.

<p style='text-indent:20px;'>Competition stems from the fact that resources are limited. When multiple competitive species are involved with spatial diffusion, the dynamics becomes even complex and challenging. In this paper, we investigate the invasive speed to a diffusive three species competition system of Lotka-Volterra type. We first show that multiple species share a common spreading speed when initial data are compactly supported. By transforming the competitive system into a cooperative system, the determinacy of the invasive speed is studied by the upper-lower solution method. In our work, for linearly predicting the invasive speed, we concentrate on finding upper solutions only, and don't care about the existence of lower solutions. Similarly, for nonlinear selection of the spreading speed, we focus only on the construction of lower solutions with fast decay rate. This greatly develops and simplifies the ideas of past references in this topic.</p>

The objective of this paper is to extend some results of pioneers for the nonlinear equationmt=(1/2)(1/mk)xxx−(1/2)(1/mk)xintroduced by Qiao. The equivalent relationship of the traveling wave solutions between the integrable equation and the generalized KdV equation is revealed. Moreover, whenk=−(p/q)  (p≠qandp,q∈ℤ+), we obtain some explicit traveling wave solutions by the bifurcation method of dynamical systems.

In this paper, we derive a new integrable equation from the KdV equation and give its Lax pair. By applying the bifurcation method of dynamical systems, cuspons and periodic cuspons of the integrable equation are presented. Numerical simulations of cuspons and periodic cuspons are given to show the correctness of our results.

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