Qualitative analysis and explicit traveling wave solutions for the Davey–Stewartson equation

In this paper, we use the bifurcation method of dynamical systems to study the traveling wave solutions for the Davey–Stewartson equation. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow‐up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended. Copyright © 2013 John Wiley & Sons, Ltd.     

Derivation of Korteweg–de Vries flow equations from the short‐wave model equation

We perform a multiple‐time scales analysis and compatibility condition to the short‐wave model equation. We derive Korteweg–de Vries flow equation in the bi‐Hamiltonian form as an amplitude equation. Copyright © 2013 John Wiley & Sons, Ltd.     

BOUNDED TRAVELING WAVE SOLUTIONS OF VARIANT BOUSSINESQ EQUATION WITH A DISSIPATION TERM AND DISSIPATION EFFECT

This article studies bounded traveling wave solutions of variant Boussinesq equation with a dissipation term and dissipation effect on them. Firstly, we make qualitative analysis to the bounded traveling wave solutions for the above equation by the theory and method of planar dynamical systems, and obtain their existent conditions, number, and general shape. Secondly, we investigate the dissipation effect on the shape evolution of bounded traveling wave solutions. We find out a critical value r*which can characterize the scale of dissipation effect, and prove that the bounded traveling wave solutions appear as kink profile waves if |r| ≥ r*; while they appear as damped oscillatory waves if |r| r*. We also obtain kink profile solitary wave solutions with and without dissipation effect. On the basis of the above discussion, we sensibly design the structure of the approximate damped oscillatory solutions according to the orbits evolution relation corresponding to the component u(ξ) in the global phase portraits, and then obtain the approximate solutions(u(ξ), H(ξ)). Furthermore,by using homogenization principle, we give their error estimates by establishing the integral equation which reflects the relation between exact and approximate solutions. Finally, we discuss the dissipation effect on the amplitude, frequency, and energy decay of the bounded traveling wave solutions.     

ERROR ESTIMATION FOR NUMERICAL METHODS USING THE ULTRA WEAK VARIATIONAL FORMULATION IN MODEL OF NEAR FIELD SCATTERING PROBLEM

In this paper, we investigate the use of ultra weak variational formulation to solve a wave scattering problem in near field optics. In order to capture the sub-scale features of waves, we utilize evanescent wave functions together with plane wave functions to approximate the local properties of the field. We analyze the global convergence and give an error estimation of the method. Numerical examples are also presented to demonstrate the effectiveness of the strategy.     

Blow up of critical semilinear wave equations on Schwarzschild spacetime

In this paper, we established the blow up theorem for critical semilinear wave equations with focusing nonlinear term on Schwarzschild spacetime. Concavity method is used to prove the main result, which was introduced by Levine–Payne in the papers Levine and Payne (1974)  and  and Levine (1973) [7] in 1970s. Also, a new auxiliary function with parameter β$\beta$ is constructed following the idea from Payne (1975) [13], in order to guarantee that the result holds without any assumption on the initial data and initial energy.     

Lifespan of classical discontinuous solutions to general quasilinear hyperbolic systems of conservation laws with small BV initial data: Rarefaction waves

In the present paper the author investigates the global structure stability of Riemann solutions for general quasilinear hyperbolic systems of conservation laws under small BV perturbations of the initial data, where the Riemann solution contains rarefaction waves, while the perturbations are in BV but they are assumed to be C¹$C^1$-smooth, with bounded and possibly large C¹$C^1$-norms. Combining the techniques employed by Li–Kong with the modified Glimm’s functional, the author obtains a lower bound of the lifespan of the piecewise C¹$C^1$ solution to a class of generalized Riemann problems, which can be regarded as a small BV perturbation of the corresponding Riemann problem. This result is also applied to the system of traffic flow on a road network using the Aw–Rascle model.     

Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas

The phenomena of concentration and cavitation and the formation of δ-shocks and vacuum states in solutions to the isentropic Euler equations for a modified Chaplygin gas are analyzed as the double parameter pressure vanishes. Firstly, the Riemann problem of the isentropic Euler equations for a modified Chaplygin gas is solved analytically. Secondly, it is rigorously shown that, as the pressure vanishes, any two-shock Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a δ-shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted δ-measure that forms the δ-shock; any two-rarefaction-wave Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a two-contact-discontinuity solution to the transport equations, the nonvacuum intermediate state between the two rarefaction waves tends to a vacuum state. Finally, some numerical results exhibiting the formation of δ-shocks and vacuum states are presented as the pressure decreases.     

Traveling wave solutions in a diffusive system with two preys and one predator

This paper is concerned with the traveling wave solutions in a diffusive system with two preys and one predator. By constructing upper and lower solutions, the existence of nontrivial traveling wave solutions is established. The asymptotic behavior of traveling wave solutions is also confirmed by combining the asymptotic spreading with the contracting rectangles. Applying the theory of asymptotic spreading, the nonexistence of traveling wave solutions is proved.     

Single-machine group scheduling with processing times dependent on position,starting time and allotted resource

This article considers scheduling problems on a single machine with learning effect, deteriorating jobs and resource allocation under group technology (GT) assumption. We assume that the actual processing time of a job depends on the job position, the group position, the starting time and the amount of resource allocated to them concurrently, and the actual setup times of groups depend on the group position and the amount of resource allocated to them concurrently. Two resource allocation functions are examined for minimizing the weighted sum of makespan and total resource cost. We prove that the problems have polynomial solutions under the condition that the number of jobs in each group are the same.     

二维气体动力学中压差方程的特征分解和简单波

利用直接的方法讨论了在自相似平面上气体动力学中二维压差方程的特征分解理论,得到了压强P和特征值A±的特征分解.进一步地,若流动来自常状态,还可得到速度（u,v）的特征分解.由此,可以得到与常状态流动相邻的流动是简单波,并说明了简单波的流动区域是被一族直线所覆盖,且沿着每条直线, （P,u,v）为常数.