On the initial-value problem to the Degasperis-Procesi equation with linear dispersion
Fei Guo Bao-Feng Feng Hongjun Gao Yue Liu
Discrete & Continuous Dynamical Systems - A 2010, 26(4): 1269-1290 doi: 10.3934/dcds.2010.26.1269
In this paper, we consider the initial-value problem for the Degasperis-Procesi equation with a linear dispersion, which is an approximation to the incompressible Euler equation for shallow water waves. We establish local well-posedness and some global existence of solutions for certain initial profiles and determine the wave breaking phenomena for the equation. Finally, we verify the occurrence of the breaking waves by numerical simulations.
keywords: Degasperis-Procesi equation Global existence. Blow-up Local well-posedness Breaking waves phenomena
Cauchy problem for the Ostrovsky equation
V. Varlamov Yue Liu
Discrete & Continuous Dynamical Systems - A 2004, 10(3): 731-753 doi: 10.3934/dcds.2004.10.731
Considered herein is an initial-value problem for the Ostrovsky equation that arises in modelling the unidirectional propagation of long waves in a rotating homogeneous incompressible fluid. Nonlinearity and dispersion are taken into account, but dissipation is ignored. Local- and global-in-time solvability is investigated. For the case of positive dispersion a fundamental solution of the Cauchy problem for the linear equation is constructed, and its asymptotics is calculated as $t\rightarrow \infty, x/t=$const. For the nonlinear problem solutions are constructed in the form of a series and the analogous long-time asymptotics is obtained.
keywords: Cauchy problem long–time asymptotics. Ostrovsky equation construction of solutions
Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation
Yue Liu
Communications on Pure & Applied Analysis 2008, 7(1): 193-209 doi: 10.3934/cpaa.2008.7.193
We establish a sharp instability theorem for the standing-wave solutions of the inhomogeneous nonlinear Schrödinger equation

$i u_t + \Delta u + V(\epsilon x ) |u|^{p-1} u = 0, \quad x \in \mathbf R^n$

with the critical power $ p = 1 + 4/n, n \ge 2, $ under certain conditions on the inhomogeneous term $ V $ with a small $ \epsilon > 0. $ We also demonstrate that these localized standing-waves converge to standing waves of the nonlinear Schrödinger equation with the homogeneous nonlinearity.

keywords: Nonlienar Schrödinger equation inhomogeneous nonlinearities standing waves stability.
Stability and weak rotation limit of solitary waves of the Ostrovsky equation
Steve Levandosky Yue Liu
Discrete & Continuous Dynamical Systems - B 2007, 7(4): 793-806 doi: 10.3934/dcdsb.2007.7.793
In this paper we study several aspects of solitary wave solutions of the Ostrovsky equation. Using variational methods, we show that as the rotation parameter goes to zero, ground state solitary waves of the Ostrovsky equation converge to solitary waves of the Korteweg-deVries equation. We also investigate the properties of the function $d(c)$ which determines the stability of the ground states. Using an important scaling identity, together with numerical approximations of the solitary waves, we are able to numerically approximate $d(c)$. These calculations suggest that $d$ is convex everywhere, and therefore all ground state solitary waves of the Ostrovsky equation are stable.
keywords: solitary waves stability KdV-like equations variational principles.
Online ordering strategy for the discrete newsvendor problem with order value-based free-shipping
Yong Zhang Huifen Zhong Yue Liu Menghu Huang
Journal of Industrial & Management Optimization 2017, 13(5): 1-14 doi: 10.3934/jimo.2018114

Suppliers always provide free-shipping for retailers whose total order value exceeds or equals an explicit promotion threshold. This paper incorporates a shipping fee in the discrete multi-period newsvendor problem and applies Weak Aggregating Algorithm (WAA) to offer explicit online ordering strategy. It further considers an extended case with salvage value and shortage cost. In particular, online ordering strategies are derived based on return loss function. Numerical examples serve to illustrate the competitive performance of the proposed ordering strategies. Results show that newsvendors' cumulative return losses are affected by the threshold of the order value-based free-shipping. Moreover, the introduction of salvage value and shortage cost greatly improves the competitive performance of online ordering strategies.

keywords: Multi-period newsvendor problem discrete order value-based free-shipping WAA online ordering strategy
On the Cauchy problem for the two-component Dullin-Gottwald-Holm system
Yong Chen Hongjun Gao Yue Liu
Discrete & Continuous Dynamical Systems - A 2013, 33(8): 3407-3441 doi: 10.3934/dcds.2013.33.3407
Considered herein is the initial-value problem for a two-component Dullin-Gottwald-Holm system. The local well-posedness in the Sobolev space $H^{s}(\mathbb{R})$ with $s>3/2$ is established by using the bi-linear estimate technique to the approximate solutions. Then the wave-breaking criteria and global solutions are determined in $H^s(\mathbb{R}), s > 3/2.$ Finally, existence of the solitary-wave solutions is demonstrated.
keywords: Two-component Dullin-Gottwald-Holm system global solutions regularization wave-breaking solitary-wave solutions.

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