On the initial-value problem to the Degasperis-Procesi equation with linear dispersion
Fei Guo Bao-Feng Feng Hongjun Gao Yue Liu
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
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
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
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.
On the Cauchy problem for the two-component Dullin-Gottwald-Holm system
Yong Chen Hongjun Gao Yue Liu
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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