## Journals

- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
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- Numerical Algebra, Control & Optimization
- Electronic Research Announcements
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- AIMS Mathematics

DCDS

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.

DCDS

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.

CPAA

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.

DCDS-B

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.

DCDS

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.

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