scholarly journals Global stability and convergence rate of traveling waves for a nonlocal model in periodic media

2012 ◽  
Vol 17 (3) ◽  
pp. 993-1007 ◽  
Author(s):  
Zigen Ouyang ◽  
◽  
Chunhua Ou ◽  
Keyword(s):  
Convergence Rate ◽  
Global Stability ◽  
Traveling Waves ◽  
Stability And Convergence ◽  
Nonlocal Model ◽  
Periodic Media

Nonlinear Periodic Systems: Bands and Localization

2009 ◽  
Author(s):  
Alexander Vakakis
Keyword(s):  
Traveling Waves ◽  
Multiple Scales ◽  
Initial Conditions ◽  
Near Field ◽  
Standing Waves ◽  
Nonlinear Resonance ◽  
Nonlinear Effects ◽  
Periodic Media ◽  
Spatially Extended ◽  
Nonlinear Standing Waves

We consider the dynamics of nonlinear mono-coupled periodic media. When coupling dominates over nonlinearity near-field standing waves and spatially extended traveling waves exist, inside stop and pass bands, respectively, of the nonlinear system. Nonlinear standing waves are analytically studied using a nonlinear normal mode formulation, whereas nonlinear traveling waves are analyzed by the method of multiple scales. When the nonlinear effects are of the same order with the coupling ones a completely different picture emerges, since nonlinear resonance interactions are unavoidable. As a result, infinite families of strongly and weakly localized nonlinear standing waves appear with frequencies lying in pass or stop bands of the corresponding linear periodic medium. Moreover, in the limit of weak coupling these solutions develop sensitive dependence on initial conditions, and the possibility of spatial chaos in the system exists. Some additional results on chaotic dynamics in linear periodic media with strongly nonlinear disorders are reviewed.

L1 scheme on graded mesh for the linearized time fractional KdV equation with initial singularity

2019 ◽  
Vol 10 (01) ◽  
pp. 1941006 ◽  
Author(s):  
Hu Chen ◽  
Xiaohan Hu ◽  
Jincheng Ren ◽  
Tao Sun ◽  
Yifa Tang
Keyword(s):  
Theoretical Analysis ◽  
Convergence Rate ◽  
Fractional Derivative ◽  
Numerical Approximation ◽  
Kdv Equation ◽  
Initial Singularity ◽  
Stability And Convergence ◽  
Fully Discrete ◽  
Graded Mesh ◽  
Galerkin Spectral Method

Numerical approximation for a linearized time fractional KdV equation with initial singularity using [Formula: see text] scheme on graded mesh is considered. It is proved that the [Formula: see text] scheme can attain order [Formula: see text] convergence rate with appropriate choice of the grading parameter, where [Formula: see text] [Formula: see text] is the order of temporal Caputo fractional derivative. A fully discrete spectral scheme is constructed combing a Petrov–Galerkin spectral method for the spatial discretization, and its stability and convergence are theoretically proved. Some numerical results are provided to verify the theoretical analysis and demonstrated the sharpness of the error analysis.

Sign in / Sign up

Export Citation Format

Share Document