Journal of Nonlinear Mathematical Physics

Volume 12, Issue Supplement 1, January 2005, Pages 1 - 12

On a "Quasi" Integrable Discrete Eckhaus Equation

Authors
M.J. Ablowitz, C.D. Ahrens
Corresponding Author
M.J. Ablowitz
Available Online 1 January 2005.
DOI
https://doi.org/10.2991/jnmp.2005.12.s1.1How to use a DOI?
Abstract
In this paper, a discrete version of the Eckhaus equation is introduced. The discretiztion is obtained by considering a discrete analog of the transformation taking the cotinuous Eckhaus equation to the continuous linear, free Schrödinger equation. The resulting discrete Eckhaus equation is a nonlinear system of two coupled second-order difference evolution equations. This nonlinear (1+1)-dimensional system is reduced to solving a first-order, ordinary, nonlinear, difference equation. In the real domain, this nonlinear difference equation is effective in reducing the complexity of the discrete Eckhaus equation. But, in the complex domain it is found that the nonlinear difference equation has a nontrivial Julia set and can actually produce chaotic dynamics. Hence, this discrete Eckhaus equation is considered to be "quasi" integrable. The chaotic behavior is numerically demonstrated in the complex plane and it is shown that the discrete Eckhaus equation retains many of the qualitative features of its continuous counterpart.
Open Access
This is an open access article distributed under the CC BY-NC license.

Download article (PDF)

Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
12 - Supplement 1
Pages
1 - 12
Publication Date
2005/01
ISBN
91-974824-3-9
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.2991/jnmp.2005.12.s1.1How to use a DOI?
Open Access
This is an open access article distributed under the CC BY-NC license.

Cite this article

TY  - JOUR
AU  - M.J. Ablowitz
AU  - C.D. Ahrens
PY  - 2005
DA  - 2005/01
TI  - On a "Quasi" Integrable Discrete Eckhaus Equation
JO  - Journal of Nonlinear Mathematical Physics
SP  - 1
EP  - 12
VL  - 12
IS  - Supplement 1
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2005.12.s1.1
DO  - https://doi.org/10.2991/jnmp.2005.12.s1.1
ID  - Ablowitz2005
ER  -