On a "Quasi" Integrable Discrete Eckhaus Equation

DOI

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.

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - M.J. Ablowitz
AU  - C.D. Ahrens
PY  - 2005
DA  - 2005/01/01
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  - 10.2991/jnmp.2005.12.s1.1
ID  - Ablowitz2005
ER  -