n this paper, a discrete version of the Eckhaus equation is introduced. The discretiza- tion is obtained by considering a discrete analog of the transformation taking the con- tinuous Eckhaus equation to the continuous linear, free Schr ̈odinger 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.

### On a "Quasi" Integrable Discrete Eckhaus Equation

#### Abstract

n this paper, a discrete version of the Eckhaus equation is introduced. The discretiza- tion is obtained by considering a discrete analog of the transformation taking the con- tinuous Eckhaus equation to the continuous linear, free Schr ̈odinger 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.
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2005
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11391/24112`
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