Abstract
In this paper, a discrete version of the Eckhaus equation is introduced. The discretization is obtained by considering a discrete analog of the transformation taking the continuous Eckhaus equation to the continuous linear, free Schrodinger 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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This article is part of the special issue published in honour of Francesco Calogero on the occasion of his 70th birthday
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Ablowitz, M.J., Ahrens, C.D. & De Lillo, S. On a “Quasi” Integrable Discrete Eckhaus Equation. J Nonlinear Math Phys 12 (Suppl 1), 1–12 (2005). https://doi.org/10.2991/jnmp.2005.12.s1.1
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DOI: https://doi.org/10.2991/jnmp.2005.12.s1.1