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C-Integrability Test for Discrete Equations via Multiple Scale Expansions
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2010, том 6
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C-Integrability Test for Discrete Equations via Multiple Scale Expansions
Інший ID:
2010 Mathematics Subject Classification: 34K99; 34E13; 37K10; 37J30
DOI:10.3842/SIGMA.2010.070
DOI:10.3842/SIGMA.2010.070
Посилання:
C-Integrability Test for Discrete Equations via Multiple Scale Expansions / C. Scimiterna, D. Levi // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 27 назв. — англ.
Підтримка:
The authors have been partly supported by the Italian Ministry of Education and Research, PRIN “Nonlinear waves: integrable finite dimensional reductions and discretizations” from 2007 to 2009 and PRIN “Continuous and discrete nonlinear integrable evolutions: from water waves to symplectic maps” from 2010.
Дата:
2010
Переглядів:
539
Завантажень:
279
Анотація:
In this paper, we are extending the well-known integrability theorems obtained by multiple scale techniques to the case of linearizable difference equations. As an example, we apply the theory to the case of a differential-difference dispersive equation of the Burgers hierarchy which via a discrete Hopf-Cole transformation reduces to a linear differential-difference equation. In this case, the equation satisfies the A₁, A₂ and A₃ linearizability conditions. We then consider its discretization. To get a dispersive equation we substitute the time derivative by its symmetric discretization. When we apply to this nonlinear partial difference equation the multiple scale expansion we find out that the lowest order non-secularity condition is given by a non-integrable nonlinear Schrödinger equation. Thus showing that this discretized Burgers equation is neither linearizable not integrable.
