### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 058, 13 pages      arXiv:1704.04078      https://doi.org/10.3842/SIGMA.2017.058
Contribution to the Special Issue on Symmetries and Integrability of Difference Equations

### Relativistic DNLS and Kaup-Newell Hierarchy

Oktay K. Pashaev a and Jyh-Hao Lee b
a) Department of Mathematics, Izmir Institute of Technology, Urla-Izmir 35430, Turkey
b) Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan

Received April 14, 2017, in final form July 18, 2017; Published online July 25, 2017

Abstract
By the recursion operator of the Kaup-Newell hierarchy we construct the relativistic derivative NLS (RDNLS) equation and the corresponding Lax pair. In the nonrelativistic limit $c \rightarrow \infty$ it reduces to DNLS equation and preserves integrability at any order of relativistic corrections. The compact explicit representation of the linear problem for this equation becomes possible due to notions of the $q$-calculus with two bases, one of which is the recursion operator, and another one is the spectral parameter.

Key words: Kaup-Newell hierarchy; relativistic DNLS; $q$-calculus; recursion operator.

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