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.

pdf (308 kb)   tex (14 kb)

References

  1. Aglietti U., Griguolo L., Jackiw R., Pi S.-Y., Seminara D., Anyons and chiral solitons on a line, Phys. Rev. Lett. 77 (1996), 4406-4409, hep-th/9606141.
  2. Francisco M.L.Y., Lee J.-H., Pashaev O.K., Dissipative hierarchies and resonance solitons for KP-II and MKP-II, Math. Comput. Simulation 74 (2007), 323-332.
  3. Jackiw R., A nonrelativistic chiral soliton in one dimension, J. Nonlinear Math. Phys. 4 (1997), 261-270, hep-th/9611185.
  4. Kaup D.J., Newell A.C., An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys. 19 (1978), 798-801.
  5. Lee J.-H., Global solvability of the derivative nonlinear Schrödinger equation, Trans. Amer. Math. Soc. 314 (1989), 107-118.
  6. Lee J.-H., Lin C.-K., Pashaev O.K., Equivalence relation and bilinear representation for derivative nonlinear Schrödinger type equations, in Proceedings of the Workshop on Nonlinearity, Integrability and All That: Twenty Years after NEEDS '79 (Gallipoli, 1999), World Sci. Publ., River Edge, NJ, 2000, 175-181.
  7. Lee J.-H., Pashaev O.K., Soliton resonances for the MKP-II, Theoret. and Math. Phys. 144 (2005), 995-1003, hep-th/0410032.
  8. Lee J.-H., Pashaev O.K., Chiral solitons in a quantum potential, Theoret. and Math. Phys. 160 (2009), 986-994.
  9. Min H., Park Q.-H., Scattering of solitons in the derivative nonlinear Schrödinger model, Phys. Lett. B 388 (1996), 621-625, hep-th/9607242.
  10. Mio K., Ogino T., Minami K., Takeda S., Modified nonlinear Schrödinger equation for Alfvén waves propagating along the magnetic field in cold plasmas, J. Phys. Soc. Japan 41 (1976), 265-271.
  11. Mjølhus E., On the modulational instability of hydromagnetic waves parallel to the magnetic field, J. Plasma Phys. 16 (1976), 321-334.
  12. Pashaev O.K., Relativistic nonlinear Schrödinger and Burgers equations, Theoret. and Math. Phys. 160 (2009), 1022-1030, arXiv:0901.1399.
  13. Pashaev O.K., Lee J.-H., Black holes and solitons of the quantized dispersionless NLS and DNLS equations, ANZIAM J. 44 (2002), 73-81.
  14. Yan Z., Liouville integrable $N$-Hamiltonian structures, involutive solutions and separation of variables associated with Kaup-Newell hierarchy, Chaos Solitons Fractals 14 (2002), 45-56.

Previous article  Next article   Contents of Volume 13 (2017)