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Nonlinear Sciences > Exactly Solvable and Integrable Systems

arXiv:1206.3210 (nlin)
[Submitted on 14 Jun 2012 (v1), last revised 12 Jul 2012 (this version, v3)]

Title:A refined and unified version of the inverse scattering method for the Ablowitz-Ladik lattice and derivative NLS lattices

Authors:Takayuki Tsuchida
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Abstract:We refine and develop the inverse scattering theory on a lattice in such a way that the Ablowitz-Ladik lattice and derivative NLS lattices as well as their matrix analogs can be solved in a unified way. The inverse scattering method for the (matrix analog of the) Ablowitz-Ladik lattice is simplified to the same level as that for the continuous NLS system. Using the linear eigenfunctions of the Lax pair for the Ablowitz-Ladik lattice, we can construct solutions of the derivative NLS lattices such as the discrete Gerdjikov-Ivanov (also known as Ablowitz-Ramani-Segur) system and the discrete Kaup-Newell system. Thus, explicit solutions such as the multisoliton solutions for these systems can be obtained by solving linear summation equations of the Gel'fand-Levitan-Marchenko type. The derivation of the discrete Kaup-Newell system from the Ablowitz-Ladik lattice is based on a new method that allows us to generate new integrable systems from known systems in a systematic manner. In an appendix, we describe the reduction of the matrix Ablowitz-Ladik lattice to a vector analog of the modified Volterra lattice from the point of view of the inverse scattering method.
Comments: 67 pages; (v2) many minor corrections (v3) typos corrected
Subjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA); Spectral Theory (math.SP)
Cite as: arXiv:1206.3210 [nlin.SI]
  (or arXiv:1206.3210v3 [nlin.SI] for this version)
  https://doi.org/10.48550/arXiv.1206.3210

Submission history

From: Takayuki Tsuchida [view email]
[v1] Thu, 14 Jun 2012 18:50:59 UTC (93 KB)
[v2] Wed, 11 Jul 2012 18:52:29 UTC (94 KB)
[v3] Thu, 12 Jul 2012 19:02:59 UTC (83 KB)
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