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The Ablowitz-Ladik Hierarchy Revisited

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Methods of Spectral Analysis in Mathematical Physics

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 186))

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Abstract

We provide a detailed recursive construction of the Ablowitz-Ladik (AL) hierarchy and its zero-curvature formalism. The two-coefficient AL hierarchy under investigation can be considered a complexified version of the discrete nonlinear Schrödinger equation and its hierarchy of nonlinear evolution equations.

Specifically, we discuss in detail the stationary. Ablowitz-Ladik formalism in connection with the underlying hyperelliptic curve and the stationary Baker-Akhiezer function and separately the corresponding time-dependent Ablowitz-Ladik formalism

Research supported in part by the Research Council of Norway, the US National Science Foundation under Grant No. DMS-0405526, and the Austrian Science Fund (FWF) under Grant No. Y330.

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Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway

    Fritz Gesztesy

  2. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Helge Holden

  3. Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090, Wien, Austria

    Johanna Michor & Gerald Teschl

  4. International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090, Wien, Austria

    Johanna Michor & Gerald Teschl

Authors
  1. Fritz Gesztesy
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  2. Helge Holden
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  3. Johanna Michor
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  4. Gerald Teschl
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Editor information

Editors and Affiliations

  1. Institut of Mathematics, Polish Academy of Sciences, ul. Sw. Tomasza 30/7, 31-027, Krakow, Poland

    Jan Janas

  2. Department of Mathematics, Lund Institute of Technology, 221 00, Lund, Sweden

    Pavel Kurasov

  3. Department of Mathematical Physics, St. Petersburg State University, Ulyanovskaya 1, 198904, St. Petersburg, Russia

    Sergei Naboko

  4. Department of Mathematics South Kensington Campus, Imperial College London, SW7 2AZ, London, UK

    Ari Laptev

  5. Department of Mathematics, Royal Institute of Technology, 100 44, Stockholm, Sweden

    Ari Laptev

  6. Department of Mathematics, University of Alabama, 452 Campbell Hall, Birmingham, AL, 35294-1170, USA

    Günter Stolz

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Gesztesy, F., Holden, H., Michor, J., Teschl, G. (2008). The Ablowitz-Ladik Hierarchy Revisited. In: Janas, J., Kurasov, P., Naboko, S., Laptev, A., Stolz, G. (eds) Methods of Spectral Analysis in Mathematical Physics. Operator Theory: Advances and Applications, vol 186. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8755-6_8

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