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SIGMA, 2010, Volume 6, 013, 52 pages (Mi sigma470)  

This article is cited in 3 scientific papers (total in 3 papers)

Bethe Ansatz, Inverse Scattering Transform and Tropical Riemann Theta Function in a Periodic Soliton Cellular Automaton for $A^{(1)}_n$

Atsuo Kunibaa, Taichiro Takagib

a Institute of Physics, University of Tokyo, Komaba, Tokyo 153-8902, Japan
b Department of Applied Physics, National Defense Academy, Kanagawa 239-8686, Japan

Abstract: We study an integrable vertex model with a periodic boundary condition associated with $U_q(A^{(1)}_n)$ at the crystallizing point $q=0$. It is an $(n+1)$-state cellular automaton describing the factorized scattering of solitons. The dynamics originates in the commuting family of fusion transfer matrices and generalizes the ultradiscrete Toda/KP flow corresponding to the periodic box-ball system. Combining Bethe ansatz and crystal theory in quantum group, we develop an inverse scattering/spectral formalism and solve the initial value problem based on several conjectures. The action-angle variables are constructed representing the amplitudes and phases of solitons. By the direct and inverse scattering maps, separation of variables into solitons is achieved and nonlinear dynamics is transformed into a straight motion on a tropical analogue of the Jacobi variety. We decompose the level set into connected components under the commuting family of time evolutions and identify each of them with the set of integer points on a torus. The weight multiplicity formula derived from the $q=0$ Bethe equation acquires an elegant interpretation as the volume of the phase space expressed by the size and multiplicity of these tori. The dynamical period is determined as an explicit arithmetical function of the $n$-tuple of Young diagrams specifying the level set. The inverse map, i.e., tropical Jacobi inversion is expressed in terms of a tropical Riemann theta function associated with the Bethe ansatz data. As an application, time average of some local variable is calculated.

Keywords: soliton cellular automaton; crystal basis;combinatorial Bethe ansatz; inverse scattering/spectral method; tropical Riemann theta function


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ArXiv: 0909.3759
MSC: 82B23; 37K15; 68R15; 37B15
Received: September 21, 2009; Published online January 31, 2010

Citation: Atsuo Kuniba, Taichiro Takagi, “Bethe Ansatz, Inverse Scattering Transform and Tropical Riemann Theta Function in a Periodic Soliton Cellular Automaton for $A^{(1)}_n$”, SIGMA, 6 (2010), 013, 52 pp.

Citation in format AMSBIB
\by Atsuo Kuniba, Taichiro Takagi
\paper Bethe Ansatz, Inverse Scattering Transform and Tropical Riemann Theta Function in a~Periodic Soliton Cellular Automaton for $A^{(1)}_n$
\jour SIGMA
\yr 2010
\vol 6
\papernumber 013
\totalpages 52

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    This publication is cited in the following articles:
    1. Taichiro Takagi, “Level Set Structure of an Integrable Cellular Automaton”, SIGMA, 6 (2010), 027, 18 pp.  mathnet  crossref  mathscinet
    2. Inoue R., Kuniba A., Takagi T., “Integrable structure of box-ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry”, Journal of Physics A-Mathematical and Theoretical, 45:7 (2012), 073001  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Kuniba A., Lyu H., Okado M., “Randomized Box-Ball Systems, Limit Shape of Rigged Configurations and Thermodynamic Bethe Ansatz”, Nucl. Phys. B, 937 (2018), 240–271  crossref  mathscinet  zmath  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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