Conserved quantities of the integrable discrete hungry systems

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October 2015, 8(5): 889-899. doi: 10.3934/dcdss.2015.8.889

Conserved quantities of the integrable discrete hungry systems

Sonomi Kakizaki ^1, , Akiko Fukuda ^2, , Yusaku Yamamoto ^3, , Masashi Iwasaki ^4, , Emiko Ishiwata ^5, and Yoshimasa Nakamura ^6,

1.	Department of Mathematical Science for Information Sciences, Graduate School of Science, Tokyo University of Science, Tokyo 162-8601, Japan
2.	Department of Mathematical Sciences, Shibaura Institute of Technology, Saitama 337-8570, Japan
3.	Department of Communication Engineering and Informatics, The University of Electro-Communications, Tokyo 182-8585, Japan/JST CREST, Tokyo, Japan
4.	Faculty of Life and Environmental Sciences, Kyoto Prefectural University, Kyoto 606-8522, Japan
5.	Department of Mathematical Information Science, Tokyo University of Science, Tokyo 162-8601, Japan
6.	Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan

Received December 2013 Revised December 2013 Published July 2015

Abstract
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In this paper, conserved quantities of the discrete hungry Lotka-Volterra (dhLV) system are derived. Our approach is based on the Lax representation of the dhLV system, which expresses the time evolution of the dhLV system as a similarity transformation on a certain square matrix. Thus, coefficients of the characteristic polynomial of this matrix constitute conserved quantities of the dhLV system. These coefficients are calculated explicitly through a recurrence relation among the characteristic polynomials of its leading principal submatrices. The conserved quantities of the discrete hungry Toda (dhToda) equation is also derived with the help of the Bäcklund transformation between the dhLV system and the dhToda equation.

Keywords: leading principal submatrix., discrete hungry Lotka-Volterra system, Conserved quantities, discrete hungry Toda equation, characteristic polynomial.

Mathematics Subject Classification: Primary: 37K10; Secondary: 39A9.

Citation: Sonomi Kakizaki, Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata, Yoshimasa Nakamura. Conserved quantities of the integrable discrete hungry systems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 889-899. doi: 10.3934/dcdss.2015.8.889

References:

[1]	LAPACK:, http://www.netlib.org/lapack/, ., ().
[2]	A. Fukuda, E. Ishiwata, M. Iwasaki and Y. Nakamura, The discrete hungry Lotka-Volterra system and a new algorithm for computing matrix eigenvalues,, Inverse Probl., 25 (2009). doi: 10.1088/0266-5611/25/1/015007.
[3]	A. Fukuda, E. Ishiwata, Y. Yamamoto, M. Iwasaki and Y. Nakamura, Integrable discrete hungry systems and their related matrix eigenvalues,, Annal. Mat. Pura Appl., 192 (2013), 423. doi: 10.1007/s10231-011-0231-0.
[4]	A. Fukuda, Y. Yamamoto, M. Iwasaki, E. Ishiwata and Y. Nakamura, A Bäcklund transformation between two integrable discrete hungry systems,, Phys. Lett. A, 375 (2011), 303. doi: 10.1016/j.physleta.2010.11.029.
[5]	R. Hirota, S. Tsujimoto and T. Imai, Difference scheme of soliton equations,, Sūrikaisekikenkyūsho Kōkyūroku, 822 (1993), 144.
[6]	M. Iwasaki and Y. Nakamura, On the convergence of a solution of the discrete Lotka-Volterra system,, Inverse Probl., 18 (2002), 1569. doi: 10.1088/0266-5611/18/6/309.
[7]	M. Iwasaki and Y. Nakamura, Accurate computation of singular values in terms of shifted integrable schemes,, Jpn. J. Indust. Appl. Math., 23 (2006), 239. doi: 10.1007/BF03167593.
[8]	T. Tokihiro, A. Nagai and J. Satsuma, Proof of solitonial nature of box and ball systems by means of inverse ultra-discretization,, Inverse Problems, 15 (1999), 1639. doi: 10.1088/0266-5611/15/6/314.

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References:

[1]	LAPACK:, http://www.netlib.org/lapack/, ., ().
[2]	A. Fukuda, E. Ishiwata, M. Iwasaki and Y. Nakamura, The discrete hungry Lotka-Volterra system and a new algorithm for computing matrix eigenvalues,, Inverse Probl., 25 (2009). doi: 10.1088/0266-5611/25/1/015007.
[3]	A. Fukuda, E. Ishiwata, Y. Yamamoto, M. Iwasaki and Y. Nakamura, Integrable discrete hungry systems and their related matrix eigenvalues,, Annal. Mat. Pura Appl., 192 (2013), 423. doi: 10.1007/s10231-011-0231-0.
[4]	A. Fukuda, Y. Yamamoto, M. Iwasaki, E. Ishiwata and Y. Nakamura, A Bäcklund transformation between two integrable discrete hungry systems,, Phys. Lett. A, 375 (2011), 303. doi: 10.1016/j.physleta.2010.11.029.
[5]	R. Hirota, S. Tsujimoto and T. Imai, Difference scheme of soliton equations,, Sūrikaisekikenkyūsho Kōkyūroku, 822 (1993), 144.
[6]	M. Iwasaki and Y. Nakamura, On the convergence of a solution of the discrete Lotka-Volterra system,, Inverse Probl., 18 (2002), 1569. doi: 10.1088/0266-5611/18/6/309.
[7]	M. Iwasaki and Y. Nakamura, Accurate computation of singular values in terms of shifted integrable schemes,, Jpn. J. Indust. Appl. Math., 23 (2006), 239. doi: 10.1007/BF03167593.
[8]	T. Tokihiro, A. Nagai and J. Satsuma, Proof of solitonial nature of box and ball systems by means of inverse ultra-discretization,, Inverse Problems, 15 (1999), 1639. doi: 10.1088/0266-5611/15/6/314.

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