Abstract
We present a general scheme for determining and studying integrable deformations of algebraic curves, based on the use of Lenard relations. We emphasize the use of several types of dynamical variables: branches, power sums, and potentials.
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REFERENCES
V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: The Inverse Scattering Method [in Russian], Nauka, Moscow (1980); English transl.: S. P. Novikov, S. V. Manakov, L. P. Pitaevsky, and V. E. Zakharov, Plenum, New York (1984); E. D. Belokolos, A. I. Bobenko, V. Z. Enolski, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, Berlin (1994); B. A. Dubrovin and S. P. Novikov, Russ. Math. Surveys, 44, No. 6, 35 (1989); H. Flaschka, M. G. Forest, and D. W. Mclauglin, Comm. Pure Appl. Math., 33, 739 (1980); B. A. Dubrovin, Comm. Math. Phys., 145, 415 (1992).
I. M. Krichever, Funct. Anal. Appl., 22, 200 (1988); Comm. Pure Appl. Math., 47, 437 (1994).
Y. Kodama and B. G. Konopelchenko, J. Phys. A, 35, L489–L500 (2002); “Deformations of plane algebraic curves and integrable systems of hydrodynamic type,” in: Nonlinear Physics: Theory and Experiment II (Proc. Intl. Workshop, Gallipoli, Lecce, Italy, 2002, M. J. Ablowitz, M. Boiti, F. Pempinelli, and B. Prinari, eds.), World Scientific, River Edge, N. J. (2003), p. 234.
B. G. Konopelchenko and L. Martinez Alonso, J. Phys. A, 37, 7859 (2004).
C. L. Siegel, Topics in Complex Function Theory, Vol. 1, Elliptic Functions and Uniformization Theory, Wiley, New York (1969).
R. Y. Walker, Algebraic Curves, Springer, Berlin (1978).
S. S. Abhyankar, Algebraic Geometry for Scientists and Engineers (Math. Surveys and Monographs, Vol. 35), Amer. Math. Soc., Providence, R. I. (1990).
B. L. van der Waerden, Algebra, Vol. 1, Springer, Berlin (1991).
L. Redei, Introduction to Algebra, Vol. 1, Pergamon, Oxford (1967).
I. G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon, Oxford (1979).
L. Schwartz, Analyse mathematique, Vol. 2, Hermann, Paris (1967).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 94–101, July, 2005.
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Kodama, Y., Konopelchenko, B.G. & Martinez Alonso, L. Integrable Deformations of Algebraic Curves. Theor Math Phys 144, 961–967 (2005). https://doi.org/10.1007/s11232-005-0123-9
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DOI: https://doi.org/10.1007/s11232-005-0123-9