Abstract
So called integrable equations have a zero measure among other equations, just like a gold section has zero measure among all possible sections. Both cases are marked out because of their beauty, and in the first case beauty show itself via a number of mathematical constructions that have appeared in soliton equations, and also via an infinite set of symmetries which are valuable not only for their own sake but also because they give us a lot of connections. Symmetries can unify different soliton equations into a single integrable hierarchy. Thus we get a large universal space which in future may appear to play a role similar to the role of complex plane in mathematics of nineteenth century. Here I present a few examples of links between different equations, which follow from symmetry approach.
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References
V.E. Zakharov, S. V. Manakov, S.P. Novikov, L.P. Pitaevsky, Soliton Theory,Nauka,Moscow 1980.
V.A. Belinsky, V.E. Zakharov, JETP 1 (1979), 1
A.Yu. Orlov, E.I. Schulman, Lett.Math.Phys.12(1986)
A.Yu. Orlov,“Vertex operators,\(\bar \partial \)-problem,variational identities and Hamiltonian formalism in 2+1 integrable systems” in: Proc.Worksh. Plasma… in Kiev 11987,ed. V.G. Bar’yahtar, World Sci.Pub., Singapure 1988.
P.G. Grinevich, A.Yu. Orlov, “KP theory and Virasoro action on Segal-Wilson τ-function” in: Proc. Work. Quantum Field THeory, Alushta 1989, ed. Belavin, A. Zamolodchikov Springer 1990;
P.G. Gurevich, A.Yu. Orlov “Flag space in KP theory and Virasoro action on det\(\bar \partial \) on Riemann surfaces”,Cornell Univ. preprint 945/89.
W. Devel, B. Fuchssteiner, Phys.Lett. 88A (1982), 7
D. David, D. Levi, D. Winternitz, Phys.Lett. 118A (1986), 890
P. Santini, A. Fokas, Comm Math.Phys,6(1987)
V.E. Zakharov, A.B. Shabat, Funk.Analis i ego Pri1.3(1974)
E. Date, M. Kashiwara, M. Jimbo, T. Miwa, “Group transformation theory of KP”, in: Proc.RIMS Symp., ed. M. Jimbo, T. Miwa, World Si Pub., Singapore 1983
S. Segal, G. Wilson, Pub.IHES, 61 (1985), 4
A.Yu. Orlov, “Symmetries for unifying different soliton systems into a single integrable hierarchy”, Preprint BINS/Oce-04/03
B. Konopelchenko, V. Strampt, A talk given at Gallipoli, June 1991
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© 1993 Springer-Verlag Berlin Heidelberg
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Orlov, A.Y. (1993). Volterra Operator Algebra for Zero Curvature Representation. Universality of KP. In: Fokas, A.S., Kaup, D.J., Newell, A.C., Zakharov, V.E. (eds) Nonlinear Processes in Physics. Springer Series in Nonlinear Dynamics . Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-77769-1_24
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DOI: https://doi.org/10.1007/978-3-642-77769-1_24
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