On KP Multi–Soliton Solutions Associated To Rational Degenerations Of Real Hyperelliptic Curves

Simonetta Abenda

Abstract


Using the technique introduced in [1], we explain the relations between the description of KP–multisolitons in the Sato Grassmannian and in finite–gap theory in the special cases GrTP(1, M) and GrTP(M − 1, M) where the multisolitons may be associated to Krichever data on rational degenerations of regular hyperelliptic M–curves of genus M − 1.

Keywords


Total positivity; KP equation; real solitons; M-curves; hyperelliptic curves; duality of Grassmann cells via space–time transformations

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References


S. Abenda, P.G. Grinevich Rational degenerations of M -curves, totally positive Grassmannians and KP–solitons, arXiv:1506.00563 submitted.

S. Chakravarty, Y. Kodama. Classification of the line-soliton solutions of KPII. J. Phys. A 41 (2008), 275209, 33 pp.

S. Chakravarty, Y. Kodama. Soliton solutions of the KP equation and application to shallow water waves. Stud. Appl. Math. 123 (2009) 83-151.

L.A, Dickey. Soliton equations and Hamiltonian systems. Second edition. Advanced Series in Mathematical Physics, 26. World Scientific Publishing Co., Inc., River Edge, NJ, 2003. xii+408 pp.

B.A. Dubrovin.Theta-functions and nonlinear equations. (Russian) With an appendix by I. M. Krichever. Uspekhi Mat. Nauk. 36 (1981), 1180.

B.A. Dubrovin, I.M. Krichever, S.P. Novikov. Integrable systems. Dynamical systems, IV, 177-332, Encyclopaedia Math. Sci., 4, Springer, Berlin, 2001.

B. A.Dubrovin, S.M. Natanzon. Real theta-function solutions of the Kadomtsev-Petviashvili equation. Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988) 267-286.

S. Fomin, A. Zelevinsky. Double Bruhat cells and total positivity. J. Amer. Math. Soc. 12 (1999) 335-380.

S. Fomin, A. Zelevinsky. Cluster algebras I: foundations. J. Am. Math. Soc. 15 (2002) 497-529.

F.R. Gantmacher, M.G. Krein. Sur les matrices oscillatoires. C.R.Acad.Sci.Paris 201 (1935) 577-579.

D.A. Gudkov. The topology of real projective algebraic varieties. Russ. Math. Surv. 29 (1974) 1-79.

A. Harnack. Über die Vieltheiligkeit der ebenen algebraischen Curven. Math. Ann. 10 (1876) 189-199.

R. Hirota. The direct method in soliton theory. Cambridge Tracts in Mathematics, 155. Cambridge University Press, Cambridge, 2004. xii+200 pp.

B.B. Kadomtsev, V.I. Petviashvili. On the stability of solitary waves in weakly dispersive media., Sov. Phys. Dokl. 15 (1970) 539-541.

Y. Kodama,L.K. Williams. The Deodhar decomposition of the Grassmannian and the regularity of KP solitons. Adv. Math. 244 (2013) 979-1032.

Y. Kodama,L.K. Williams. KP solitons and total positivity for the Grassmannian. Invent. Math. 198 (2014) 637-699.

I. M. Krichever. An algebraic-geometric construction of the Zakharov-Shabat equations and their periodic solutions. (Russian) Dokl. Akad. Nauk SSSR 227 (1976) 291-294.

I. M. Krichever. Integration of nonlinear equations by the methods of algebraic geometry. (Russian) Funkcional. Anal. i Priloen. 11 (1977) 15-31, 96.

G. Lusztig. A survey of total positivity. Milan J. Math. 76 (2008) 125-134.

V. B. Matveev, M.A. Salle. Darboux transformations and solitons. Springer Series in Nonlinear Dynamics. Springer-Verlag, Berlin, 1991.

T. Miwa, M. Jimbo, E. Date. Solitons. Differential equations, symmetries and infinite-dimensional algebras. Cambridge Tracts in Mathematics, 135. Cambridge University Press, Cambridge, 2000. x+108 pp.

A. Pinkus. Totally positive matrices. Cambridge Tracts in Mathematics, 181. Cambridge University Press, Cambridge, 2010. xii+182 pp.

A. Postnikov. Total positivity, Grassmannians, and networks., arXiv:math/0609764 [math.CO]

M. Sato. Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold. in: Nonlinear PDEs in Applied Sciences (US-Japan Seminar, Tokyo), P. Lax and H. Fujita eds., North-Holland, Amsterdam (1982) 259-271.

J. Schoenberg. Über variationsvermindende lineare Transformationen. Mathematishe Zeitschrift 32 (1930) 321-328.

Y. Zarmi. Vertex dynamics in multi-soliton solutions of Kadomtsev-Petviashvili II equation. Non-linearity 27 (2014), 14991523.




DOI: 10.6092/issn.2240-2829/5976

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