Summary
Pole expansions of certain solutions of various nonlinear partial differential equations are investigated. The most interesting results obtain for the Korteweg-de Vries and especially for the Burgers-Hopf equations. The motion of the poles is shown to correspond formally to the motion of one-dimensional particles interacting via simple two-body potentials, such that the corresponding many-body problems are integrable.
Riassunto
Si analizzano sviluppi polari di certe soluzioni di varie equazioni differenziali parziali non lineari. I risultati più interessanti si ottengono per l’equazione di Korteweg-de Vries e specialmente per quella di Burgers-Hopf. Si mostra che il moto dei poli corrisponde formalmente al moto di particelle unidimensionali interagenti attraverso semplici potenziali a due corpi in modo che i corrispondenti problemi a molti corpi sono integrabili.
Резюме
Исследуетса разложение по полюсам мероморфных решений различных нелинейных уравнений в частных производных. Наиболее полные резулбтаты получены дла уравнеий Кортевега-де Фриза и Бюргерса-Хопфа, дла которых показано сведение задачи о движении полюсов к механическим задачам о движении частиц с потенциаламиx −2,x −4. Показано, что движение полюсов дла ряда уравнений описываетса гамильтоновыми системами с эллиптическим потенциалом взаимодействиа.
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References
B. A. Dubrovin, V. B. Matveev andS. P. Novikov:Usp. Math. Nauk,31, 55 (1976).
M. A. Olshanetzky andA. M. Perelomov:Invent. Math.,37, 93 (1976). See alsoM. A. Olshanetzky andA. M. Perelomov:Lett. Math. Phys.,1, 187 (1976);Lett. Nuovo Cimento,16, 333 (1976);17, 97 (1976).
J. Moser:Adv. Math.,16, 197 (1975). See also the article byJ. Moser: in theBattelle Seattle 1974 Rencontres Proceedings (J. Moser, Editor:Theory and applications, inLecture Notes in Physics, Vol.38 (Berlin, 1975)).
F. Calogero, C. Marchioro andO. Ragnisco:Lett. Nuovo Cimento,13, 383 (1975).
F. Calogero:Lett. Nuovo Cimento,13, 411 (1975).
In ref. (1) a formula, that constitutes an indirect indication of this connection, was however given; see below.
W. R. Thikstun:Journ. Math. Anal. Appl.,55, 335 (1976).
M. D. Kruskal:Lectures in Appl. Math., Amer. Math. Soc.,15, 61 (1974).
H. Airault, H. P. McKean andJ. Moser:Rotational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem (preprint, to be published).
F. Calogero:Lett. Nuovo Cimento,16, 77 (1976).
B. Sutherland:Phys. Rev. A,5, 1372 (1972).
F. Calogero:Lett. Nuovo Cimento,13, 507 (1975).
A. Erdelyi, Editor:Higher Transcendental Functions, Vol.2 (New York, N. Y., 1953).
M. Kac andP. van Moerbecke:Proc. Nat. Acad. Sci.,72, 1627 (1975).
J. Ford: inLectures in Statistical Physics, Lecture Notes in Physics, Vol.28 (Berlin, 1974), p. 204; inFundamental Problems in Statistical Mechanics, Vol.3 (Amsterdam, 1975).
I. M. Krichever:Funct. Anal.,10, 75 (1976).
F. Calogero:Lett. Nuovo Cimento,16, 35 (1976).
J. Cole:Quart. Appl. Math.,9, 225 (1951).
See, for instance, the paper byR. Hirota:Direct method of finding exact solutions of nonlinear evolution equations, inBacklund Transformations, edited byR. Miura,Lecture Notes in Mathematics, Vol.515 (Berlin, 1976), p. 61.
Exact treatment of nonlinear lattice waves, Suppl. Prog. Theor. Phys.,59 (1976) (see p. 97).
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Choodnovsky, D.V., Choodnovsky, G.V. Pole expansions of nonlinear partial differential equations. Nuovo Cim B 40, 339–353 (1977). https://doi.org/10.1007/BF02728217
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DOI: https://doi.org/10.1007/BF02728217