Existence of conservation laws and characterization of recursion operators for completely integrable systems

Grifone, Joseph; Mehdi, Mohamad

doi:10.1090/S0002-9947-97-01974-0

Existence of conservation laws and characterization of recursion operators for completely integrable systems
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by Joseph Grifone and Mohamad Mehdi PDF

Trans. Amer. Math. Soc. 349 (1997), 4609-4633 Request permission

Abstract:

Using the Spencer-Goldschmidt version of the Cartan-Kähler theorem, we give conditions for (local) existence of conservation laws for analytical quasi-linear systems of two independent variables. This result is applied to characterize the recursion operator (in the sense of Magri) of completely integrable systems.

References

R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, Exterior differential systems, Mathematical Sciences Research Institute Publications, vol. 18, Springer-Verlag, New York, 1991. MR 1083148, DOI 10.1007/978-1-4613-9714-4
Patrick Cabau, Joseph Grifone, and Mohamad Mehdi, Existence de lois de conservation dans le cas cyclique, Ann. Inst. H. Poincaré Phys. Théor. 55 (1991), no. 3, 789–803 (French, with English summary). MR 1141714
P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
Leon Ehrenpreis, Victor W. Guillemin, and Shlomo Sternberg, On Spencer’s estimate for $\delta$-Poincaré, Ann. of Math. (2) 82 (1965), 128–138. MR 187256, DOI 10.2307/1970565
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
J. Gasqui, Formal integrability of systems of partial differential equations, Nonlinear equations in classical and quantum field theory (Meudon/Paris, 1983/1984) Lecture Notes in Phys., vol. 226, Springer, Berlin, 1985, pp. 21–36. MR 802098, DOI 10.1007/3-540-15213-X_{6}8
Hubert Goldschmidt, Integrability criteria for systems of nonlinear partial differential equations, J. Differential Geometry 1 (1967), 269–307. MR 226156
J. Grifone, M. Mehdi, Existence of Conservation Laws and Characterisation of Recursion Operators for Completely Integrable Systems. Prépublication n. 27 du Laboratoire de Topologie et Geometrie, Université de Toulouse III, Juin 1993
P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
Bernard Malgrange, Equations de Lie. II, J. Differential Geometry 7 (1972), 117–141 (French). MR 326785
F. Magri, C. Morosi, A geometrical characterization of integrable hamiltonian systems, Quaderno 5/19 Università di Milano (1984)
M. Mehdi, Existence de lois de conservation et de systèmes bihamiltoniens Thèse, Toulouse (1991)
Howard Osborn, The existence of conservation laws. I, Ann. of Math. (2) 69 (1959), 105–118. MR 101958, DOI 10.2307/1970096
Howard Osborn, Les lois de conservation, Ann. Inst. Fourier (Grenoble) 14 (1964), no. fasc. 1, 71–81 (French). MR 172205, DOI 10.5802/aif.162
D. C. Spencer, Overdetermined systems of linear partial differential equations, Bull. Amer. Math. Soc. 75 (1969), 179–239. MR 242200, DOI 10.1090/S0002-9904-1969-12129-4
Francisco-Javier Turiel, Classification locale d’un couple de formes symplectiques Poisson-compatibles, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), no. 20, 575–578 (French, with English summary). MR 1001810
Francisco-Javier Turiel, Classification locale des tenseurs de type $(1,1)$ presque plats, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 5, 471–474 (French, with English and French summaries). MR 1296775