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Các Chuỗi Thủy Động Egorov, Phương Trình Chazy và SL(2,ℂ)
Tóm tắt
Giải pháp tổng quát của hệ phương trình vi phân mô tả các chuỗi thủy động Egorov được xây dựng. Giải pháp được thể hiện qua hàm sigma elliptic. Các bất biến của hàm sigma được diễn tả dưới dạng đa thức vi phân trong một giải pháp của phương trình Chazy. Các quỹ đạo của hành động được sinh ra từ SL(2,ℂ) và các toán tử suy thoái trong không gian các giải pháp cũng được mô tả.
Từ khóa
#chuỗi thủy động #phương trình Chazy #hàm sigma elliptic #bất biến #SL(2 #ℂ)Tài liệu tham khảo
V. E. Zakharov, “Benney equations and quasiclassical approximation in the inverse problem method,” Funkts. Anal. Prilozhen., 14, No. 2, 15–24 (1980); English transl. Funct. Anal. Appl., 14, pp. 89–98 (1980).
I. M. Krichever, “The averaging method for two-dimensional 'integrable' equations,” Funkts. Anal. Prilozhen., 22, No. 3, 37–52 (1988); English transl. Funct. Anal. Appl., 22, No. 3, pp. 200–213 (1988).
I. M. Krichever, “Spectral theory of two-dimensional periodic operators and its applications,” Usp. Mat. Nauk, 44, No. 2, 121–184 (1989); English transl. Russian Math. Surveys, 44, No. 2, 145–225 (1989).
B. A. Kupershmidt and Yu. I. Manin, “Long wave equations with a free surface. II. The Hamiltonian structure and the higher equations,” Funkts. Anal. Prilozhen., 12, No. 1, 25–37 (1978); English transl. Funct. Anal. Appl., 12, No. 1, 20–29 (1978).
M. V. Pavlov, “New integrable (2+1)-equations of hydrodynamic type,” Usp. Mat. Nauk, 58, No. 2, 171–172 (2003); English transl. Russian Math. Surveys, 58, No. 2, pp. 384–385 (2003).
M. V. Pavlov, “The classification of integrable Egorov hydrodynamic chains,” Teor. Mat. Fiz., to appear (2004).
M. V. Pavlov and S. P. Tsarev, “Tri-Hamiltonian structures of Egorov systems of hydrodynamic type,” Funkts. Anal. Prilozhen., 37, No. 1, 38–54 (2003); English transl. Funct. Anal. Appl., 37, No. 1, 32–45 (2003).
S. P. Tsarev, “Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type,” Dokl. Akad. Nauk SSSR, 282, No. 3, 534–537 (1985); English transl. Soviet Math. Dokl., 31, 488–491 (1985).
S. P. Tsarev, “The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method,” Izv. Akad. Nauk SSSR, Ser. Mat., 54, No. 5, 1048–1068 (1990); English transl. Math. USSR Izvestiya, 37, No. 2, pp. 397–419 (1991).
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (M. Abramowitz and I. A. Stegun, eds.), John Wiley & Sons, New York, 1984.
D. J. Benney, “Some properties of long non-linear waves,” Stud. Appl. Math., 52, 45–50 (1973).
M. Błaszak, “Classical R-matrices on Poisson algebras and related dispersionless systems,” Phys. Lett. A, 297, 191–195 (2002).
M. Błaszak and B. M. Szablikowski, “Classical R-matrix theory of dispersionless systems: I. (1+1)-dimension theory,” J. Phys. A: Math. Gen., 35 10325–10344 (2002).
M. Błaszak and B. M. Szablikowski, “Classical R-matrix theory of dispersionless systems: II. (2+1)-dimension theory,” J. Phys. A: Math. Gen., 35, 10345–10364 (2002).
A. Boyarsky, A. Marshakov, O. Ruchayskiy, P. Wiegmann, and A. Zabrodin, “Associativity equations in dispersionless integrable hierarchies,” Phys. Lett. B, 515, 483–492 (2001).
C. P. Boyer and J. D. Finley, “Killing vectors in self-dual Euclidean Einstein spaces,” J. Math. Phys., 23, 1126–1130 (1982).
C. P. Boyer, J. D. Finley, J. F. Plebarnski, “Complex general relativity, H and HH spaces-a survey of one approach,” In: General Relativity and Gravitation, Vol. 2, Plenum, New York-London, 1980, pp. 241–281.
R. Carroll and Y. Kodama, “Solutions of the dispersionless Hirota equations,” J. Phys. A: Math. Gen., 28, 6373 (1995).
P. A. Clarkson and P. J. Olver, “Symmetry and the Chazy equation,” J. Diff. Eq., 124, 225–246 (1996).
J. Chazy, “Sur les équations différentiellles dont l'intégrale générale possède un coupure essentielle mobile,” C. R. Acad. Sci. Paris, 150, 456–458 (1910).
E. V. Ferapontov and K. R. Khusnutdinova, “On integrability of (2+1)-dimensional quasilinear systems,” Comm. Math. Phys., to appear; arXiv: nlin.SI/0305044. 262
E. V. Ferapontov, D. A. Korotkin, and V. A. Shramchenko, “Boyer-Finley equation and systems of hydrodynamic type,” Class. Quantum Grav., 19, No. 24, L205-L210 (2002).
E. V. Ferapontov and M. V. Pavlov, “Hydrodynamic reductions of the heavenly equation,” Class. Quantum Grav., 20, No. 11, 2429–2441 (2003).
F. Frobenius and L. Stickelberger, “Ñber die Differentiation der elliptischen Funktionen nach den Perioden und Invarianten,” Crelle's Journal, XCII, 311–337 (1882).
J. Gibbons, “Collisionless Boltzmann equations and integrable moment equations,” Phys. D, 3, No. 3, 503–511 (1981).
J. Gibbons and Y. Kodama, “A method for solving the dispersionless KP hierarchy and its exact solutions. II,” Phys. Lett. A, 135, No. 3, 167–170 (1989).
J. Gibbons and S. P. Tsarev, “Reductions of the Benney equations,” Phys. lett. A, 211, 19–24 (1996).
J. Gibbons and S. P. Tsarev, “Conformal maps and reductions of the Benney equations,” Phys. Lett. A, 258, 263–270 (1999).
I. M. Krichever, A. Marshakov, and A. Zabrodin, “Integrable structure of the Dirichlet boundary problem in multiply-connected domains,” arXiv:hep-th/0309010.