Abstract
New types of reduction of the Kadomtsev-Petviashvili (KP) hierarchy are considered on the basis of Sato's approach. As a result, we obtain a new multicomponent nonlinear integrable system. Bi-Hamiltonian structures for the new equations are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, “Operator approach to the Kadomtsev-Petviashvili equation. Transformation groups for soliton equations, III,”J. Phys. Soc. Jpn.,50, No. 11, 3806–3812 (1981).
M. Jimbo, “Theory of τ-functions in integrable systems,”Springer Lect. Notes Phys.,153, 232–237 (1982).
E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, “Vertex operators and t-functions. Transformation groups for soliton equations, III,”Proc. Jpn. Acad. A.,57, No. 8., 387–392 (1981).
B. Yu. Konopel'chenko, “On generalization of the Bäcklund-Calogero transformations for integrable equations,”J. Pkys. A.: Math. Gen.,21, L743-L747 (1988).
A. S. Fokas and P. M. Santini, “The recursion operator of the Kadomtsev-Petviashvili equation and the squared eigenfunctions of the Schrödinger operator,”Stud. Appl. Math.,75, 179–186 (1986).
P. M. Santini and A. S. Fokas, “Recursion operators and bi-Hamiltonian structures in multidimensions. I,”Commun. Math. Phys.,115, 375–419 (1988).
A. I. Orlov,Symmetries for Unifying Different Soliton Systems into a Single Integrable Hierarchy, Preprint IINS-Oce-04/03 Moscow (1991).
J. Matsu Kidaira, J. Satsuma, and W. Strampp, “Conserved quantities and symmetries of KP hierarchy,”J. Math. Phys.,31, No. 6, 1426–1434 (1990).
P. M. Santini, “The algebraic structure underlying integrability,”Inverse Problems, 99–114 (1990).
B. Yu. Konopel'chenko and W. Strampp, “On the structure and properties of the singularity manifold equations of the KP hierarchy,”Inverse Problems,7, L17-L28 (1991).
B. Yu. Konopel'chenko, Yu. N. Sidorenko, and W. Strampp, “(1+1)-dimensional integrable systems as symmetry constraints of (2 + 1)-dimensional systems,”Phys. Lett. A,157, No. 1, 17–21 (1991).
Yu. N. Sidorenko and W. Strampp, “Symmetry constraints of the KP hierarchy,”Inverse Problems,7, L37-L43 (1991).
Y. Cheng and T. Lee “A method for solving the Kadomtsev-Petviashvili equation,”Phys. Lett. A,157, No. 4, 22–26 (1991).
Y. Zeng and T. Lee “Integrable Hamiltonian systems related to the polynomial eigenvalue problem,”J. Math. Phys.,31, No. 2, 2835–2839 (1990).
V. K. Mel'nikov, “Wave emission and absorption in a nonlinear integrable system,”Phys. Lett. A,118, No.1, 22–24 (1986).
H. Flaschka, “Relations between infinite-dimensional and finite-dimensional isospectral equations,” in:Proc. RIMS Symp. on Nonlinear Integrable Systems-Classical and Quantum Theory (M. Jimbo and T. Miwa, eds.), World Scientific, Singapore (1983), pp. 219–240.
C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Korteweg-de Vries equation and generalizations. VI. Methods for exact solution,”Commun. Pure Appl. Math.,27, 97–133 (1974).
Yu. Zhuquan, “Classical Liouville completely integrable systems associated with the solutions of Boussinesq-Borgers' hierarchy,”J. Math. Phys.,31, No. 6, 1374–1380 (1990).
M. Antonowitcz and S. Rauch-Wojciechowski,How to Construct Finite-Dimensional Bi-Hamiltonian Equations: Jacobi Integrable Potentials, Preprint Linkoping Univ. (LITH-MAT-R-90-35), Linkoping (1991).
W. Oevel and B. Fuchssteiner “Explicit formulas for symmetries and conservation laws of the KP equation,”Phys. Lett. A,88, No. 7, 323–327 (1982).
H. H. Chen, Y. C. Lee, and J. E. Lin, “On a new hierarchy of symmetries for the Kadomtsev-Petviashvili equation,”Physica D,11, No. 3, 439–445 (1983).
N. Yajima and M. Oikawa, “Formation and interaction of sonic-Langmuir solitons,”Prog. Theor. Phys.,56, No. 6, 1719–1739 (1976).
G. Wilson, “The affine Lie algebraC (1)2 and an equation of Hirota and Satsuta,”Prog. Theor. Phys.,89, No. 7, 332–334 (1982).
L. D. Faddeev and L. A. Takhtadzhan,Hamiltonian Methods in the Theory of Solitons, Springer, Berlin (1987).
Author information
Authors and Affiliations
Additional information
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 1, pp. 91–104, January, 1993.
Rights and permissions
About this article
Cite this article
Sidorenko, Y.M. KP hierarchy and (1+1)-dimensional multicomponent integrable systems. Ukr Math J 45, 100–115 (1993). https://doi.org/10.1007/BF01062043
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01062043