Abstract
The general resolvent scheme for solving nonlinear integrable evolution equations is formulated. Special attention is paid to the problem of nontrivial dressing and the corresponding transformation of spectral data. The Kadomtsev-Petviashvili equation is considered as the standard example of integrable models in 2+1 dimensions. Properties of the solutionu(t, x, y) of the Kadomtsev-Petviashvili I equation as well as the corresponding Jost solutions and spectral data with given initial datau(0, x, y) belonging to the Schwartz space are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Boiti, F. Pempinelli, A. K. Pogrebkov, and M. C. Polivanov, “Resolvent approach for the nonstationary Schrödinger equation (standard case of rapidly decreasing potential),” in Proceedings of the Seventh Workshop on Nonlinear Evolution Equations and Dynamical Systems (NEEDS'91), World Scientific Publ. Co., Singapore (1992).
M. Boiti, F. Pempinelli, A. K. Pogrebkov, and M. C. Polivanov,Theor. Math. Phys. 93 (1992) 1200–1224.
M. Boiti, F. Pempinelli, A. K. Pogrebkov, and M. C. Polivanov,Inverse Problems 8 (1992) 331.
B. B. Kadomtsev and V. I. Petviashvili,Sov. Phys. Doklady 192 (1970) 539.
V. Dryuma,Sov. Phys. J. Exp. Theor. Phys. Lett. 19 (1974) 381.
V. E. Zakharov and S. V. Manakov,Sov. Sci. Rev.-Phys. Rev. 1 (1979) 133; S. V. Manakov,Physica D3 (1981) 420.
M. J. Ablowitz and H. Segur,J. Fluid Mech. 92 (1979) 691.
A. S. Fokas and M. J. Ablowitz,Stud. Appl. Math. 69 (1983) 211.
R. Beals and R. R. Coifman,Commun. Pure. Appl. Math. 37 (1984) 39;38 (1985) 29.
S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov,Theory of Solitons. The Method of Inverse Scattering 1984 (New York: Plenum).
M. Boiti, J. Léon, and F. Pempinelli,Phys. Lett. A 141 (1989) 96.
V. G. Bakurov,Phys. Lett. A 160 (1991) 367.
M. J. Ablowitz and J. Villarroel,Stud. Appl. Math. 85 (1991) 195.
A. S. Fokas and V. E. Zakharov,J. Nonlinear Sci. 2 (1992) 109–134.
M. Boiti, F. Pempinelli, and A. Pogrebkov,Solutions of the KPI equation with smooth initial data, submitted toInverse Problems (1993).
M. Boiti, F. Pempinelli, and A. Pogrebkov,Properties of solutions of the KPI equation, to be published.
M. Boiti, F. Pempinelli, A. K. Pogrebkov, and M. C. Polivanov,Inverse Problems 7 (1991) 43–56.
S. V. Manakov, private communication.
M. J. Ablowitz, S. V. Manakov, and C. L. Shultz,Phys. Lett. A 148 (1990) 50.
Additional information
Dipartimento di Fisica dell'Università and Sezione, INFN 73100 Lecce, ITALIA. E-mail: boiti@lecce.infn.it and pempi@lecce.infn.it. Steklov Mathematical Institute, Vavilov Str. 42, Moscow 117966, GSP-1, RUSSIA. E-mail: progreb@qft.mian.su. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 2, pp. 185–200, May, 1994.
Rights and permissions
About this article
Cite this article
Boiti, M., Pempinelli, F. & Pogrebkov, A.K. Some new methods and results in the theory of (2+1)-dimensional integrable equations. Theor Math Phys 99, 511–522 (1994). https://doi.org/10.1007/BF01016132
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01016132