Abstract
The nonlinear evolution equations and their inhomogeneous versions related through the inverse scattering method to the generalized Zakharov-Shabat systemL=id/dx+q(x)−λJ are studied. Here we assume that the potentialq(x)=[J,Q(x)] takes values in the simple Lie algebra g and thatJ is a nonregular element of the Cartan subalgebra ν. The corresponding systems of equations for the scattering data ofL are derived. These can be applied to the study of soliton perturbations of such equations as the matrix nonlinear Schrödinger equation, the matrixn-wave equations, etc.
Similar content being viewed by others
References
M. Ablowitz and H. Seegur, Solitons and the Inverse Scattering Transform, SIAM Studies in Applied Mathematics (Philadelphia: SIAM) 1981.
R. Beals and D. Sattinger, Commun. Math. Phys.,138, 409 (1991).
N. Bourbaki, Lie Groups and Lie Algebras, Reading Mass., Addison-Wesley Publ. Co. (1975).
F. Calogero and A. Degasperis, Spectral Transform and Solitons (Studies in Mathematics and its Applications). Amsterdam, North Holland, 1982.
F. Calogero and A. Degasperis, In: Solitons, Ed. R. K. Bullough and P. J. Caudrey, Berlin, Springer-Verlag, 1980.
F. Calogero and Xiaodin, J. Math. Phys.,32, 875, 2703 (1991); F. Calogero and C. Nucci, J. Math. Phys.,32, 72 (1991).
L. D. Faddeev and L. A. Takhtadjan, Hamiltonian Method in the Theory of Solitons, Berlin, Springer-Verlag, 1986.
A. Fokas and M. Ablowitz, Stud. Appl. Math.,80, 253 (1989).
A. P. Fordy and P. P. Kulish, Commun. Math. Phys.,89, 427 (1983).
V. S. Gerdjikov, Lett. Math. Phys.,6, 315 (1982).
V. S. Gerdjikov, Inverse Problems,2, 51 (1986).
V. S. Gerdjikov, Phys. Lett. A.,126A, 184 (1987).
V. S. Gerdjikov, Teor. Math. Fiz.,92, 374 (1992); Lax representation does not mean complete integrability. Internal Report of ICTP IC/91/274, 1991.
V. S. Gerdjikov and E. Kh. Khristov, Bulgarian J. Phys.,7, 28 119 [In Russian] 1980.
V. S. Gerdjikov and M. I. Ivanov, Inverse Problems,8, 831 (1992).
V. S. Gerdjikov and A. B. Yanovski, Phys. Lett. A,103A, 232 (1984).
V. S. Gerdjikov and A. B. Yanovski, Commun. Math. Phys.,103, 549 (1986).
Harnad J., Y. Saint-Auben, and S. Schneider, Commun. Math. Phys.,89, 329 (1984).
Kaup, D. J., A. Reiman, and A. Bers, Rev. Mod. Phys.,51, 275 (1979).
Lalomed B, Phys. Rev. A,43A, 410, 3114 (1991).
V. E. Zakharov and S. V. Manakov, ZhETP,69, 1654 (1975).
V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. I. Pitaevskii, Soliton Theory. The Inverse Problem Method, Moscow, Nauka, 1980.
Additional information
Institute for Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 2, pp. 292–299, May, 1994.
Rights and permissions
About this article
Cite this article
Gerdjikov, V.S. The generalized Zakharov-Shabat system and the soliton perturbations. Theor Math Phys 99, 593–598 (1994). https://doi.org/10.1007/BF01016144
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01016144