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Boundary conditions for nonlinear equations compatible with integrability. (English. Russian original) Zbl 0802.35138
Theor. Math. Phys. 96, No. 1, 845-853 (1993); translation from Teor. Mat. Fiz. 96, No. 1, 109-122 (1993).
Summary: Infinite series of local boundary conditions, completely compatible with the inverse scattering method, are presented for the nonlinear Schrödinger equation and the sine-Gordon equation.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q53 KdV equations (Korteweg-de Vries equations)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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References:
[1] E. K. Sklyanin,Funktsional Analiz i Ego Prilozhen.,21, 86 (1987).
[2] A. S. Fokas, ?Initial-boundary value problems for a class of nonlinear evolution equations,? INS-81 Preprint, New York (1987), in:Physica (Utrecht) D,35, 167 (1989).
[3] P. N. Bibikov and V. O. Tarasov,Teor. Mat. Fiz.,79, 334 (1989).
[4] I. T. Habibulin,Proc. Fourth International Workshop: Nonlinear and Turbulent Processes in Physics, Kiev, 1989, Singapore (1990), Vol. 1, p. 259.
[5] I. T. Khabibullin,Mat. Zam. 49, 130 (1991).
[6] I. T. Khabibullin,Teor. Mat. Fiz.,86, 43 (1991).
[7] R. F. Bikbaev and V. O. Tarasov,J. Phys. A,24, 2507 (1991). · Zbl 0753.35090
[8] R. A. Sharipov and R. I. Yamilov, in:Problems of Mathematical Physics and the Asymptotic Behavior of their Solutions [in Russian], Institute of Mathematics, Russian Academy of Sciences, Ufa (1991), p. 66.
[9] I. T. Khabibullin,Teor. Mat. Fiz.,91, 363 (1992).
[10] V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii,The Theory of Solitons. The Inverse Scattering Method [in Russian], Nauka, Moscow (1980). · Zbl 0598.35002
[11] M. J. Ablowitz and H. Segur,J. Math. Phys. 16, 1054 (1975). · Zbl 0299.35076
[12] M. J. Ablowitz and H. Segur,Solitons and the Inverse Scattering Transform (SIAM Studies in Applied Maths., Vol. 4), Philadelphia (1981). · Zbl 0472.35002
[13] A. B. Shabat,Problems of Mechanics and Mathematical Physics [in Russian], Nauka, Moscow (1976). · Zbl 0352.34020
[14] L. A. Takhtadzhyan and L. D. Faddeev,The Hamiltonian Approach in the Theory of Solitons [in Russian], Nauka, Moscow (1986). · Zbl 0632.58003
[15] R. F. Bikbaev and V. O. Tarasov,Algebra i Analiz,3, 78 (1991).
[16] A. B. Shabat and R. I. Yamilov,Phys. Lett. A.,130, 271 (1988).
[17] I. T. Habibullin,Boundary Condition Completely Compatible with the Integrability, Nonlinear Evolution Equations and Dynamical Systems (NEEDS-92) [in Russian], JINR, Dubna (1992).
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