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References
V. E. Zakharov and A. B. Shabat. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP, 34:62–69, 1972.
M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur. The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math., 53:249–315, 1974.
H. Segur and M. J. Ablowitz. Solitons and the Inverse Scattering Transform. Society for Industrial & Applied Mathematics, Philadelphia, PA 1981.
M. J. Ablowitz, A. D. Trubatch, and B. Prinari. Discrete and Continuous Nonlinear Schrodinger Systems. Cambridge University Press, Cambridge, 2003.
L. D. Faddeev and L. A. Takhtajan. Hamiltonian Methods in the Theory of Solitons. Springer-Verlag, Berlin, 1987.
V. A. Marchenko. Sturm-Liouville operators and applications. Birkhäuser, Basel, 1987.
B. M. Levitan. Inverse Sturm-Liouville Problems. VSP Architecture, Zeist, 1987.
F. Guerra and R. Marra. Origin of the quantum observable operator algebra in the frame of stochastic mechanics. Phys. Rev. D, 28(8):1916–1921, 1983.
F. Guerra and R. Marra. Discrete stochastic variational principles and quantum mechanics. Phys. Rev. D, 29(8):1647–1655, 1984.
F. Guerra and L. M. Morato. Quantization of dynamical systems and stochastic control theory. Phys. Rev. D, 27(8):1774–1786, 1983.
F. Calogero, editor. Nonlinear Evolution Equations Solvable by the Spectral Transform, volume 26 of Res. Notes in Math. Pitman, London, 1978.
V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. I. Pitaevskii. Theory of Solitons: The Inverse Scattering Method. Plenum, New York, 1984.
M. J. Ablowitz and P. A. Clarkson. Solitons, Nonlinear Evolution Equations and Inverse Scattering, volume 149 of London Mathematical Society Lecture Notes Series. ,n.Cambridge University Press, Cambridge, 1991.
I. D. Iliev, E. Kh. Christov, and K. P. Kirchev. Spectral Methods in Soliton Equations, volume 73 of Pitman Monographs and Surveys in Pure and Applied Mathematics. John Wiley & Sons, New York, 1991.
M. A. Naimark. Linear Differential Operators. Nauka, Moskow, 1969.
N. Dunford and J. T. Schwartz. Linear Operators. Part 1, 2, 3. Wiley Interscience Publications, New York, 1971.
A. B. Shabat. Inverse-scattering problem for a system of differential equations. Funct. Anal. Appl., 9(3):244–247, 1975.
A. B. Shabat. An inverse scattering problem. Diff. Equ., 15(10):1299–1307, 1979.
A. V. Mikhailov. Reduction in integrable systems. The reduction group. JETP Lett., 32:174, 1980.
P. J. Caudrey. The inverse problem for the third order equation u xxx+ q(x) ux+ r(x)u=-iζ3 u. Phys. Lett. A, 79(4):264–268, 1980.
P. J. Caudrey. The inverse problem for a general N×N spectral equation. Physica D: Nonl. Phen., 6(1):51–66, 1982.
R. Beals and R. R. Coifman. Scattering and inverse scattering for first order systems. Commun. Pure Appl. Math., 37:39–90, 1984.
R. Beals and R. R. Coifman. Inverse scattering and evolution equations. Commun. Pure Appl. Math., 38(1):29–42, 1985.
R. Beals and R. R. Coifman. The D-bar approach to inverse scattering and nonlinear evolutions. Physica D, 18(1-3):242–249, 1986.
R. Beals and R. R. Coifman. Scattering and inverse scattering for first-order systems: II. Inverse Probl., 3(4):577–593, 1987.
R. Beals and R. R. Coifman. Linear spectral problems, non-linear equations and the overline partial-method. Inverse Probl., 5(2):87–130, 1989.
V. S. Gerdjikov and A. B. Yanovski. Completeness of the eigenfunctions for the Caudrey–Beals–Coifman system. J. Math. Phys., 35:3687–3721, 1994.
V. S. Gerdjikov. On the spectral theory of the integro-ifferential operator generating nonlinear evolution equations. Lett. Math. Phys., 6:315–324, 1982.
L. D. Faddeev. Properties of the S-matrix of the one-dimensional Schrödinger equation. Amer. Math. Soc. Transl. (Ser. 2), 65:139–166, 1967.
L. D. Faddeev. Inverse problem of quantum scattering theory. II. J. Math. Sci., 5(3):334–396, 1976. In “Contemporary Mathematical Problems”, English translation from: VINITI, 3, 93–180 (1974).
I. S. Frolov. Inverse scattering problem for a Dirac system on the whole axis. Soviet Math. Dokl, 13:1468–1472, 1972.
J. Satsuma and N. Yajima. Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media. Prog. Theor. Phys. Suppl., 55:284, 1974.
A. C. Newell. The general structure of integrable evolution equations. Proc. R. Soc. Lond. A, Math. Phys. Sci., 365(1722):283–311, 1979.
E. C. Titchmarsch. Eigenfunctions Expansions Associated with Second Order Differential Equations. Part I. Clarendon Press, Oxford, 1983.
T. Kawata. Inverse scattering transform of the higher order eigenvalue problem. J. Phys. Soc. Japan, 57(2):422–435, 1988.
X. Zhou. Direct and inverse scattering transforms with arbitrary spectral singularities. Commun. Pure Appl. Math., 42:895–938, 1989.
A. C. Scott, F. Y. F. Chu, and D. W. McLaughlin. The soliton: A new concept in applied science. Proc. IEEE, 61(10):1443–1483, 1973.
P. G. Drazin and R. S. Johnson. Solitons: An Introduction. Cambridge texts in Applied Mathematics. Cambridge University Press, Cambridge, 1989.
V. S. Gerdjikov and M. I. Ivanov. A quadratic pencil of general type and nonlinear evolution equations. II. Hierarchies of Hamiltonian structures. Russ. Bulg. J. Phys. 10, 130–143, 1983.
V. S. Gerdjikov and M. I. Ivanov. The quadratic bundle of general form and the nonlinear evolution equations. II. Hierarchies of Hamiltonian structures. Bulg. J. Phys., 10:130–143, 1983.
I. T. Gadjiev, V. S. Gerdjikov, and M. I. Ivanov. Hamiltonian structures of the nonlinear evolution equations related to the polynomial bundle. Notes of LOMI Sci., 120:55–68, 1982.
L. A. Bordag and A. B. Yanovski. Algorithmic construction of O(3) chiral field equation hierarchy and the Landau–Lifshitz equation hierarchy via polynomial bundle. J. Phys. A: Math. Gen., 29(17):5575–5590, 1996.
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Gerdjikov, V., Vilasi, G., Yanovski, A. (2008). The Direct Scattering Problem for theZakharov–Shabat System. In: Gerdjikov, V., Vilasi, G., Yanovski, A. (eds) Integrable Hamiltonian Hierarchies. Lecture Notes in Physics, vol 748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77054-1_3
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