This is a preview of subscription content, log in via an institution to check access.
Literature Cited
C. Gardner, J. Green, M. Kruskal, and R. Miura, "A method for solving the Korteweg—de Vries equation," Phys. Rev. Lett.,19, 1095–1098 (1967).
P. Lax, "Integrals of nonlinear equations of evolution and solitary waves," Comm. Pure Appl. Math.,21, No. 2, 467–490 (1968).
I. M. Gel'fand and B. M. Levitan, "The determination of a differential equation by its spectral function," Izv. Akad. Nauk SSSR, Ser. Matem.,15, 309–360 (1951).
V. A. Marchenko, "Some problems of the theory of one-dimensional differential operators. I," Tr. Mosk. Matem., Ob-va,1, 327–420 (1952).
L. D. Faddeev, "S-matrix properties of the one-dimensional Schrödinger equation," Tr. Matem. V. A. Steklov Inst.,73, 314–336 (1964).
V. E. Zakharov, "Kinetic equation for solitons," Zh. Éksp. Teor. Fiz.,60, No. 3, 993–1000 (1971).
S. P. Novikov, "Periodic problem for the Korteweg—de Vries equation. I," Funktsional'. Analiz i Ego Prilozhen.,8, No. 3, 54–66 (1974).
B. A. Dubrovin, "Inverse-scattering problem for periodic finite-band potentials," Funktsional'. Analiz i Ego Prilozhen.,9, No. 1, 65–66 (1975).
A. R. Its and V. B. Matveev, "Hill operators with a finite number of gaps," Funktsional Analiz i Ego Prilozhen.9, No. 1, 69–70 (1975).
V. A. Marchenko, "The periodic Korteweg—de Vriesproblem," Dokl. Akad. Nauk SSSR,217, No. 2, 276–279 (1974).
E. L. Ince, "Further investigations into the periodic Lamé functions," Proc. Roy. Soc. Edinburgh,60, 83–99 (1940).
N. I. Akhiezer, "Continual analog of orthogonal polynomials on a system of intervals," Dokl. Akad. Nauk SSSR,141, No. 2, 263–266 (1971).
H. Hochstadt, "On the determination of a Hill's equation from its spectrum," Arch. Rat. Mech. and Anal.,19, No. 5, 353–362 (1965).
E. C. Titchmarsh, Eigenfunction Expansion Associated with Second-Order Differential Equations, Oxford University Press (1958).
A. B. Shabat, "Potentials with vanishing reflection coefficients," in: Dynamics of Continuous Media, No. 5, Novosibirsk (1970), pp. 130–156.
R. Miura, C. Gardner, and M. Kruskal, "Korteweg—de Vries equation and generalizations," J. Math. Phys.,9, 1202–1209 (1968).
V. A. Zakharov and L. D. Faddeev, "The Korteweg—de Vries equations, a completely integrable Hamiltonian system," Funktsional'. Analiz i Ego Prilozhen.,5, No. 4, 18–27 (1971).
B. A. Dubrovin and S. P. Novikov, "Periodic problem for the Korteweg—de Vries and Sturm—Liouville equations; their connection with algebraic geometry," Dokl. Akad. Nauk SSSR,219, No. 3, 19–22 (1974).
B. A. Dubrovin and S. P. Novikov, "Periodic and conditionally periodic analogs of many-soliton solutions of the KdV equation," Zh. Éksp. Teor. Fiz.,12, 2131–2144 (1974).
P. Lax, "Periodic solutions of the KdV equations," Lectures in Appl. Math.,15, 85–96 (1974).
Additional information
Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 9, No. 3, pp. 41–51, July–September, 1975.
Rights and permissions
About this article
Cite this article
Dubrovin, B.A. Periodic problems for the Korteweg — de Vries equation in the class of finite band potentials. Funct Anal Its Appl 9, 215–223 (1975). https://doi.org/10.1007/BF01075598
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01075598