Literature Cited
C. S. Gardner, J. M. Green, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg-de Vries equation,” Phys. Rev. Lett.19, 1095–1097 (1967).
V. A. Marchenko, “Periodic Korteweg-de Vries problem,” Mat. Sb.,95, 331–336 (1974).
S. P. Novikov, “Periodic problem for the Korteweg-de Vries equation,” Funktsion. Anal.,8, No. 3, 54–66 (1974).
A. R. Its and V. B. Matveev, “Schrödinger operators with finite-zone spectrum and N-soliton solutions of Korteweg-de Vries,” Teor. Mat. Fiz.,23, No. 1, 51–68 (1975).
P. D. Lax, “Periodic solutions of the KdV equation,” Commun. Pure Appl. Math.28, 141–188 (1975).
B. A. Bubnov, “Cauchy problem for the Korteweg-de Vries equation,” Dokl. Akad. Nauk SSSR,251, No. 4, 777 (1980).
A. V. Gurevich and A. P. Pitaevskii, “Decay of an initial discontinuity in the Korteweg-de Vries equation,” Pis'ma Zh. Eksp. Teor. Fiz.,17, No. 5, 268–271 (1973).
A. V. Gurevich and A. P. Pitaevskii, “Nonstationary structure of collisionless shock wave,” Zh. Eksp. Teor. Fiz.,65, No. 2, 590–604 (1973).
E. Ya. Khruslov, “Asymptotics of the solution of the Cauchy problem for the Korteweg-de Vries equation with initial data of piecewise type,” Mat. Sb.,99, 261–281 (1976).
Mayumi Ohmiya, “On the reflectionless solutions of the modified Korteweg-de Vries equation,” J. Math. Tokushima Univ.,12, 9–17 (1978).
J. L. Bona and R. Smith, “The initial-value problem for the Korteweg-de Vries equation,” Philos. Trans. R. Soc. London,A, 278, 555 (1975).
T. B. Benjamin, J. L. Bona, and J. J. Mahony, “Model equations for long waves in nonlinear dispersive systems,” Philos. Trans. R. Soc. London,A 272, 47 (1972).
E. P. Zhidkov and K. P. Kirchev, “Cauchy problem for the nonlinear modified Korteweg-de Vries equation on the whole line,” P5-81-130, Preprint JINR, Dubna (1981).
R. M. Miura, C. S. Gardner, and M. D. Kruskal, “Existence of conservation laws and constants of motion,” J. Math. Phys.,9, 1204 (1968).
T. B. Benjamin, “The stability of solitary waves,” Proc. R. Soc. London,A 328, 153 (1972).
E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Pt. 1, Oxford Univ. Press (1962).
Theory of Solitons. Method of the Inverse Problem [in Russian], Nauka, Moscow (1980).
Additional information
Dubna, Province of Moscow. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 25, No. 5, pp. 30–41, September–October, 1984.
Rights and permissions
About this article
Cite this article
Zhidkov, E.P., Kirchev, K.P. Cauchy problem for modified Korteweg-de Vries equation with piecewise-type initial data. Sib Math J 25, 710–719 (1984). https://doi.org/10.1007/BF00968683
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00968683