We study the Cauchy problem for the Korteweg–de-Vries equation in the class of functions approaching a finite-zone periodic solution of the Korteweg–de-Vries equation as x→−∞ and 0 as x→+∞. We prove the existence of infinitely many “regularized” integrals of motion for the solutions u(x, t) of the Cauchy problem with explicit dependence on time.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg–de Vries equation,” Phys. Rev. Lett., 19, 1095–1097 (1967).
R. M. Miura, C. S. Gardner, and M. D. Kruskal, “Korteweg–de Vries equation and generalizations, II. Existence of conservation laws and constants of motion,” J. Math. Phys., 9, No. 8, 1204–1209 (1968).
M. D. Kruskal, R. M. Miura, C. S. Gardner, and N. J. Zabusky, “Korteweg–de Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws,” J. Math. Phys., 11, No. 3, 952–960 (1970).
P. D. Lax, “Integrals of nonlinear equations and solitary waves,” Comm. Pure Appl. Math., 21, No. 2, 467–490 (1968).
V. E. Zakharov and L. D. Faddeev, “Korteweg–de Vries equation is a completely integrable Hamiltonian system,” Funkts. Anal. Prilozhen., 5, No. 4, 18–27 (1971).
E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations [Russian translation], Vol. 2, Inostr. Lit., Moscow (1961).
V. A. Marchenko, Sturm–Liouville Operators and Their Applications [in Russian], Naukova Dumka, Kiev (1977).
V. D. Ermakova, Inverse Scattering Problem for the Schr¨odinger Equation with Nondecreasing Potential and Its Application to the Integration of the Korteweg–de-Vries Equation [in Russian], Candidate-Degree Thesis (Physics and Mathematics), Kharkov (1983).
N. E. Firsova, “Inverse scattering problem for the perturbed Hill operator,” Mat. Zametki, 18, No. 6, 831–843 (1974).
I. Egorova and G. Teshl, “On the Cauchy problem for the Korteweg–de-Vries equation with steplike finite-gap initial data II. Perturbations with finite moments,” J. d’Anal. Math., 115, No. 1, 71–101 (2011).
S. P. Novikov, “Periodic problem for the Korteweg–de-Vries equation,” Funkts. Anal. Prilozhen., 8, Issue 3, 54–66 (1974).
V. A. Marchenko, “Periodic Korteweg–de-Vries problem,” Mat. Sb., 8, Issue 3, 331–356 (1974).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 12, pp. 1587–1601, December, 2015.
Rights and permissions
About this article
Cite this article
Andreev, K.N., Khruslov, E.Y. Regularized Integrals of Motion for the Korteweg–De-Vries Equation in the Class of Nondecreasing Functions. Ukr Math J 67, 1793–1809 (2016). https://doi.org/10.1007/s11253-016-1191-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-016-1191-8