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## Archive of Issues

Uzbekistan Urgench
Year
2021
Volume
31
Issue
2
Pages
285-295
 Section Mathematics Title Integration of the Harry Dym equation with an integral type source Author(-s) Urazboev G.U.ab, Babadjanova A.K.a, Saparbaeva D.R.b Affiliations Institute of Mathematics, Khorezm Branch, Uzbekistan Academy of Sciencesa, Urgench State Universityb Abstract In the work, we deduce the evolution of scattering data for a spectral problem associated with the nonlinear evolutionary equation of Harry Dym with a self-consistent source of integral type. The obtained equalities completely determine the scattering data for any $t$, which makes it possible to apply the method of the inverse scattering problem to solve the Cauchy problem for the Harry Dym equation with an integral type source. Keywords nonlinear evolution equation, Harry Dym equation, integral source, inverse scattering method, Gelfand–Levitan–Marchenko equation UDC 517.957 MSC 34L25, 35Q41, 37K10, 35R30, 34M46 DOI 10.35634/vm210209 Received 24 December 2020 Language English Citation Urazboev G.U., Babadjanova A.K., Saparbaeva D.R. Integration of the Harry Dym equation with an integral type source, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 2, pp. 285-295. References Kruskal M.D. Lecture notes in physics, vol. 38, Berlin: Springer, 1975. Sabatier P.C. On some spectral problems and isospectral evolutions connected with the classical string problem. II: evolution equation, Lettere al Nuovo Cimento, 1979, vol. 26, no. 15, pp. 483-486. https://doi.org/10.1007/BF02750261 Yi-Shen L. Evolution equations associated with the eigenvalue problem based on the equation $\varphi_{xx} =[v(x)-k$$2 \rho$$2$ $(x)]\varphi$, Lettere al Nuovo Cimento, 1982, vol. 70, no. 1, pp. 1-12. https://doi.org/10.1007/BF02814006 Qiao Zh. A completely integrable system associated with the Harry Dym hierarchy, Journal of Nonlinear Mathematical Physics, 1994, vol. 1, no. 1, pp. 65-74. https://doi.org/10.2991/jnmp.1994.1.1.5 Qiao Zh. Commutator representations of nonlinear evolution equations: Harry Dym and Kaup-Newell cases, Journal of Nonlinear Mathematical Physics, 1995, vol. 2, no. 2, pp. 151-157. https://doi.org/10.2991/jnmp.1995.2.2.5 Calogero F., Degasperis A. Spectral transform and solitons: Tools to solve and investigate nonlinear evolution equations, vol. 1, Amsterdam: North-Holland, 1982. Wadati M., Ichikawa Y.H., Shimizu T. Cusp soliton of a new integrable nonlinear evolution equation, Progress of Theoretical Physics, 1980, vol. 64, no. 6, pp. 1959-1967. https://doi.org/10.1143/PTP.64.1959 Wadati M., Konno K., Ichikawa Y.H. New integrable nonlinear evolution equations, Journal of the Physical Society of Japan, 1979, vol. 47, no. 5, pp. 1698-1700. https://doi.org/10.1143/JPSJ.47.1698 Shimizu T. Properties of cusp soliton solution in Harry Dym Equation, Advanced Studies in Theoretical Physics, 2020, vol. 14, no. 5, pp. 227-235. https://doi.org/10.12988/astp.2020.91469 Hongyu L., Jian X. On the double-pole and two-soliton solutions of the Harry Dym equation, Applied Mathematics Letters, 2020, vol. 104, 106276. https://doi.org/10.1016/j.aml.2020.106276 Wen-Xiu M. An extended Harry Dym hierarchy, Journal of Physics A: Mathematical and Theoretical, 2010, vol. 43, no. 16, 165202. https://doi.org/10.1088/1751-8113/43/16/165202 Mel'nikov V.K. Integration method of the Korteweg-de Vries equation with a self-consistent source, Physics Letters A., 1988, vol. 133, issue 9, pp. 493-496. https://doi.org/10.1016/0375-9601(88)90522-1 Full text