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Integration of mKdV Equation with a Self-Consistent Source in the Class of Finite Density Functions in the Case of Moving Eigenvalues

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Abstract

In this paper, we prove the possibility of using the inverse scattering problem method for integrating an mKdV equation with a self-consistent source in the class of finite density functions in the case when the corresponding spectral problem has moving simple eigenvalues.

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REFERENCES

  1. Wadati, M. “The Exact Solution of the Modified Korteweg–de Vries Equation”, J. Phys. Soc. Japan 32, 1681–1681 (1972).

  2. Gardner, S.S., Green, I.M., Kruskal, M.D., Miura, R.M. “Method for Solving the Korteweg–de Vries Equation”, Phys. Rev. Lett. 19, 1095–1097 (1967).

  3. Gasymov, M.G., Levitan, B.M. “The Inverse Problem for the Dirac System”, Dokl. Akad. Nauk SSSR 167 (5), 967–970 (1966) [in Russian].

  4. Zakharov, V.E., Shabat, A.B. “Exact Theory of Two-Dimensional Self-Focusing and One-Dimensional Self-Modulation of Wave in Nonlinear Media”, J. Experim. Theor. Phys. 61 (1), 118–134 (1971).

  5. Frolov, I.S. “Inverse Scattering Problem for a Dirac System on the Whole Axis”, Dokl. Akad. Nauk SSSR 207 (1) 44–47 (1972) [in Russian].

  6. Nizhnik, L.P., Fam Loi Wu. “The Inverse Scattering Problem on the Semiaxis with Non-Self-Adjoint Potential Matrix”, Ukr. Matem. Zhurn. 26 (4), 469–486 (1974).

  7. Khasanov, A.B. “An Inverse Problem in Scattering Theory for a System of Two First-Order Nonselfadjoint Differential Equations”, Dokl. Akad. Nauk SSSR 277 (3), 559–562 (1984).

  8. Takhtadzhyan, L.A., Faddeev, L.D. Hamiltonian Methods in the Theory of Solitons (Nauka, Moscow, 1986) [in Russian].

  9. Ablowitz, M., Kaup, D., Newell, A., Segur, H. “The Inverse Scattering Transform-Fourier Analysis for Nonlinear Problems”, Stud. Appl. Math. 53 (4), 249–315 (1974).

  10. Zakharov, V.E., Takhtadzhyan, L.A., Faddeev, L.D. “Complete Description of Solutions of the “sin-Gordon” Equation”, Dokl. Akad. Nauk SSSR 219 (6), 1334–1337 (1974).

  11. Mel'nikov, V.K. “Exact Solutions of the Korteweg–de Vries Equation with a Self-Consistent Source”, Phys. Lett. A 128, 488–4924 (1988).

  12. Mel'nikov, V.K. “Creation and Annihilation of Solitons in the System Described by the KdV Equation with a Self-Consistent Source”, Inverse Probl. 6, 809–823 (1990).

  13. Mel'nikov, V.K. “Integration of the Nonlinear Schrodinger Equation with a Source”, Inverse Probl. 8, 133–147 (1992).

  14. Urazboev, G.U., Khasanov, A.B. “Integrating the Korteweg–de Vries Equation with a Self-Consistent Source and “Steplike” Initial Data”, Teor. i Matem. Fiz. 129 (1), 38–54 (2001).

  15. Leon, J., Latifi, A. “Solution of an Initial-Boundary Value Problem for Coupled Nonlinear Waves”, J. Phys. A: Math. Gen. 23, 1385–1403 (1990).

  16. Yakhshimuratov, A.B., Khasanov, M.M. “Integration of the Modified Korteweg-de Vries Equation with a Self-Consistent Source in the Class of Periodic Functions”, Diff. Equat. 50 (4), 533–540 (2014).

  17. Romanova, N.N. “N-Soliton Solution “on a Pedestal” of the Modified Korteweg–de Vries Equation”, Teor. i Matem. Fiz. 39 (2), 205–215 (1979).

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Correspondence to K. A. Mamedov.

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Russian Text © The Author(s), 2020, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2020, No. 10, pp. 73–85.

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Mamedov, K.A. Integration of mKdV Equation with a Self-Consistent Source in the Class of Finite Density Functions in the Case of Moving Eigenvalues. Russ Math. 64, 66–78 (2020). https://doi.org/10.3103/S1066369X20100072

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  • DOI: https://doi.org/10.3103/S1066369X20100072

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