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Integrable nonlinear equations on a half-axis


A study is made of the nonlinear Schrödinger equation and the modified Kortewegde Vries equation on the half-axis 0≤x<∞. Classes of solutions are identified when there is sufficient initial data for unique solvability of a corresponding problem. The class of solutions investigated is characterized in terms of the Weyl-Titchmarsh function of an auxiliary linear problem. A solution is constructed in explicit form when the corresponding Weyl-Titchmarsh function v(t, z) is a rational function of z.

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Literature cited

  1. 1.

    S. P. Novikov (ed.), Theory of Solitons, Method of the Inverse Problem [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  2. 2.

    R. Bullaf and F. Kodri (eds.), Solitons [Russian translation], Mir, Moscow (1983).

    Google Scholar 

  3. 3.

    L. A. Sakhnovich, “Factorization problems and operator identities,” Usp. Mat. Nauk,41, No. 1, 3–55 (1986).

    Google Scholar 

  4. 4.

    L. A. Sakhnovich, Nonlinear Equations and Inverse Problems on a Half-Axis [in Russian], Insitute of Mathematics, Academy of Sciences of the Ukraine, Preprint 87.30, Kiev (1987).

    Google Scholar 

  5. 5.

    L. A. Sakhnovich, “Procedure of solving nonlinear equations on a half-axis,” Nonlinear World,2, 314–317 (1989).

    Google Scholar 

  6. 6.

    V. A. Marchenko, Integrability and Kinetic Equations for Solitons [in Russian], Naukova Dumka, Kiev (1980), pp. 168–212.

    Google Scholar 

  7. 7.

    L. A. Sakhnovich, “Explicit formulas for spectral characteristics and solutions of the sine-Gordon equation,” Ukr. Mat. Zh.,42, No. 11, 1517–1523 (1990).

    Google Scholar 

  8. 8.

    L. A. Sakhnovich and I. F. Tydnyuk, “Effective solution of the sine-Gordon equation”, Paper deposited in UkrNIINTI, No. 2015-90, Kiev (1990).

  9. 9.

    L. A. Sakhnovich and I. F. Tydnyuk, “Effective solution of the sine-Gordon equation,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 9, 20–25 (1990).

    Google Scholar 

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Additional information

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 11, pp. 1578–1584, November, 1991.

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Sakhnovich, L.A. Integrable nonlinear equations on a half-axis. Ukr Math J 43, 1470–1476 (1991).

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  • Initial Data
  • Rational Function
  • Nonlinear Equation
  • Explicit Form
  • Linear Problem