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Explicit formulas for spectral characteristics and solutions of the sh-Gordon equation

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Ukrainian Mathematical Journal Aims and scope

Abstract

To the sh-Gordon equation

$$\frac{{\partial ^2 \varphi }}{{\partial t\partial x}} = 4sh\varphi , 0 \leqslant x, t< \infty ,$$

corresponds a linear system and its Weyl-Titchmarsh functionυ(i, z). If υ (0, z) is a rational function, then υ(i,z) is also a rational function in z. For this case a procedure is given for the explicit construction of ϕ (x, i) of the equation.

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Literature cited

  1. Yu. M. Berezanskii, “Integration of nonlinear difference equations by the method of the inverse problem,” Dokl. Akad. Nauk SSSR,281, No. 1, 16–19 (1985).

    Google Scholar 

  2. L. A. Sakhnovich, “Evolution of spectral data and nonlinear equations,” Ukr. Mat. Zh.,40, No. 4, 533–535 (1988).

    Google Scholar 

  3. L. A. Sakhnovich, “Nonlinear equations and inverse problems on the half-line,” Preprint, Inst. Mat., Akad. Nauk UkrSSR, Kiev (1987).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  6. L. A. Sakhnovich and I. F. Tydnyuk, “Effective solution of the sh-Gordon equation,” Dep. UkrNIINTI, No. 2015-90, Kiev (1990).

    Google Scholar 

  7. V. A. Marchenko, Nonlinear Equations and Operator Algebras [in Russian], Naukova Dumka, Kiev (1986).

    Google Scholar 

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Authors and Affiliations

  1. Odessa Electrical Engineering Communications Institute, USSR

    L. A. Sakhnovich

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  1. L. A. Sakhnovich
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 11, pp. 1517–1523, November, 1990.

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Sakhnovich, L.A. Explicit formulas for spectral characteristics and solutions of the sh-Gordon equation. Ukr Math J 42, 1359–1365 (1990). https://doi.org/10.1007/BF01066192

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  • DOI: https://doi.org/10.1007/BF01066192

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