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Integration of the Modified Korteweg–de Vries–Liouville Equation in the Class of Periodic Infinite-Gap Functions

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Abstract

In this paper, the inverse spectral problem method is used to integrate the nonlinear mKdV–L equation in the class of periodic infinite-gap functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of \(6\) times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent function series constructed by solving the system of Dubrovin equations and by using the first trace formula satisfies the mKdV–L equations. Moreover, we prove that if the initial function is a real-valued \(\pi\)-periodic analytic function, then the solution of the Cauchy problem for the mKdV–L equation is a real-valued analytic function in the variable \(x\) as well; and if the number \(\frac{\pi}{2}\) is a period (respectively, antiperiod) of the initial function, then the number \(\frac{\pi}{2}\) is the period (respectively, antiperiod) in the variable \(x\) of the solution of the Cauchy problem for the mKdV–L equations.

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Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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Authors and Affiliations

  1. Samarkand State University, Samarkand, 140104, Uzbekistan

    A. B. Khasanov

  2. Samarkand State University of Architecture and Construction, Samarkand, 140143, Uzbekistan

    U. O. Khudayorov

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  1. A. B. Khasanov
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  2. U. O. Khudayorov
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Correspondence to A. B. Khasanov.

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Translated from Matematicheskie Zametki, 2023, Vol. 114, pp. 894–908 https://doi.org/10.4213/mzm13907.

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Khasanov, A.B., Khudayorov, U.O. Integration of the Modified Korteweg–de Vries–Liouville Equation in the Class of Periodic Infinite-Gap Functions. Math Notes 114, 1247–1259 (2023). https://doi.org/10.1134/S0001434623110573

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