Abstract:
The method of inverse spectral problem is used to integrate the nonlinear Hirota equation in the class of periodic infinite-zone functions. The evolution of spectral data is introduced for the periodic Dirac operator whose coefficient is the solution of the nonlinear Hirota equation. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of five times continuously differentiable periodic infinite-zone functions is shown. In addition, it is proved that if the initial function is real-analytic and π-periodic, then the solution of the Cauchy problem for the Hirota equation is also a real-analytic function of the variable x; next, if the number π/2 is a period (antiperiod) of the initial function, then the number π/2 is a period (antiperiod) in the variable x for the solution of the Cauchy problem for the Hirota equation.
Citation:
G. A. Mannonov, A. B. Khasanov, “The Cauchy problem for a nonlinear Hirota equation in the class of periodic infinite-zone functions”, Algebra i Analiz, 34:5 (2022), 139–172; St. Petersburg Math. J., 34:5 (2023), 821–845
\Bibitem{ManKha22}
\by G.~A.~Mannonov, A.~B.~Khasanov
\paper The Cauchy problem for a nonlinear Hirota equation in the class of periodic infinite-zone functions
\jour Algebra i Analiz
\yr 2022
\vol 34
\issue 5
\pages 139--172
\mathnet{http://mi.mathnet.ru/aa1833}
\transl
\jour St. Petersburg Math. J.
\yr 2023
\vol 34
\issue 5
\pages 821--845
\crossref{https://doi.org/10.1090/spmj/1780}
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https://www.mathnet.ru/eng/aa1833
https://www.mathnet.ru/eng/aa/v34/i5/p139
This publication is cited in the following 12 articles:
A. B. Khasanov, Kh. N. Normurodov, T. G. Khasanov, “Integration of a Nonlinear Sine-Gordon–Liouville-Type Equation in the Class of Periodic Infinite-Gap Functions”, Ukr Math J, 76:8 (2025), 1381
A. B. Khasanov, R. Kh. Eshbekov, T. G. Hasanov, “Integration of a non-linear Hirota type equation with additional terms”, Izv. Math., 89:1 (2025), 196–219
A. B. Khasanov, Kh. N. Normurodov, “Integrirovanie uravneniya tipa sinus-Gordona s dopolnitelnym chlenom v klasse periodicheskikh beskonechnozonnykh funktsii”, Izv. vuzov. Matem., 2024, no. 3, 70–83
A. B. Khasanov, T. G. Khasanov, “The Cauchy Problem for the Nonlinear Complex Modified Korteweg-de Vries Equation with Additional Terms in the Class of Periodic Infinite-Gap Functions”, Sib Math J, 65:4 (2024), 846
A. B. Khasanov, Kh. N. Normurodov, “Integration of a Sine-Gordon Type Equation with an Additional Term in the Class of Periodic Infinite-Gap Functions”, Russ Math., 68:3 (2024), 58
A. B. Khasanov, T. G. Khasanov, “Zadacha Koshi dlya nelineinogo kompleksnogo modifitsirovannogo uravneniya Kortevega — de Friza (kmKdF) s dopolnitelnymi chlenami v klasse periodicheskikh beskonechnozonnykh funktsii”, Sib. matem. zhurn., 65:4 (2024), 735–759
A. B. Khasanov, Kh. N. Normurodov, T. G. Khasanov, “Integration of a nonlinear sine-Gordon–Liouville-type equation in the class of periodic infinite-gap functions”, Ukr. Mat. Zhurn., 76:8 (2024), 1217
A. B. Khasanov, A. A. Reyimberganov, “On the Hirota equation with a self-consistent source”, Theoret. and Math. Phys., 221:2 (2024), 1852–1866
A. B. Khasanov, A. A. Abdivokhidov, R. Kh. Eshbekov, “Negative Order Modified Korteweg–de Vries–Liouville (nmKdV-L) Equation in the Class of Periodic Infinite-gap Functions”, Lobachevskii J Math, 45:12 (2024), 6497
A. B. Khasanov, Kh. N. Normurodov, U. O. Hudayerov, “Integrating the modified Korteweg–de Vries–sine-Gordon equation in the class of periodic infinite-gap functions”, Theoret. and Math. Phys., 214:2 (2023), 170–182
A. B. Khasanov, U. O. Xudayorov, “Integration of the Modified Korteweg–de Vries–Liouville Equation in the Class of Periodic Infinite-Gap Functions”, Math. Notes, 114:6 (2023), 1247–1259
A. Khasanov, R. Eshbekov, Kh. Normurodov, “Integration of a Nonlinear Hirota Type Equation with Finite Density in the Class of Periodic Functions”, Lobachevskii J Math, 44:10 (2023), 4329
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