Geodesic
Integration of the Korteweg–de Vries type equations with a loaded term in the class of periodic functions
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 64 (2024), pp. 60-69.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we study the integrability of a Korteweg–de Vries type equation with a loaded term in the class of periodic functions using direct and inverse spectral problems posed for the Sturm–Liouville operator on the entire axis. We present some information about the Sturm–Liouville operator with a periodic coefficient and its application to solving a loaded Korteweg–de Vries type equation. We show that the function constructed using the obtained system of Dubrovin differential equations and trace formulas is a solution to the problem posed. We present important consequences about the period of the solution with respect to $x$ and about the analyticity of the solution with respect to $x$ for a Korteweg–de Vries type equation with a loaded term.
Keywords: Korteweg–de Vries equation, Sturm–Liouville operator, inverse spectral problem, Dubrovin’s system of equations
Mots-clés : trace formula 
@article{IIMI_2024_64_a4,
     author = {M. M. Matyoqubov},
     title = {Integration of the {Korteweg{\textendash}de} {Vries} type equations with a loaded term in the class of periodic functions},
     journal = {Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta},
     pages = {60--69},
     publisher = {mathdoc},
     volume = {64},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IIMI_2024_64_a4/}
}
TY  - JOUR
AU  - M. M. Matyoqubov
TI  - Integration of the Korteweg–de Vries type equations with a loaded term in the class of periodic functions
JO  - Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
PY  - 2024
SP  - 60
EP  - 69
VL  - 64
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IIMI_2024_64_a4/
LA  - ru
ID  - IIMI_2024_64_a4
ER  - 
%0 Journal Article
%A M. M. Matyoqubov
%T Integration of the Korteweg–de Vries type equations with a loaded term in the class of periodic functions
%J Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
%D 2024
%P 60-69
%V 64
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IIMI_2024_64_a4/
%G ru
%F IIMI_2024_64_a4
M. M. Matyoqubov. Integration of the Korteweg–de Vries type equations with a loaded term in the class of periodic functions. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 64 (2024), pp. 60-69. http://geodesic.mathdoc.fr/item/IIMI_2024_64_a4/

[1] Novikov S.P., “The periodic problem for the Korteweg–de Vries equation”, Functional Analysis and Its Applications, 8:3 (1974), 236–246 | DOI | MR | Zbl

[2] Dubrovin B.A. Novikov S.P., “Periodic and conditionally periodic analogs of the many-soliton solutions of the Korteweg–de Vries equation”, Soviet Physics JETP, 40:6 (1974), 1058–1063 | MR

[3] Marchenko V.A., “The periodic Korteweg–de Vries problem”, Mathematics of the USSR–Sbornik, 24:3 (1974), 319–344 | DOI | MR | Zbl

[4] Dubrovin B.A., “Periodic problems for the Korteweg–de Vries equation in the class of finite band potentials”, Functional Analysis and Its Applications, 9:3 (1975), 215–223 | DOI | MR | Zbl

[5] Its A.R., Matveev V.B., “Schrödinger operators with finite-gap spectrum and $N$-soliton solutions of the Korteweg–de Vries equation”, Theoretical and Mathematical Physics, 23:1 (1975), 343–355 | DOI | MR

[6] Lax P.D., “Periodic solutions of the KdV equations”, Lectures in Applied Mathematics, 15 (1974), 85–96 | MR

[7] Lax P.D., “Periodic solutions of the KdV equation”, Communications on Pure and Applied Mathematics, 28:1 (1975), 141–188 | DOI | MR | Zbl

[8] McKean H.P., Trubowitz E., “Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points”, Communications on Pure and Applied Mathematics, 29:2 (1976), 143–226 | DOI | MR | Zbl

[9] Levitan B.M., Sargsyan I.S., Sturm–Liouville and Dirac operators, Nauka, Moscow, 1988 | MR

[10] Stankevich I.V., “A certain inverse spectral analysis problem for Hill’s equation”, Doklady Akademii Nauk SSSR, 192:1 (1970), 34–37 (in Russian) | Zbl

[11] Trubowitz E., “The inverse problem for periodic potentials”, Communications on Pure and Applied Mathematics, 30:3 (1977), 321–337 | DOI | MR | Zbl

[12] Hochstadt H., “A generalization of Borg’s inverse theorem for Hill’s equations”, Journal of Mathematical Analysis and Applications, 102:2 (1984), 599–605 | DOI | MR | Zbl

[13] Khasanov A.B., Matyakubov M.M., “Integration of the nonlinear Korteweg–De Vries equation with an additional term”, Theoretical and Mathematical Physics, 203:2 (2020), 596–607 | DOI | DOI | MR | Zbl

[14] Yakhshimuratov A.B., Matyokubov M.M., “Integration of a loaded Korteweg–de Vries equation in a class of periodic functions”, Russian Mathematics, 60:2 (2016), 72–76 | DOI | MR | Zbl

[15] Khasanov A.B., Khasanov T.G., “Integration of the nonlinear Korteweg–de Vries equation with a loaded term and sourse”, Journal of Applied and Industrial Mathematics, 16:2 (2022), 227–239 | DOI | MR | Zbl

[16] Khasanov A.B., Hoitmetov U.A., “Integration of the general loaded Korteweg–de Vries equation with an integral type source in the class of rapidly decreasing complex-valued functions”, Russian Mathematics, 65:7 (2021), 43–57 | DOI | MR | Zbl

[17] Rodriguez M., Li J., Qiao Z., “Negative order KdV equation with no solitary traveling waves”, Mathematics, 10:1 (2022), 48 | DOI

[18] Urazboev G.U., Khasanov M.M., Baltaeva I.I., “Integration of the negative order Korteweg–de Vries equation with a special source”, Bulletin of Irkutsk State University. Series Mathematics, 44 (2023), 31–43 | DOI | MR | Zbl