Geodesic
Integration of the nonlinear Korteweg---de Vries equation with loaded term and source
Sibirskij žurnal industrialʹnoj matematiki, Tome 25 (2022) no. 2, pp. 127-142

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A simple algorithm for deriving an analog of the system of Dubrovin differential equations is proposed. It is shown that the sum of a uniformly convergent functional series constructed by solving the system of Dubrovin equations and the first trace formula really satisfies the loaded nonlinear Korteweg—de Vries equation with a source. In addition, it has been proven that if the initial function is a real $\pi$-periodic analytic function, then the solution of the Cauchy problem is also a real analytic function with respect to the variable $x$; and if the number $\pi/n$ is the period of the initial function, then the number $\pi/n$ is the period for solving the Cauchy problem with respect to the variable $x$. Here $n$ is a natural number, $n\geqslant 2$.
Keywords: Korteweg—de Vries equation, trace formulas, inverse spectral problem, Hill operator, Dubrovin’s system of equations. . 
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     author = {A. B. Khasanov and T. G. Hasanov},
     title = {Integration of the nonlinear {Korteweg---de} {Vries} equation with loaded term and source},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {127--142},
     publisher = {mathdoc},
     volume = {25},
     number = {2},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2022_25_2_a8/}
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A. B. Khasanov; T. G. Hasanov. Integration of the nonlinear Korteweg---de Vries equation with loaded term and source. Sibirskij žurnal industrialʹnoj matematiki, Tome 25 (2022) no. 2, pp. 127-142. http://geodesic.mathdoc.fr/item/SJIM_2022_25_2_a8/