Geodesic
On the integration of the periodic Camassa--Holm equation with a self-consistent source
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 15 (2022) no. 6, pp. 785-796

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Recently, much attention has been paid to non-linear equations with a self-consistent source that have soliton solutions. Sources arise in solitary waves with a variable speed and lead to a variety of physical models. Such models are usually used to describe interactions between solitary waves. The Cauchy problem for the Camassa–Holm equation with a source in the class of periodic functions is considered in this paper. The main result of this work is a theorem on the evolution of the spectral data of the weighted Sturm–Liouville operator where potential of the operator is a solution of the periodic Camassa–Holm equation with a source. The obtained relations allow one to apply the method of the inverse spectral transform to solve the Cauchy problem for the periodic Camassa–Holm equation with a source.
Keywords: self-consistent source, trace formulas, inverse spectral problem, weighted Sturm–Liouville operator.
Mots-clés : Camassa–Holm equation 
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     title = {On the integration of the periodic {Camassa--Holm} equation with a self-consistent source},
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Aknazar B. Khasanov; Bazar A. Babajanov; Dilshod O. Atajonov. On the integration of the periodic Camassa--Holm equation with a self-consistent source. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 15 (2022) no. 6, pp. 785-796. http://geodesic.mathdoc.fr/item/JSFU_2022_15_6_a11/