Geodesic
Multisoliton solutions of the~two-component Camassa--Holm equation and its reductions
Teoretičeskaâ i matematičeskaâ fizika, Tome 214 (2023) no. 3, pp. 359-386

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The Bäcklund transformation for an integrable two-component Camassa–Holm ($2$CH) equation is presented and studied. It involves both dependent and independent variables. A nonlinear superposition formula is given for constructing multisoliton, multiloop, and multikink solutions of the $2$CH equation. We also present solutions of the Camassa–Holm equation, the two-component Hunter–Saxton ($2$HS) equation, and the Hunter–Saxton equation, which all arise from solutions of the $2$CH equation. By appropriate limit procedures, a solution of the $2$HS equation is successfully obtained from that of the $2$CH equation, which is worked out with the method of Bäcklund transformations. By analyzing the solution, we obtain the soliton and loop solutions for $2$HS equation.
Keywords: two-component Camassa–Holm equation, two-component Hunter–Saxton equation, Bäcklund transformation, reduction.
Mots-clés : soliton 
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     author = {Gaihua Wang},
     title = {Multisoliton solutions of the~two-component {Camassa--Holm} equation and its reductions},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {359--386},
     publisher = {mathdoc},
     volume = {214},
     number = {3},
     year = {2023},
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     url = {http://geodesic.mathdoc.fr/item/TMF_2023_214_3_a1/}
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Gaihua Wang. Multisoliton solutions of the~two-component Camassa--Holm equation and its reductions. Teoretičeskaâ i matematičeskaâ fizika, Tome 214 (2023) no. 3, pp. 359-386. http://geodesic.mathdoc.fr/item/TMF_2023_214_3_a1/