Geodesic
Solving the~modified Camassa--Holm equation via the~inverse scattering transform
Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 2, pp. 326-349

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With the aid of the reciprocal transformation and the associated equation, we study the inverse scattering transform with a matrix Riemann–Hilbert problem for the modified Camassa–Holm (mCH) equation with nonzero boundary conditions (NZBC) at infinity. In terms of a suitable uniformization variable, the direct and inverse scattering problems are presented for the associated modified Camassa–Holm (amCH) equation. By means of the reciprocal transformation and the reconstruction formula for the potential of the amCH equation, we present the $N$-soliton solution for the mCH equation with NZBC. As applications, various solutions including both bright and dark types, smooth soliton solutions, singular soliton solutions, and multi-valued singular soliton solutions of the mCH equation and their interactions are exhibited.
Keywords: modified Camassa–Holm equation, reciprocal transformation, inverse scattering transform
Mots-clés : soliton solutions. 
@article{TMF_2023_216_2_a9,
     author = {Hui Mao and Yu Qian and Yuanyuan Miao},
     title = {Solving the~modified {Camassa--Holm} equation via the~inverse scattering transform},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {326--349},
     publisher = {mathdoc},
     volume = {216},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2023_216_2_a9/}
}
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Hui Mao; Yu Qian; Yuanyuan Miao. Solving the~modified Camassa--Holm equation via the~inverse scattering transform. Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 2, pp. 326-349. http://geodesic.mathdoc.fr/item/TMF_2023_216_2_a9/