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Multisoliton solutions of the two-component Camassa–Holm equation and its reductions

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Abstract

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.

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Funding

This work is supported by the National Natural Science Foundation of China (grant Nos. 11905110 and 12001560).

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  1. School of Mathematics and Physics, Nanjing Institute of Technology, Nanjing, China

    Gaihua Wang

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  1. Gaihua Wang
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Correspondence to Gaihua Wang.

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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 214, pp. 359–386 https://doi.org/10.4213/tmf10366.

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Wang, G. Multisoliton solutions of the two-component Camassa–Holm equation and its reductions. Theor Math Phys 214, 308–333 (2023). https://doi.org/10.1134/S0040577923030029

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