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On Riemann–Hilbert problem and multiple high-order pole solutions to the cubic Camassa–Holm equation

Part of: Representations of solutions General higher-order equations and systems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  25 November 2024

Wen-Yu Zhou ,
Shou-Fu Tian  and
Zhi-Qiang Li
Wen-Yu Zhou
Affiliation:
School of Mathematics, China University of Mining and Technology, Xuzhou 221116, People’s Republic of China ([email protected])
Shou-Fu Tian
Affiliation:
School of Mathematics, China University of Mining and Technology, Xuzhou 221116, People’s Republic of China ([email protected], [email protected]) (corresponding author)
Zhi-Qiang Li
Affiliation:
School of Mathematics, China University of Mining and Technology, Xuzhou 221116, People’s Republic of China ([email protected])
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Abstract

In this work, the Riemann–Hilbert (RH) problem is employed to study the multiple high-order pole solutions of the cubic Camassa–Holm (cCH) equation with the term characterizing the effect of linear dispersion under zero boundary conditions and nonzero boundary conditions. Under the reflectionless situation, we generalize the residue theorem and obtain the multiple high-order pole solutions of cCH equation by solving an algebraic system. During the process of establishing the solution of RH problem, to simplify the calculations involving the implicitly expressed of variables (x, t) in the solution, we introduce a new scale (y, t) to ensure the solution of RH problem is explicitly expressed with respect to it. Finally, the exact solutions are obtained for cases involving one high-order pole and N high-order poles.

Keywords

multiple high-order polesRiemann–Hilbert problemthe cubic Camassa–Holm equationzero and nonzero boundary conditions

MSC classification

Primary: 35Q51: Soliton-like equations 35Q15: Riemann-Hilbert problems 35C08: Soliton solutions 35G25: Initial value problems for nonlinear higher-order equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 51
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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Footnotes

*

All authors contributed equally as the first author to this work.

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