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Stability of Periodic Peakons for a Nonlinear Quartic \(\mu \)-Camassa–Holm Equation

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Abstract

In this paper, we prove the orbital stability of periodic peaked traveling waves (peakons) for a nonlinear quartic \(\mu \)-Camassa–Holm equation. The equation is a \(\mu \)-version of the nonlinear quartic Camassa–Holm equation which was proposed by Anco and Recio (J Phys A Math Theor 52:125–203, 2019). The equation admits the periodic peakons. It is shown that the periodic peakons are orbitally stable under small perturbations in the energy space by finding inequalities related to the three conservation laws with global maximum and minimum of the solution.

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Acknowledgements

The author would like to thank the anonymous referee for her/his valuable comments which has lead to significant improvement of the manuscript.

Funding

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2020R1F1A1A01048468).

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  1. Department of Mathematics, Incheon National University, Incheon, 22012, Republic of Korea

    Byungsoo Moon

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Appendix

Appendix

In this appendix we verify explicitly that the functional \(H_1\) and \(H_2\) defined in Proposition 2 is conserved under the flow of a nonlinear quartic \(\mu \)-Camassa–Holm equation (2.1). The following three forms of a nonlinear quartic \(\mu \)-Camassa–Holm equation (2.1) are used:

$$\begin{aligned}&m_t+\left[ \frac{1}{4}\left( 2\mu (u)u-u_x^2\right) ^2+u(2\mu (u)u-u_x^2)m\right] _x=0,\\&u_t+\left( 2\mu (u)u^2u_x-\frac{1}{3}uu_x^3\right) +\partial _x(\mu -\partial _x^2)^{-1}\left[ 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right] \\&\quad +\frac{1}{3}\mu (uu_x^3)=0,\\&u_{tx}+\partial _x\left( 2\mu (u)u^2u_x-\frac{1}{3}uu_x^3\right) -\left[ 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right] \\&\quad +\mu \left[ 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right] =0. \end{aligned}$$

We also use the identities \(\mu \left[ (\mu -\partial _x^2)^{-1}u\right] =\mu (u)\) and \(\partial _x^2(\mu -\partial _x^2)^{-1}u=-u+\mu (u).\) We directly compute

$$\begin{aligned}&\quad \frac{dH_1}{dt}\\ {}&=\int _{\mathbb {S}}m_tu+mu_tdx\\ {}&=-\int _{\mathbb {S}}u\left[ \frac{1}{4}\left( 2\mu (u)u-u_x^2\right) ^2+u(2\mu (u)u-u_x^2)m\right] _xdx\\ {}&\quad -\int _{\mathbb {S}}m\left[ \left( 2\mu (u)u^2u_x-\frac{1}{3}uu_x^3\right) +\partial _x(\mu -\partial _x^2)^{-1}\left[ 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right] \right. \\ {}&\quad \qquad \quad \qquad +\left. \frac{1}{3}\mu (uu_x^3)\right] dx\\ {}&=\int _{\mathbb {S}}\left[ \frac{1}{4}\left( 2\mu (u)u-u_x^2\right) ^2u_x+uu_x\left( 2\mu (u)u-u_x^2\right) m\right] dx\\ {}&\quad -\int _{\mathbb {S}}\left[ 2\mu (u)u^2u_xm-\frac{1}{3}uu_x^3m+m\partial _x(\mu -\partial _x^2)^{-1}\left[ 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right] \right. \\ {}&\quad \qquad \quad \qquad \left. +\frac{1}{3}\mu (uu_x^3)m\right] dx\\ {}&=-\int _{\mathbb {S}}\left[ -2\mu (u)u^2u_xu_{xx}-\frac{1}{3}\mu (u)uu_x^3+\frac{1}{3}uu_x^3u_{xx} \right. \\ {}&\quad \qquad \quad -u_{xx}\partial _x(\mu -\partial _x^2)^{-1}\left[ 3\mu (u)^2u^2+ 2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right] +\left. \frac{1}{3}\mu (u)\mu (uu_x^3)\right] dx\\ {}&=-\int _{\mathbb {S}}\left[ \frac{5}{3}\mu (u)uu_x^3+\frac{1}{3}uu_x^3u_{xx}-u_x\left[ 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right] \right. \\ {}&\quad \qquad \quad +\left. \frac{1}{3}\mu (u)\mu (uu_x^3)\right] dx\\ {}&=-\int _{\mathbb {S}}\left[ -\frac{1}{3}\mu (u)uu_x^3+\frac{1}{3}\mu (u)\mu (uu_x^3)\right] dx=0, \end{aligned}$$

where we use the following identities

$$\begin{aligned} \left[ \frac{1}{4}\left( 2\mu (u)u-u_x^2\right) ^2u\right] _x=\frac{1}{4}\left( 2\mu (u)u-u_x^2\right) ^2u_x+uu_x\left( 2\mu (u)u-u_x^2\right) m \end{aligned}$$

and

$$\begin{aligned}&\qquad -\int _{\mathbb {S}}u_x\partial _x^2(\mu -\partial _x^2)^{-1}\left[ 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right] dx\\ {}&\quad =-\int _{\mathbb {S}}\left[ -u_x\left[ 3\mu (u)^2u^2+2\mu (u)uu_x^2{-}u_x^4\right] +u_x\mu \left[ 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right] \right] dx. \end{aligned}$$
$$\begin{aligned} \frac{dH_2}{dt}&=\int _{\mathbb {S}}\left[ 3\mu (u)^2u^2u_t+2\mu (u)uu_x^2u_t+2\mu (u)u^2u_xu_{xt}-\frac{1}{12}u_x^4u_t-\frac{1}{3}uu_x^3u_{xt}\right] dx\\ {}&=\int _{\mathbb {S}}\left[ \left( 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right) u_t+\left( 2\mu (u)u^2u_x-\frac{1}{3}uu_x^3\right) u_{xt}\right] dx\\ {}&=\int _{\mathbb {S}}\left[ \left( 3\mu (u)^2u^2-2\mu (u)uu_x^2-2\mu (u)u^2u_{xx}+\frac{1}{4}u_x^4+uu_x^2u_{xx}\right) u_t\right] dx\\ {}&=-\int _{\mathbb {S}}\left( \left( 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right) -\partial _x\left( 2\mu (u)u^2u_x-\frac{1}{3}uu_x^3\right) \right) \\ {}&\quad \times \left[ \left( 2\mu (u)u^2u_x-\frac{1}{3}uu_x^3\right) +\partial _x(\mu -\partial _x^2)^{-1}\left[ 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right] \right. \\ {}&\quad \qquad \quad \left. +\frac{1}{3}\mu (uu_x^3)\right] dx\\ {}&=-\int _{\mathbb {S}}\left( 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right) \\ {}&\quad \qquad \times \partial _x(\mu -\partial _x^2)^{-1}\left[ 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right] dx\\ {}&\quad -\int _{\mathbb {S}}\frac{1}{3}\mu (uu_x^3)\left( 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right) dx\\ {}&\quad -\int _{\mathbb {S}}\left( 2\mu (u)u^2u_x-\frac{1}{3}uu_x^3\right) \mu \left( 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right) dx\\ {}&=-\int _{\mathbb {S}}\left( 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right) \\ {}&\quad \quad \qquad \times \partial _x(\mu -\partial _x^2)^{-1} \left[ 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right] dx\\ {}&\quad -\int _{\mathbb {S}}\left[ \frac{1}{3}\mu (uu_x^3)\left( 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right) \right. \\ {}&\quad \qquad -\left. \frac{1}{3}uu_x^3\mu \left( 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right) \right] dx\\ {}&=\int _{\mathbb {S}}(\mu -\partial _x^2)^{-1}\left[ 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right] \\ {}&\quad \quad \times \partial _x\left( 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right) dx\\ {}&=0, \end{aligned}$$

where we use

$$\begin{aligned}&-\int _{\mathbb {S}}\left( 2\mu (u)u^2u_x-\frac{1}{3}uu_x^3\right) \partial _x^2(\mu -\partial _x^2)^{-1}\left[ 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right] dx\\&\quad =-\int _{\mathbb {S}}\left[ -\left( 2\mu (u)u^2u_x-\frac{1}{3}uu_x^3\right) \left[ 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right] \right. \\&\qquad +\left. \left( 2\mu (u)u^2u_x-\frac{1}{3}uu_x^3\right) \mu \left[ 3\mu (u)^2u^2+2\mu (u)uu_x^2-\frac{1}{12}u_x^4\right] \right] dx. \end{aligned}$$

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Moon, B. Stability of Periodic Peakons for a Nonlinear Quartic \(\mu \)-Camassa–Holm Equation. J Dyn Diff Equat 36, 703–725 (2024). https://doi.org/10.1007/s10884-022-10156-z

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