# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022105
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## Stability in the energy space of the sum of $N$ solitary waves for an equation modelling shallow water waves of moderate amplitude

 School of mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China

* Corresponding author: Xingxing Liu

Received  November 2021 Revised  March 2022 Early access June 2022

Fund Project: The author is supported by National Nature Science Foundation of China under Grant 12001528

Considered herein is an equation modelling the propagation of surface waves of moderate amplitude in shallow water regime, which admits blow-up solutions and solitary waves. Using modulation argument, combining the result of the stability of a single solitary wave with a property of almost monotonicity of local energy norm, we show that the decoupled sum of $N$ solitary waves is orbitally stable in the energy space $H^1({{\mathbb{R}}})$.

Citation: Xingxing Liu. Stability in the energy space of the sum of $N$ solitary waves for an equation modelling shallow water waves of moderate amplitude. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022105
##### References:
 [1] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661. [2] A. Constantin and J. Escher, Well-posedness, global existence and blow-up phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504.  doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. [3] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586. [4] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2. [5] A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear. Sci., 12 (2002), 415-422.  doi: 10.1007/s00332-002-0517-x. [6] N. Duruk Mutlubaş, On the Cauchy problem for a model equation for shallow water waves of moderate amplitude, Nonlinear Anal. RWA, 14 (2013), 2022-2026.  doi: 10.1016/j.nonrwa.2013.02.006. [7] N. Duruk Mutlubaş, Local well-posedness and wave breaking results for periodic solutions of a shallow water equation for waves of moderate amplitude, Nonlinear Anal. TMA, 97 (2014), 145-154.  doi: 10.1016/j.na.2013.11.021. [8] N. Duruk Mutlubaş and A. Geyer, Orbital stability of solitary waves of moderate amplitude in shallow water, J. Differential Equations, 255 (2013), 254-263.  doi: 10.1016/j.jde.2013.04.010. [9] K. El Dika and L. Molinet, Exponential decay of $H^1$-localized solutions and stability of the train of $N$ solitary waves for the Camassa-Holm equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2313-2331.  doi: 10.1098/rsta.2007.2011. [10] B. Fuchssteiner and A. Fokas, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D, 4 (1981), 47-66.  doi: 10.1016/0167-2789(81)90004-X. [11] A. Geyer, Solitary traveling water waves of moderate amplitude, J. Nonlinear Math. Phys., 19 (2012), 1240010, 12 pp. doi: 10.1142/S1402925112400104. [12] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9. [13] X. Liu, Stability of the train of $N$ solitary waves for the two-component Camassa-Holm shallow water system, J. Differential Equations, 260 (2016), 8403-8427.  doi: 10.1016/j.jde.2016.02.028. [14] X. Liu and J. Liu, On the low regularity solutions and wave breaking for an equation modelling shallow water waves of moderate amplitude, Nonlinear Anal. TMA, 107 (2014), 1-11.  doi: 10.1016/j.na.2014.04.021. [15] Y. Martel, F. Merle and T.-P. Tsai, Stability and asymptotic stability in the energy space of the sum $N$ solitons for subcritical gKdV eqautions, Comm. Math. Phys., 231 (2002), 347-373.  doi: 10.1007/s00220-002-0723-2. [16] Y. Mi and C. Mu, On the solutions of a model equation for shallow water waves of moderate amplitude, J. Differential Equations, 255 (2013), 2101-2129.  doi: 10.1016/j.jde.2013.06.008. [17] S. Yang, Wave breaking for a model equation for shallow water waves of moderate amplitude, Commun. Math. Sci., 19 (2021), 1799-1807.  doi: 10.4310/CMS.2021.v19.n7.a2. [18] S. Zhou and C. Mu, Global conservative solutions for a model equation for shallow water waves of moderate amplitude, J. Differential Equations, 256 (2014), 1793-1816.  doi: 10.1016/j.jde.2013.11.011.

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##### References:
 [1] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661. [2] A. Constantin and J. Escher, Well-posedness, global existence and blow-up phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504.  doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5. [3] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586. [4] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2. [5] A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear. Sci., 12 (2002), 415-422.  doi: 10.1007/s00332-002-0517-x. [6] N. Duruk Mutlubaş, On the Cauchy problem for a model equation for shallow water waves of moderate amplitude, Nonlinear Anal. RWA, 14 (2013), 2022-2026.  doi: 10.1016/j.nonrwa.2013.02.006. [7] N. Duruk Mutlubaş, Local well-posedness and wave breaking results for periodic solutions of a shallow water equation for waves of moderate amplitude, Nonlinear Anal. TMA, 97 (2014), 145-154.  doi: 10.1016/j.na.2013.11.021. [8] N. Duruk Mutlubaş and A. Geyer, Orbital stability of solitary waves of moderate amplitude in shallow water, J. Differential Equations, 255 (2013), 254-263.  doi: 10.1016/j.jde.2013.04.010. [9] K. El Dika and L. Molinet, Exponential decay of $H^1$-localized solutions and stability of the train of $N$ solitary waves for the Camassa-Holm equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2313-2331.  doi: 10.1098/rsta.2007.2011. [10] B. Fuchssteiner and A. Fokas, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D, 4 (1981), 47-66.  doi: 10.1016/0167-2789(81)90004-X. [11] A. Geyer, Solitary traveling water waves of moderate amplitude, J. Nonlinear Math. Phys., 19 (2012), 1240010, 12 pp. doi: 10.1142/S1402925112400104. [12] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9. [13] X. Liu, Stability of the train of $N$ solitary waves for the two-component Camassa-Holm shallow water system, J. Differential Equations, 260 (2016), 8403-8427.  doi: 10.1016/j.jde.2016.02.028. [14] X. Liu and J. Liu, On the low regularity solutions and wave breaking for an equation modelling shallow water waves of moderate amplitude, Nonlinear Anal. TMA, 107 (2014), 1-11.  doi: 10.1016/j.na.2014.04.021. [15] Y. Martel, F. Merle and T.-P. Tsai, Stability and asymptotic stability in the energy space of the sum $N$ solitons for subcritical gKdV eqautions, Comm. Math. Phys., 231 (2002), 347-373.  doi: 10.1007/s00220-002-0723-2. [16] Y. Mi and C. Mu, On the solutions of a model equation for shallow water waves of moderate amplitude, J. Differential Equations, 255 (2013), 2101-2129.  doi: 10.1016/j.jde.2013.06.008. [17] S. Yang, Wave breaking for a model equation for shallow water waves of moderate amplitude, Commun. Math. Sci., 19 (2021), 1799-1807.  doi: 10.4310/CMS.2021.v19.n7.a2. [18] S. Zhou and C. Mu, Global conservative solutions for a model equation for shallow water waves of moderate amplitude, J. Differential Equations, 256 (2014), 1793-1816.  doi: 10.1016/j.jde.2013.11.011.
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