# American Institute of Mathematical Sciences

November  2019, 18(6): 2961-2981. doi: 10.3934/cpaa.2019132

## Existence results of solitary wave solutions for a delayed Camassa-Holm-KP equation

 1 School of Mathematics, Jilin University, Changchun, Jilin 130012, China 2 School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China 3 School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, Jilin 130024, China

* Corresponding author

Received  August 2018 Revised  January 2019 Published  May 2019

Fund Project: This work is supported by the Natural Science Foundation of China (Grant Nos. 11871251, 11771185, 11671071 and 11871140), the Fundamental Research Funds for the Central Universities at Jilin University (no. 2017TD–18), and the Special Funds of Provincial Industrial Innovation in Jilin Province (no. 2017C028–1).

This paper is concerned with the Camassa-Holm-KP equation, which is a model for shallow water waves. By using the analysis of the phase space, we obtain some qualitative properties of equilibrium points and existence results of solitary wave solutions for the Camassa-Holm-KP equation without delay. Furthermore we show the existence of solitary wave solutions for the equation with a special local delay convolution kernel by combining the geometric singular perturbation theory and invariant manifold theory. In addition, we discuss the existence of solitary wave solutions for the Camassa-Holm-KP equation with strength $1$ of nonlinearity, and prove the monotonicity of the wave speed by analyzing the ratio of the Abelian integral.

Citation: Xiaowan Li, Zengji Du, Shuguan Ji. Existence results of solitary wave solutions for a delayed Camassa-Holm-KP equation. Communications on Pure and Applied Analysis, 2019, 18 (6) : 2961-2981. doi: 10.3934/cpaa.2019132
##### References:
 [1] M. Ablowitz, P. Clarkson and So litons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge Univ. Press, Cambridge, 1991.  doi: 10.1017/CBO9780511623998. [2] S. Ai, Traveling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differential Equations, 232 (2007), 104-133.  doi: 10.1016/j.jde.2006.08.015. [3] S. Ai, S. Chow and Y. Yi, Travelling wave solutions in a tissue interaction model for skin pattern formation, J. Dynam. Differential Equations, 15 (2003), 517-534.  doi: 10.1023/B:JODY.0000009746.52357.28. [4] W. Bates and J. Shi, Existence and instability of spike layer solutions to singular perturbation problems, J. Funct. Anal., 196 (2002), 211-264.  doi: 10.1016/S0022-1236(02)00013-7. [5] A. Biswas, 1-Soliton solution of the generalized Camassa-Holm Kadomtsev-Petviashvili equation, Commun. Nonlinear Sci. Numer. Simulat., 14 (2009), 2524-2527.  doi: 10.1016/j.cnsns.2008.09.023. [6] R. Camassa and D. Holm, An integrable shallow water equation with peaked soliton, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661. [7] A. Chen, L. Guo and X. Deng, Existence of solitary waves and periodic waves for a perturbed generalized BBM equation, J. Differential Equations, 261 (2016), 5324-5349.  doi: 10.1016/j.jde.2016.08.003. [8] Z. Du, J. Li and X. Li, The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005. [9] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9. [10] B. Fuchssteiner and A. Fokas, Symplectic structures, their backland transformations and hereditary symmetries, Phys. D, 4 (1981), 47-66.  doi: 10.1016/0167-2789(81)90004-X. [11] A. Gourley and G. Ruan, Convergence and traveling wave fronts in functional differential equations with nonlocal terms: A competition model, SIAM. J. Math. Anal., 35 (2003), 806-822.  doi: 10.1137/S003614100139991. [12] G. Hek, Geometrical singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347-386.  doi: 10.1007/s00285-009-0266-7. [13] C. Hsu, T. Yang and C. Yang, Diversity of traveling wave solutions in FitzHugh-Nagumo type equations, J. Differential Equations, 247 (2009), 1185-1205.  doi: 10.1016/j.jde.2009.03.023. [14] P. Isaza and J. Mejia, On the support of solutions to the Kadomtsev-Petviashvili (KP-Ⅱ) equation, Commun. Pure Appl. Anal., 10 (2011), 1239-1255.  doi: 10.3934/cpaa.2011.10.1239. [15] J. Isaza, L. Mejia and N. Tzvetkov, A smoothing effect and polynomial growth of the Sobolev norms for the KP-Ⅱ equation, J. Differential Equations, 220 (2006), 1-17.  doi: 10.1016/j.jde.2004.10.002. [16] C. 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Zeng, Normally elliptic singular perturbations and persistence of homoclinic orbits, J. Differential Equations, 250 (2011), 4124-4176.  doi: 10.1016/j.jde.2011.02.001. [27] P. Maesschalck and F. Dumortier, Canard solutions at non-generic turning points, Trans. Amer. Math. Soc., 358 (2006), 2291-2334.  doi: 10.1090/S0002-9947-05-03839-0. [28] L. Molinet, J. Saut and N. Tzvetkov, Global well-posedness for the KP-Ⅱ equation on the background of a non-localized solution, Ann. I. H. Poincare-AN, 28 (2011), 653-676.  doi: 10.1016/j.anihpc.2011.04.004. [29] E. Novruzov and A. Hagverdiyev, On the behavior of the solution of the dissipative Camassa-Holm equation with the arbitrary dispersion coefficient, J. Differential Equations, 257 (2014), 4525-4541.  doi: 10.1016/j.jde.2014.08.016. [30] C. Ou and J. Wu, Persistence of wave fronts in delayed nonlocal reaction-diffusion equations, J. Differential Equations, 238 (2007), 219-261.  doi: 10.1016/j.jde.2006.12.010. [31] V. Petviashvili and V. Yan'kov, Solitons and turbulence, Rev. Plasma Phys., XIV (1989), 1-62. [32] C. Qu, Y. Fu and Y. Liu, Well-posedness, wave breaking and peakons for a modified $\mu$-Camassa-Holm equation, J. Funct. Anal., 266 (2014), 433-477.  doi: 10.1016/j.jfa.2013.09.021. [33] J. Saut and N. Tzvetkov, The cauchy problem for higher-order KP equations, J. Differential Equations, 153 (1999), 196-222.  doi: 10.1006/jdeq.1998.3534. [34] P. Szmolyan, Transversal heteroclinic and homoclinic orbits in singular perturbation problems, J. Differential Equations, 92 (1991), 252-281.  doi: 10.1016/0022-0396(91)90049-F. [35] A. Tovbis, Breaking of symmetrical periodic solutions in a singularly perturbed KdV model, SIAM J. Math. Anal., 40 (2008), 1516-1549.  doi: 10.1137/070694053. [36] A. Wazwaz, The Camassa-Holm-KP equations with compact and noncompact travelling wave solutions, Appl Math Comput, 170 (2005), 347-360.  doi: 10.1016/j.amc.2004.12.002. [37] M. Wei, X. Sun and S. Tang, Single peak solitary wave solutions for the CH-KP(2, 1) equation under boundary condition, J. Differential Equations, 259 (2015), 628-641.  doi: 10.1016/j.jde.2015.02.015. [38] L. Yang, Z. Rong, S. Zhou and C. Mu, Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics, Discrete. Contin. Dyn. Syst., 38 (2018), 5205-5220.  doi: 10.3934/dcds.2018230. [39] S. Yang and T. Xu, Symmetry analysis, persistence properties and unique continuation for the cross-coupled Camassa-Holm system, Discrete. Contin. Dyn. Syst., 38 (2018), 329-341.  doi: 10.3934/dcds.2018016. [40] V. Zakharov and E. Kuznetsov, On three-dimensional solitons, Sov. Phys., 39 (1974), 285-288. [41] K. Zhang, S. Tang and Z. Wang, Bifurcation of travelling wave solutions for the generalized Camassa-Holm-KP equations, Commun. Nonlinear Sci. Numer. Simulat., 15 (2010), 564-572.  doi: 10.1016/j.cnsns.2009.04.027. [42] L. Zhang and B. Liu, Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system, Discrete. Contin. Dyn. Syst., 38 (2018), 2655-2685.  doi: 10.3934/dcds.2018112.

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##### References:
 [1] M. Ablowitz, P. Clarkson and So litons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge Univ. Press, Cambridge, 1991.  doi: 10.1017/CBO9780511623998. [2] S. Ai, Traveling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differential Equations, 232 (2007), 104-133.  doi: 10.1016/j.jde.2006.08.015. [3] S. Ai, S. Chow and Y. Yi, Travelling wave solutions in a tissue interaction model for skin pattern formation, J. Dynam. Differential Equations, 15 (2003), 517-534.  doi: 10.1023/B:JODY.0000009746.52357.28. [4] W. Bates and J. Shi, Existence and instability of spike layer solutions to singular perturbation problems, J. Funct. Anal., 196 (2002), 211-264.  doi: 10.1016/S0022-1236(02)00013-7. [5] A. Biswas, 1-Soliton solution of the generalized Camassa-Holm Kadomtsev-Petviashvili equation, Commun. Nonlinear Sci. Numer. Simulat., 14 (2009), 2524-2527.  doi: 10.1016/j.cnsns.2008.09.023. [6] R. Camassa and D. Holm, An integrable shallow water equation with peaked soliton, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661. [7] A. Chen, L. Guo and X. Deng, Existence of solitary waves and periodic waves for a perturbed generalized BBM equation, J. Differential Equations, 261 (2016), 5324-5349.  doi: 10.1016/j.jde.2016.08.003. [8] Z. Du, J. Li and X. Li, The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005. [9] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9. [10] B. Fuchssteiner and A. Fokas, Symplectic structures, their backland transformations and hereditary symmetries, Phys. D, 4 (1981), 47-66.  doi: 10.1016/0167-2789(81)90004-X. [11] A. Gourley and G. Ruan, Convergence and traveling wave fronts in functional differential equations with nonlocal terms: A competition model, SIAM. J. Math. Anal., 35 (2003), 806-822.  doi: 10.1137/S003614100139991. [12] G. Hek, Geometrical singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347-386.  doi: 10.1007/s00285-009-0266-7. [13] C. Hsu, T. Yang and C. Yang, Diversity of traveling wave solutions in FitzHugh-Nagumo type equations, J. Differential Equations, 247 (2009), 1185-1205.  doi: 10.1016/j.jde.2009.03.023. [14] P. Isaza and J. Mejia, On the support of solutions to the Kadomtsev-Petviashvili (KP-Ⅱ) equation, Commun. Pure Appl. Anal., 10 (2011), 1239-1255.  doi: 10.3934/cpaa.2011.10.1239. [15] J. Isaza, L. Mejia and N. Tzvetkov, A smoothing effect and polynomial growth of the Sobolev norms for the KP-Ⅱ equation, J. Differential Equations, 220 (2006), 1-17.  doi: 10.1016/j.jde.2004.10.002. [16] C. Jones, Geometrical singular perturbation theory, In Dynamical Systems, Lecture Notes in Mathematics (R. Johnson ed.), vol. 1609. Springer, New York, 1995. doi: 10.1007/BFb0095239. [17] B. Kadomtsev and V. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl., 15 (1970), 539-541. [18] D. Korteweg and G. Vries, On the change of form of long waves advancing in a rectangular channel and on a new type of long stationary waves, Philos. Mag. R Soc. London, 39 (1895), 422-443.  doi: 10.1080/14786449508620739. [19] S. Lai and Y. Xu, The compact and noncompact structures for two types of generalized Camassa-Holm-KP equations, Math. Comput. Model., 28 (2008), 1089-1098.  doi: 10.1016/j.mcm.2007.06.020. [20] J. Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Differential Equations, 271 (2005), 393-430.  doi: 10.1016/j.jde.2004.09.007. [21] C. Li and K. Lu, Slow divergence integral and its application to classical Liénard equations of degree 5, J. Differential Equations, 257 (2014), 4437-4469.  doi: 10.1016/j.jde.2014.08.015. [22] C. Li and H. Zhu, Canard cycles for predator-prey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910.  doi: 10.1016/j.jde.2012.10.003. [23] X. Liu, Orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity, Discrete. Contin. Dyn. Syst., 38 (2018), 5505-5521. [24] X. Liu, Z. Qiao and Z. Yin, On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity, Commun. Pure Appl. Anal., 13 (2014), 1283-1304.  doi: 10.3934/cpaa.2014.13.1283. [25] W. Liu and E. Vleck, Turning points and traveling waves in FitzHugh-Nagumo type equations, J. Differential Equations, 225 (2006), 381-410.  doi: 10.1016/j.jde.2005.10.006. [26] N. Lu and C. Zeng, Normally elliptic singular perturbations and persistence of homoclinic orbits, J. Differential Equations, 250 (2011), 4124-4176.  doi: 10.1016/j.jde.2011.02.001. [27] P. Maesschalck and F. Dumortier, Canard solutions at non-generic turning points, Trans. Amer. Math. Soc., 358 (2006), 2291-2334.  doi: 10.1090/S0002-9947-05-03839-0. [28] L. Molinet, J. Saut and N. Tzvetkov, Global well-posedness for the KP-Ⅱ equation on the background of a non-localized solution, Ann. I. H. Poincare-AN, 28 (2011), 653-676.  doi: 10.1016/j.anihpc.2011.04.004. [29] E. Novruzov and A. Hagverdiyev, On the behavior of the solution of the dissipative Camassa-Holm equation with the arbitrary dispersion coefficient, J. Differential Equations, 257 (2014), 4525-4541.  doi: 10.1016/j.jde.2014.08.016. [30] C. Ou and J. Wu, Persistence of wave fronts in delayed nonlocal reaction-diffusion equations, J. Differential Equations, 238 (2007), 219-261.  doi: 10.1016/j.jde.2006.12.010. [31] V. Petviashvili and V. Yan'kov, Solitons and turbulence, Rev. Plasma Phys., XIV (1989), 1-62. [32] C. Qu, Y. Fu and Y. Liu, Well-posedness, wave breaking and peakons for a modified $\mu$-Camassa-Holm equation, J. Funct. Anal., 266 (2014), 433-477.  doi: 10.1016/j.jfa.2013.09.021. [33] J. Saut and N. Tzvetkov, The cauchy problem for higher-order KP equations, J. Differential Equations, 153 (1999), 196-222.  doi: 10.1006/jdeq.1998.3534. [34] P. Szmolyan, Transversal heteroclinic and homoclinic orbits in singular perturbation problems, J. Differential Equations, 92 (1991), 252-281.  doi: 10.1016/0022-0396(91)90049-F. [35] A. Tovbis, Breaking of symmetrical periodic solutions in a singularly perturbed KdV model, SIAM J. Math. Anal., 40 (2008), 1516-1549.  doi: 10.1137/070694053. [36] A. Wazwaz, The Camassa-Holm-KP equations with compact and noncompact travelling wave solutions, Appl Math Comput, 170 (2005), 347-360.  doi: 10.1016/j.amc.2004.12.002. [37] M. Wei, X. Sun and S. Tang, Single peak solitary wave solutions for the CH-KP(2, 1) equation under boundary condition, J. Differential Equations, 259 (2015), 628-641.  doi: 10.1016/j.jde.2015.02.015. [38] L. Yang, Z. Rong, S. Zhou and C. Mu, Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics, Discrete. Contin. Dyn. Syst., 38 (2018), 5205-5220.  doi: 10.3934/dcds.2018230. [39] S. Yang and T. Xu, Symmetry analysis, persistence properties and unique continuation for the cross-coupled Camassa-Holm system, Discrete. Contin. Dyn. Syst., 38 (2018), 329-341.  doi: 10.3934/dcds.2018016. [40] V. Zakharov and E. Kuznetsov, On three-dimensional solitons, Sov. Phys., 39 (1974), 285-288. [41] K. Zhang, S. Tang and Z. Wang, Bifurcation of travelling wave solutions for the generalized Camassa-Holm-KP equations, Commun. Nonlinear Sci. Numer. Simulat., 15 (2010), 564-572.  doi: 10.1016/j.cnsns.2009.04.027. [42] L. Zhang and B. Liu, Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system, Discrete. Contin. Dyn. Syst., 38 (2018), 2655-2685.  doi: 10.3934/dcds.2018112.
The phase portraits of system (19) for n is odd
The phase portraits of system (19) for n is even
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