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doi: 10.3934/dcdsb.2021098
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The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China

* Corresponding author: Zengji Du

Received  October 2020 Revised  January 2021 Early access March 2021

Fund Project: This work is supported by the Natural Science Foundation of China (Grant Nos. 11871251, 12090011 and 11771185)

In this paper, we are concerned with the existence of solitary waves for a generalized Kawahara equation, which is a model equation describing solitary-wave propagation in media. We obtain some qualitative properties of equilibrium points and existence results of solitary wave solutions for the generalized Kawahara equation without delay and perturbation by employing the phase space analysis. Furthermore the existence of solitary wave solutions for the equation with two types of special delay convolution kernels is proved by combining the geometric singular perturbation theory, invariant manifold theory and Fredholm orthogonality. We also discuss the asymptotic behaviors of traveling wave solutions by means of the asymptotic theory. Finally, some examples are given to illustrate our results.

Citation: Shuting Chen, Zengji Du, Jiang Liu, Ke Wang. The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021098
References:
[1]

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform, London Mathematical Society Lecture Note Series, 149. Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511623998.  Google Scholar

[2]

V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk., 18 (1963), 91-192.   Google Scholar

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations, Ⅰ. Schr$\ddot{o}$dinger equations, Ⅱ. The KdV equation, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.  Google Scholar

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T. J. Bridges and G. Derks, Linear instability of solitary wave solutions of the Kawahara equation and its generalizations, SIAM J. Math. Anal., 33 (2002), 1356-1378.  doi: 10.1137/S0036141099361494.  Google Scholar

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N. F. Britton, Spatial structures and periodic travelling waves in an integral differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099.  Google Scholar

[6]

W. Craig and J. Goodman, Linear dispersive equations of Airy Type, J. Differential Equations, 87 (1990), 38-61.  doi: 10.1016/0022-0396(90)90014-G.  Google Scholar

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L. L. Dawson, Uniqueness properties of higher order dispersive equations, J. Differential Equations, 236 (2007), 199-236.  doi: 10.1016/j.jde.2007.01.015.  Google Scholar

[8]

M. V. Demina and N. A. Kudryashov, From Laurent series to exact meromorphic solutions: the Kawahara equation, Phys. Lett. A, 374 (2010), 4023-4029.  doi: 10.1016/j.physleta.2010.08.013.  Google Scholar

[9]

G. Derks and S. Gils, On the uniqueness of traveling waves in perturbed Korteweg-de Vries equations, Jpn. J. Ind. Appl. Math., 10 (1993), 413-430.  doi: 10.1007/BF03167282.  Google Scholar

[10]

Z. DuZ. Feng and X. Zhang, Traveling wave phenomena of $n$-dimensional diffusive predator-prey systems, Nonlinear Anal. Real World Appl., 41 (2018), 288-312.  doi: 10.1016/j.nonrwa.2017.10.012.  Google Scholar

[11]

Z. DuJ. 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.  Google Scholar

[12]

Z. DuJ. Liu and L. Ren, Traveling pulse solutions of a generalized Keller-Segel system with small cell diffusion via a geometric approach, J. Differential Equations, 270 (2021), 1019-1042.  doi: 10.1016/j.jde.2020.09.009.  Google Scholar

[13]

Z. Du and Q. Qiao, The Dynamics of traveling waves for a nonlinear Belousov-Zhabotinskii system, J. Differential Equations, 269 (2020), 7214-7230.  doi: 10.1016/j.jde.2020.05.033.  Google Scholar

[14]

L. Escauriaza, C.bE. Kenig and G. Ponce, et al. On uniqueness properties of solutions of the k-generalized KdV equations, J. Funct. Anal., 244 (2007), 504–535. doi: 10.1016/j.jfa.2006.11.004.  Google Scholar

[15]

G. Faye and A. Scheel, Existence of pulses in excitable media with nonlocal coupling, Adv. Math., 270 (2015), 400-456.  doi: 10.1016/j.aim.2014.11.005.  Google Scholar

[16]

N. Fenichel, Geometric singular perturbation theory for ordinary differential, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

[17]

L. Guo and Y. Zhao, Existence of periodic waves for a perturbed quintic BBM equation, Discrete Contin. Dyn. Syst., 40 (2020), 4689-4703.  doi: 10.3934/dcds.2020198.  Google Scholar

[18]

G. Hek, Geometric singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347-386.  doi: 10.1007/s00285-009-0266-7.  Google Scholar

[19]

J. K. Hunter and J. Scheule, Existence of perturbed solitary wave solutions to a model equation for water waves, Physica D, 32 (1988), 253-268.  doi: 10.1016/0167-2789(88)90054-1.  Google Scholar

[20]

J. M. Hyman and B. Nicolaenko, The Kuramoto-Sivashinsky equation: A bridge between PDEs and dynamical systems, Physica D, 18 (1986), 113-126.  doi: 10.1016/0167-2789(86)90166-1.  Google Scholar

[21]

Y. Jia and Z. Huo, Well-posedness for the fifth-order shallow water equations, J. Differential Equations, 246 (2009), 2448-2467.  doi: 10.1016/j.jde.2008.10.027.  Google Scholar

[22]

C. K. R. T. Jones, Geometric singular perturbation theory, In Dynamical systems (ed. R. Johnson). Lecture Notes in Mathematics, 1609 (1995), 44-118. doi: 10.1007/BFb0095239.  Google Scholar

[23]

T. Kato, Local well-posedness for Kawahara equation, Adv. Differential Equations, 16 (2011), 257-287.   Google Scholar

[24]

Ka wahara and Ta kuji, Oscillatory Solitary Waves in Dispersive Media, Journal of the Physical Society of Japan, 33 (1972), 260-264.   Google Scholar

[25]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math, 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.  Google Scholar

[26]

A. N. Kolmogorov, On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian, Dokl. Akad. Nauk SSSR, 98 (1954), 527-530.   Google Scholar

[27]

S. Krantz and H. Parks, The Implicit Function Theorem: History, Theory, and Applications, , Birkh$\ddot{a}$user Boston, Inc., Boston, MA, 2002. doi: 10.1007/978-1-4612-0059-8.  Google Scholar

[28]

C. Kwak, Well-posedness issues on the periodic modified Kawahara equation, Ann. I. H. Poincar$\acute{e}$-AN, 37 (2020), 373–416. doi: 10.1016/j.anihpc.2019.09.002.  Google Scholar

[29]

S. Kwon, Well-posedness and ill-posedness of the fifth-order modifified KdV equation, J. Differential Equations, 2008 (2008), 1-15.   Google Scholar

[30]

L. Molinet and Y. Wang, Dispersive limit from the Kawahara to the KdV equation, J. Differential Equations, 255 (2013), 2196-2219.  doi: 10.1016/j.jde.2013.06.012.  Google Scholar

[31]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nach. Akad. Wiss. G$\ddot{o}$ttingen, Math. Phys. Kl., II, 1962 (1962), 1–20.  Google Scholar

[32]

K. NakanishH. Takaoka and Y. Tsutsumi, Local well-posedness in low regularity of the mKdV equation with periodic boundary condition, Discrete Contin. Dyn. Syst., 28 (2010), 1635-1654.  doi: 10.3934/dcds.2010.28.1635.  Google Scholar

[33]

T. Ogawa, Travelling wave solutions to a perturbed Korteweg-de Vries equation, Hiroshima Math. J., 24 (1994), 401-422.  doi: 10.32917/hmj/1206128032.  Google Scholar

[34]

G. Ponce, Lax pairs and higher order models for water waves, J. Differential Equations, 102 (1993), 360-381.  doi: 10.1006/jdeq.1993.1034.  Google Scholar

[35]

X. Sun and P. Yu, Periodic Traveling waves in a generalized BBM equation with weak backward diffusion and dissipation, Discrete Contin. Dyn. Syst., 24 (2019), 965-987.  doi: 10.3934/dcdsb.2018341.  Google Scholar

[36]

H. Takaoka and Y. Tsutsumi, Well-posedness of the Cauchy problem for the modifified KdV equation with periodic boundary condition, Int. Math. Res. Not., 2004 (2004), 3009-3040.  doi: 10.1155/S1073792804140555.  Google Scholar

[37]

T. Tao, Scattering for the quartic generalised Korteweg-de Vries equation, J. Differential Equations, 232 (2007), 623-651.  doi: 10.1016/j.jde.2006.07.019.  Google Scholar

[38]

O. P. V. Villag$\acute{r}$an, Gain of regularity for a korteweg-de vries-kawahara equation, J. Differential Equations, 71 (2004), 1-24.   Google Scholar

show all references

References:
[1]

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform, London Mathematical Society Lecture Note Series, 149. Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511623998.  Google Scholar

[2]

V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk., 18 (1963), 91-192.   Google Scholar

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations, Ⅰ. Schr$\ddot{o}$dinger equations, Ⅱ. The KdV equation, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.  Google Scholar

[4]

T. J. Bridges and G. Derks, Linear instability of solitary wave solutions of the Kawahara equation and its generalizations, SIAM J. Math. Anal., 33 (2002), 1356-1378.  doi: 10.1137/S0036141099361494.  Google Scholar

[5]

N. F. Britton, Spatial structures and periodic travelling waves in an integral differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099.  Google Scholar

[6]

W. Craig and J. Goodman, Linear dispersive equations of Airy Type, J. Differential Equations, 87 (1990), 38-61.  doi: 10.1016/0022-0396(90)90014-G.  Google Scholar

[7]

L. L. Dawson, Uniqueness properties of higher order dispersive equations, J. Differential Equations, 236 (2007), 199-236.  doi: 10.1016/j.jde.2007.01.015.  Google Scholar

[8]

M. V. Demina and N. A. Kudryashov, From Laurent series to exact meromorphic solutions: the Kawahara equation, Phys. Lett. A, 374 (2010), 4023-4029.  doi: 10.1016/j.physleta.2010.08.013.  Google Scholar

[9]

G. Derks and S. Gils, On the uniqueness of traveling waves in perturbed Korteweg-de Vries equations, Jpn. J. Ind. Appl. Math., 10 (1993), 413-430.  doi: 10.1007/BF03167282.  Google Scholar

[10]

Z. DuZ. Feng and X. Zhang, Traveling wave phenomena of $n$-dimensional diffusive predator-prey systems, Nonlinear Anal. Real World Appl., 41 (2018), 288-312.  doi: 10.1016/j.nonrwa.2017.10.012.  Google Scholar

[11]

Z. DuJ. 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.  Google Scholar

[12]

Z. DuJ. Liu and L. Ren, Traveling pulse solutions of a generalized Keller-Segel system with small cell diffusion via a geometric approach, J. Differential Equations, 270 (2021), 1019-1042.  doi: 10.1016/j.jde.2020.09.009.  Google Scholar

[13]

Z. Du and Q. Qiao, The Dynamics of traveling waves for a nonlinear Belousov-Zhabotinskii system, J. Differential Equations, 269 (2020), 7214-7230.  doi: 10.1016/j.jde.2020.05.033.  Google Scholar

[14]

L. Escauriaza, C.bE. Kenig and G. Ponce, et al. On uniqueness properties of solutions of the k-generalized KdV equations, J. Funct. Anal., 244 (2007), 504–535. doi: 10.1016/j.jfa.2006.11.004.  Google Scholar

[15]

G. Faye and A. Scheel, Existence of pulses in excitable media with nonlocal coupling, Adv. Math., 270 (2015), 400-456.  doi: 10.1016/j.aim.2014.11.005.  Google Scholar

[16]

N. Fenichel, Geometric singular perturbation theory for ordinary differential, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

[17]

L. Guo and Y. Zhao, Existence of periodic waves for a perturbed quintic BBM equation, Discrete Contin. Dyn. Syst., 40 (2020), 4689-4703.  doi: 10.3934/dcds.2020198.  Google Scholar

[18]

G. Hek, Geometric singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347-386.  doi: 10.1007/s00285-009-0266-7.  Google Scholar

[19]

J. K. Hunter and J. Scheule, Existence of perturbed solitary wave solutions to a model equation for water waves, Physica D, 32 (1988), 253-268.  doi: 10.1016/0167-2789(88)90054-1.  Google Scholar

[20]

J. M. Hyman and B. Nicolaenko, The Kuramoto-Sivashinsky equation: A bridge between PDEs and dynamical systems, Physica D, 18 (1986), 113-126.  doi: 10.1016/0167-2789(86)90166-1.  Google Scholar

[21]

Y. Jia and Z. Huo, Well-posedness for the fifth-order shallow water equations, J. Differential Equations, 246 (2009), 2448-2467.  doi: 10.1016/j.jde.2008.10.027.  Google Scholar

[22]

C. K. R. T. Jones, Geometric singular perturbation theory, In Dynamical systems (ed. R. Johnson). Lecture Notes in Mathematics, 1609 (1995), 44-118. doi: 10.1007/BFb0095239.  Google Scholar

[23]

T. Kato, Local well-posedness for Kawahara equation, Adv. Differential Equations, 16 (2011), 257-287.   Google Scholar

[24]

Ka wahara and Ta kuji, Oscillatory Solitary Waves in Dispersive Media, Journal of the Physical Society of Japan, 33 (1972), 260-264.   Google Scholar

[25]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math, 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.  Google Scholar

[26]

A. N. Kolmogorov, On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian, Dokl. Akad. Nauk SSSR, 98 (1954), 527-530.   Google Scholar

[27]

S. Krantz and H. Parks, The Implicit Function Theorem: History, Theory, and Applications, , Birkh$\ddot{a}$user Boston, Inc., Boston, MA, 2002. doi: 10.1007/978-1-4612-0059-8.  Google Scholar

[28]

C. Kwak, Well-posedness issues on the periodic modified Kawahara equation, Ann. I. H. Poincar$\acute{e}$-AN, 37 (2020), 373–416. doi: 10.1016/j.anihpc.2019.09.002.  Google Scholar

[29]

S. Kwon, Well-posedness and ill-posedness of the fifth-order modifified KdV equation, J. Differential Equations, 2008 (2008), 1-15.   Google Scholar

[30]

L. Molinet and Y. Wang, Dispersive limit from the Kawahara to the KdV equation, J. Differential Equations, 255 (2013), 2196-2219.  doi: 10.1016/j.jde.2013.06.012.  Google Scholar

[31]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nach. Akad. Wiss. G$\ddot{o}$ttingen, Math. Phys. Kl., II, 1962 (1962), 1–20.  Google Scholar

[32]

K. NakanishH. Takaoka and Y. Tsutsumi, Local well-posedness in low regularity of the mKdV equation with periodic boundary condition, Discrete Contin. Dyn. Syst., 28 (2010), 1635-1654.  doi: 10.3934/dcds.2010.28.1635.  Google Scholar

[33]

T. Ogawa, Travelling wave solutions to a perturbed Korteweg-de Vries equation, Hiroshima Math. J., 24 (1994), 401-422.  doi: 10.32917/hmj/1206128032.  Google Scholar

[34]

G. Ponce, Lax pairs and higher order models for water waves, J. Differential Equations, 102 (1993), 360-381.  doi: 10.1006/jdeq.1993.1034.  Google Scholar

[35]

X. Sun and P. Yu, Periodic Traveling waves in a generalized BBM equation with weak backward diffusion and dissipation, Discrete Contin. Dyn. Syst., 24 (2019), 965-987.  doi: 10.3934/dcdsb.2018341.  Google Scholar

[36]

H. Takaoka and Y. Tsutsumi, Well-posedness of the Cauchy problem for the modifified KdV equation with periodic boundary condition, Int. Math. Res. Not., 2004 (2004), 3009-3040.  doi: 10.1155/S1073792804140555.  Google Scholar

[37]

T. Tao, Scattering for the quartic generalised Korteweg-de Vries equation, J. Differential Equations, 232 (2007), 623-651.  doi: 10.1016/j.jde.2006.07.019.  Google Scholar

[38]

O. P. V. Villag$\acute{r}$an, Gain of regularity for a korteweg-de vries-kawahara equation, J. Differential Equations, 71 (2004), 1-24.   Google Scholar

Figure 1.  The graph of the traveling wave $ \Phi(\xi) $
Figure 2.  The graph of the traveling wave $ \Phi(\xi) $
Figure 3.  The graph of the traveling wave $ \Phi(\xi) $
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