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March  2022, 27(3): 1471-1496. doi: 10.3934/dcdsb.2021098

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 Published  March 2022 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 and Continuous Dynamical Systems - B, 2022, 27 (3) : 1471-1496. 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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[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.

[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. 

[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.

[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.

[29]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[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.

[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. 

[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.

[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.

[29]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

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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