June  2022, 21(6): 1987-2003. doi: 10.3934/cpaa.2021118

Dynamics of solitary waves and periodic waves for a generalized KP-MEW-Burgers equation with damping

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

*Corresponding author

(Dedicated to Professor Goong Chen on the occasion of his 70th birthday)

Received  January 2021 Revised  June 2021 Published  June 2022 Early access  July 2021

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

This paper discusses the existence of solitary waves and periodic waves for a generalized (2+1)-dimensional Kadomtsev-Petviashvili modified equal width-Burgers (KP-MEW-Burgers) equation with small damping and a weak local delay convolution kernel by using the dynamical systems approach, specifically based on geometric singular perturbation theory and invariant manifold theory. Moreover, the monotonicity of the wave speed is proved by analyzing the ratio of Abelian integrals. The upper and lower bounds of the limit wave speed are given. In addition, the upper and lower bounds and monotonicity of the period $ T $ of traveling wave when the small positive parameter $ \tau\rightarrow 0 $ are also obtained. Perhaps this paper is the first discussion on the solitary waves and periodic waves for the delayed KP-MEW-Burgers equations and the Abelian integral theory may be the first application to the study of the (2+1)-dimensional equation.

Citation: Zengji Du, Xiaojie Lin, Yulin Ren. Dynamics of solitary waves and periodic waves for a generalized KP-MEW-Burgers equation with damping. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1987-2003. doi: 10.3934/cpaa.2021118
References:
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J. CarrS. N. Chow and J. K. Hale, Abelian integrals and bifurcation theory, J. Differ. Equ., 59 (1985), 413-436.  doi: 10.1016/0022-0396(85)90148-2.

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S. Chakravarty and Y. Kodama, Soliton solutions of the KP equation and application to shallow water waves, Stud. Appl. Math., 123 (2009), 83-151.  doi: 10.1111/j.1467-9590.2009.00448.x.

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A. ChenL. Guo and X. Deng, Existence of solitary waves and periodic waves for a perturbed generalized BBM equation, J. Differ. Equ., 261 (2016), 5324-5349.  doi: 10.1016/j.jde.2016.08.003.

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F. Chen and J. Li, Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay, Discrete Contin. Dyn. Syst., 41 (2021), 967-985.  doi: 10.3934/dcds.2020305.

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M. Darwich, On the well-posedness for Kadomtsev-Petviashvili-Burgers I equation, J. Differ. Equ., 253 (2012), 1584-1603.  doi: 10.1016/j.jde.2012.05.013.

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A. DegasperisS. Lombardo and M. Sommaca, Integrability and Linear Stability of NonlinearWaves, J. Nonlinear Sci., 28 (2018), 1251-1291.  doi: 10.1007/s00332-018-9450-5.

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Z. DuJ. Li and X. Li, The existence of solitary wave solutions of delayed Camassa-Holm via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005.

[9]

Z. Du and Z. Feng, Existence and asymptotic behavior of traveling waves in a modified vector-disease model, Commun. Pur. Appl. Anal., 17 (2018), 1899-1920.  doi: 10.3934/cpaa.2018090.

[10]

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

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

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G. GuiY. Liu and T. Luo, Model equations and traveling wave solutions for shallow-water waves with the coriolis effect, J. Nonlinear Sci., 29 (2019), 993-1039.  doi: 10.1007/s00332-018-9510-x.

[13]

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.

[14]

M. HolzerA. Doelman and T. J. Kaper, Existence and stability of traveling pulses in a reaction-diffusion-mechanics system, J Nonlinear Sci., 23 (2013), 129-177.  doi: 10.1007/s00332-012-9147-0.

[15]

J. IsazaL. Mejia and N. Tzvetkov, A smoothing effect and polynomial growth of the Sobolev norms for the KP-II equation, J. Differ. Equ., 220 (2006), 1-17.  doi: 10.1016/j.jde.2004.10.002.

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C. K. R. T. Jones, Geometrical singular perturbation theory, in: R. Johnson (Ed.), Dynamical Systems, in: Lecture Notes in Mathematics, Springer, New York, 1995. doi: 10.1007/BFb0095239.

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B. Kojok, Sharp well-posedness for Kadomtsev-Petviashvili-Burgers (KPBII) equation in $R^{2}$, J. Differ. Equ., 242 (2007), 211-247.  doi: 10.1016/j.jde.2007.08.010.

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[20]

J. LiK. Lu and P. Bates, Invariant foliations for random dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 3639-3666.  doi: 10.3934/dcds.2014.34.3639.

[21]

J. LiK. Lu and P. Bates, Geometric singular perturbation theory with real noise, J. Differ. Equ., 259 (2015), 5137-5167.  doi: 10.1016/j.jde.2015.06.023.

[22]

L. MolinetJ. 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.

[23]

L. MolinetJ. Saut and N. Tzvetkov, Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation, Duke Math. J., 115 (2002), 353-384.  doi: 10.1215/S0012-7094-02-11525-7.

[24]

T. Ogawa, Travelling wave solutions to a perturbed Korteweg-de Vries equation, Hiroshima Math. J., 24 (1994), 401-422. 

[25]

S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model, Proc. R. Soc. Edinb., 134A (2004), 991-1011.  doi: 10.1017/S0308210500003590.

[26]

A. Saha, Dynamics of the generalized KP-MEW-Burgers equation with external periodic perturbation, Comput. Math. Appl., 73 (2017), 1879-1885.  doi: 10.1016/j.camwa.2017.02.017.

[27]

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

[28]

C. Wang and X. Zhang, Canards, heteroclinic and homoclinic orbits for a slow-fast predator-prey model of generalized Holling type III, J. Differ. Equ., 267 (2019), 3397-3441.  doi: 10.1016/j.jde.2019.04.008.

[29]

S. I. Zaki, Solitary wave interactions for the modified equal width equation, Comput. Phys. Commun., 126 (2000), 219-231. 

show all references

References:
[1]

S. Ai, Traveling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differ. Equ., 232 (2007), 104-133.  doi: 10.1016/j.jde.2006.08.015.

[2]

J. CarrS. N. Chow and J. K. Hale, Abelian integrals and bifurcation theory, J. Differ. Equ., 59 (1985), 413-436.  doi: 10.1016/0022-0396(85)90148-2.

[3]

S. Chakravarty and Y. Kodama, Soliton solutions of the KP equation and application to shallow water waves, Stud. Appl. Math., 123 (2009), 83-151.  doi: 10.1111/j.1467-9590.2009.00448.x.

[4]

A. ChenL. Guo and X. Deng, Existence of solitary waves and periodic waves for a perturbed generalized BBM equation, J. Differ. Equ., 261 (2016), 5324-5349.  doi: 10.1016/j.jde.2016.08.003.

[5]

F. Chen and J. Li, Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay, Discrete Contin. Dyn. Syst., 41 (2021), 967-985.  doi: 10.3934/dcds.2020305.

[6]

M. Darwich, On the well-posedness for Kadomtsev-Petviashvili-Burgers I equation, J. Differ. Equ., 253 (2012), 1584-1603.  doi: 10.1016/j.jde.2012.05.013.

[7]

A. DegasperisS. Lombardo and M. Sommaca, Integrability and Linear Stability of NonlinearWaves, J. Nonlinear Sci., 28 (2018), 1251-1291.  doi: 10.1007/s00332-018-9450-5.

[8]

Z. DuJ. Li and X. Li, The existence of solitary wave solutions of delayed Camassa-Holm via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005.

[9]

Z. Du and Z. Feng, Existence and asymptotic behavior of traveling waves in a modified vector-disease model, Commun. Pur. Appl. Anal., 17 (2018), 1899-1920.  doi: 10.3934/cpaa.2018090.

[10]

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

[11]

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

[12]

G. GuiY. Liu and T. Luo, Model equations and traveling wave solutions for shallow-water waves with the coriolis effect, J. Nonlinear Sci., 29 (2019), 993-1039.  doi: 10.1007/s00332-018-9510-x.

[13]

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.

[14]

M. HolzerA. Doelman and T. J. Kaper, Existence and stability of traveling pulses in a reaction-diffusion-mechanics system, J Nonlinear Sci., 23 (2013), 129-177.  doi: 10.1007/s00332-012-9147-0.

[15]

J. IsazaL. Mejia and N. Tzvetkov, A smoothing effect and polynomial growth of the Sobolev norms for the KP-II equation, J. Differ. Equ., 220 (2006), 1-17.  doi: 10.1016/j.jde.2004.10.002.

[16]

C. K. R. T. Jones, Geometrical singular perturbation theory, in: R. Johnson (Ed.), Dynamical Systems, in: Lecture Notes in Mathematics, Springer, New York, 1995. doi: 10.1007/BFb0095239.

[17]

B. Kojok, Sharp well-posedness for Kadomtsev-Petviashvili-Burgers (KPBII) equation in $R^{2}$, J. Differ. Equ., 242 (2007), 211-247.  doi: 10.1016/j.jde.2007.08.010.

[18]

S. Krantz and H. Parks, The Implicit Function Theorem: History, Theory, and Applications, Birkh$\ddot{a}$user Boston, 2003. doi: 10.1007/978-1-4612-0059-8.

[19]

J. LiK. Lu and P. Bates, Normally hyperbolic invariant manifolds for random dynamical systems, Trans. Amer. Math. Soci., 365 (2013), 5933-5966.  doi: 10.1090/S0002-9947-2013-05825-4.

[20]

J. LiK. Lu and P. Bates, Invariant foliations for random dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 3639-3666.  doi: 10.3934/dcds.2014.34.3639.

[21]

J. LiK. Lu and P. Bates, Geometric singular perturbation theory with real noise, J. Differ. Equ., 259 (2015), 5137-5167.  doi: 10.1016/j.jde.2015.06.023.

[22]

L. MolinetJ. 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.

[23]

L. MolinetJ. Saut and N. Tzvetkov, Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation, Duke Math. J., 115 (2002), 353-384.  doi: 10.1215/S0012-7094-02-11525-7.

[24]

T. Ogawa, Travelling wave solutions to a perturbed Korteweg-de Vries equation, Hiroshima Math. J., 24 (1994), 401-422. 

[25]

S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model, Proc. R. Soc. Edinb., 134A (2004), 991-1011.  doi: 10.1017/S0308210500003590.

[26]

A. Saha, Dynamics of the generalized KP-MEW-Burgers equation with external periodic perturbation, Comput. Math. Appl., 73 (2017), 1879-1885.  doi: 10.1016/j.camwa.2017.02.017.

[27]

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

[28]

C. Wang and X. Zhang, Canards, heteroclinic and homoclinic orbits for a slow-fast predator-prey model of generalized Holling type III, J. Differ. Equ., 267 (2019), 3397-3441.  doi: 10.1016/j.jde.2019.04.008.

[29]

S. I. Zaki, Solitary wave interactions for the modified equal width equation, Comput. Phys. Commun., 126 (2000), 219-231. 

Figure 1.  The homoclinic orbit within $ u\leq-\frac{3}{2a} $
Figure 2.  The graph of the function $ v(z) $
Figure 3.  The graph of the function $ v(z) $
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