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doi: 10.3934/dcdss.2022048
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Limit speed of traveling wave solutions for the perturbed generalized KdV equation

1. 

Department of Mathematics, Hunan First Normal University, Changsha, Hunan 410205, China

2. 

School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China

3. 

School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China

Dedicated to Professor Jibin Li on the occasion of his 80th birthday

Received  December 2021 Revised  January 2022 Early access March 2022

Fund Project: This work have been supported by the Natural Science Foundation of Hunan Province (No. 2021JJ30166) and the National Natural Science Foundation of China (No. 11971163, and No. 12061016)

The existence of solitary waves and periodic waves for a perturbed generalized KdV equation is established by using geometric singular perturbation theory. It is proven that the limit wave speed $ c_{0}(h) $ is decreasing by analyzing the ratio of Abelian integrals for $ n = 2 $ and $ n = 3 $. The upper and lower bounds of the limit wave speed are given. Moreover, the relation between the wave speed and the wavelength of traveling waves is obtained. Our results answer partially an open question proposed by Yan, Liu and Liang [Math. Model. Anal., 19 (2014), pp. 537-555].

Citation: Aiyong Chen, Chi Zhang, Wentao Huang. Limit speed of traveling wave solutions for the perturbed generalized KdV equation. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022048
References:
[1]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive system, Philos. Trans. R. Soc. Lond. Ser. A., 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.

[2]

R. Camassa and D. D. Holm, An integrable shallow wave equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.

[3]

J. Carr, Applications of the Center Manifold Theory, Applied Mathematieal Sciences, vol. 35. Springer, New York., 1981.

[4]

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.

[5]

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.

[6]

A. ChenL. Guo and W. Huang, Existence of kink waves and periodic waves for a perturbed defocusing mKdV equation, Qual. Theory Dyn. Syst., 17 (2018), 495-517.  doi: 10.1007/s12346-017-0249-9.

[7]

A. ChenJ. Li and W. Huang, The monotonicity and critical periods of periodic waves of the $\phi^{6}$ field model, Nonlinear Dyn., 63 (2011), 205-215.  doi: 10.1007/s11071-010-9797-0.

[8]

A. Chen, C. Zhang and W. Huang, Monotonicity of limit wave speed of traveling wave solutions for a perturbed generalized KdV equation, Appl. Math. Lett., 121 (2021), Paper No. 107381, 7 pp. doi: 10.1016/j.aml.2021.107381.

[9]

X. ChenV. G. Romanovski and W. Zhang, Critical periods of perturbations of reversible rigidly isochronous centers, J. Differ. Equ., 251 (2011), 1505-1525.  doi: 10.1016/j.jde.2011.05.022.

[10]

C. Chicone and M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Am. Math. Soc., 312 (1989), 433-486.  doi: 10.1090/S0002-9947-1989-0930075-2.

[11]

S.-N. ChowC. Li and Y. Yi, The cyclicity of period annuli of degenerate quadratic Hamiltonian systems with elliptic segment loops, Ergod. Th. Dynam. Sys., 22 (2002), 349-374.  doi: 10.1017/S0143385702000184.

[12]

S.-N. Chow and J. A. Sanders, On the number of critical points of the period, J. Differ. Equ., 64 (1986), 51-66.  doi: 10.1016/0022-0396(86)90071-9.

[13]

R. Cushman and J. A. Sanders, A codimension two bifurcations with a third order Picard-Fuchs equation, J. Differ. Equ., 59 (1985), 243-256.  doi: 10.1016/0022-0396(85)90156-1.

[14]

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

[15]

F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (Ⅰ) saddle loop and two saddle cycle, J. Differ. Equ., 176 (2001), 114-157.  doi: 10.1006/jdeq.2000.3977.

[16]

F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (Ⅱ) cuspidal loop, J. Differ. Equ., 175 (2001), 209-243.  doi: 10.1006/jdeq.2000.3978.

[17]

F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (Ⅲ) global center, J. Differ. Equ., 188 (2003), 473-511.  doi: 10.1016/S0022-0396(02)00110-9.

[18]

F. Dumortier and C. Li, Perturbation from an elliptic Hamiltonian of degree four: (Ⅳ) figure eight-loop, J. Differ. Equ., 188 (2003), 512-554.  doi: 10.1016/S0022-0396(02)00111-0.

[19]

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

[20]

A. Garijo and J. Villadelprat, Algebraic and analytical tools for the study of the period function, J. Differ. Equ., 257 (2014), 2464-2484.  doi: 10.1016/j.jde.2014.05.044.

[21]

A. GasullC. Liu and J. Yang, On the number of critical periods for planar polynomial systems of arbitrary degree, J. Differ. Equ., 249 (2010), 684-692.  doi: 10.1016/j.jde.2010.01.002.

[22]

A. Geyer and J. Villadelprat, On the wave length of smooth periodic traveling waves of the Camassa-Holm equation, J. Differ. Equ., 259 (2015), 2317-2332.  doi: 10.1016/j.jde.2015.03.027.

[23]

A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246. 

[24]

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.

[25]

D. J. Korteweg and G. de Vries, On the change of form of the long waves advancing in a rectangular canal, and on a new type of stationary waves, Philos. Mag., 39 (1895), 422-443.  doi: 10.1080/14786449508620739.

[26]

C. Li and K. Lu, The period function of hyperelliptic Hamiltonians of degree 5 with real critical points, Nonlinearity., 21 (2008), 465-483.  doi: 10.1088/0951-7715/21/3/006.

[27]

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

[28]

M. Sabatini, On the period function of Liénard systems, J. Differ. Equ., 152 (1999), 467-487.  doi: 10.1006/jdeq.1998.3520.

[29]

W. YanZ. Liu and Y. Liang, Existence of solitary waves and periodic waves to a perturbed generalized KdV equation, Math. Model. Anal., 19 (2014), 537-555.  doi: 10.3846/13926292.2014.960016.

[30]

Y. Zhao, The monotonicity of the period function for codimension four quadratic system $Q_{4}$, J. Differ. Equ., 185 (2002), 370-387.  doi: 10.1006/jdeq.2002.4175.

show all references

References:
[1]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive system, Philos. Trans. R. Soc. Lond. Ser. A., 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.

[2]

R. Camassa and D. D. Holm, An integrable shallow wave equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.

[3]

J. Carr, Applications of the Center Manifold Theory, Applied Mathematieal Sciences, vol. 35. Springer, New York., 1981.

[4]

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.

[5]

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.

[6]

A. ChenL. Guo and W. Huang, Existence of kink waves and periodic waves for a perturbed defocusing mKdV equation, Qual. Theory Dyn. Syst., 17 (2018), 495-517.  doi: 10.1007/s12346-017-0249-9.

[7]

A. ChenJ. Li and W. Huang, The monotonicity and critical periods of periodic waves of the $\phi^{6}$ field model, Nonlinear Dyn., 63 (2011), 205-215.  doi: 10.1007/s11071-010-9797-0.

[8]

A. Chen, C. Zhang and W. Huang, Monotonicity of limit wave speed of traveling wave solutions for a perturbed generalized KdV equation, Appl. Math. Lett., 121 (2021), Paper No. 107381, 7 pp. doi: 10.1016/j.aml.2021.107381.

[9]

X. ChenV. G. Romanovski and W. Zhang, Critical periods of perturbations of reversible rigidly isochronous centers, J. Differ. Equ., 251 (2011), 1505-1525.  doi: 10.1016/j.jde.2011.05.022.

[10]

C. Chicone and M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Am. Math. Soc., 312 (1989), 433-486.  doi: 10.1090/S0002-9947-1989-0930075-2.

[11]

S.-N. ChowC. Li and Y. Yi, The cyclicity of period annuli of degenerate quadratic Hamiltonian systems with elliptic segment loops, Ergod. Th. Dynam. Sys., 22 (2002), 349-374.  doi: 10.1017/S0143385702000184.

[12]

S.-N. Chow and J. A. Sanders, On the number of critical points of the period, J. Differ. Equ., 64 (1986), 51-66.  doi: 10.1016/0022-0396(86)90071-9.

[13]

R. Cushman and J. A. Sanders, A codimension two bifurcations with a third order Picard-Fuchs equation, J. Differ. Equ., 59 (1985), 243-256.  doi: 10.1016/0022-0396(85)90156-1.

[14]

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

[15]

F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (Ⅰ) saddle loop and two saddle cycle, J. Differ. Equ., 176 (2001), 114-157.  doi: 10.1006/jdeq.2000.3977.

[16]

F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (Ⅱ) cuspidal loop, J. Differ. Equ., 175 (2001), 209-243.  doi: 10.1006/jdeq.2000.3978.

[17]

F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (Ⅲ) global center, J. Differ. Equ., 188 (2003), 473-511.  doi: 10.1016/S0022-0396(02)00110-9.

[18]

F. Dumortier and C. Li, Perturbation from an elliptic Hamiltonian of degree four: (Ⅳ) figure eight-loop, J. Differ. Equ., 188 (2003), 512-554.  doi: 10.1016/S0022-0396(02)00111-0.

[19]

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

[20]

A. Garijo and J. Villadelprat, Algebraic and analytical tools for the study of the period function, J. Differ. Equ., 257 (2014), 2464-2484.  doi: 10.1016/j.jde.2014.05.044.

[21]

A. GasullC. Liu and J. Yang, On the number of critical periods for planar polynomial systems of arbitrary degree, J. Differ. Equ., 249 (2010), 684-692.  doi: 10.1016/j.jde.2010.01.002.

[22]

A. Geyer and J. Villadelprat, On the wave length of smooth periodic traveling waves of the Camassa-Holm equation, J. Differ. Equ., 259 (2015), 2317-2332.  doi: 10.1016/j.jde.2015.03.027.

[23]

A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246. 

[24]

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.

[25]

D. J. Korteweg and G. de Vries, On the change of form of the long waves advancing in a rectangular canal, and on a new type of stationary waves, Philos. Mag., 39 (1895), 422-443.  doi: 10.1080/14786449508620739.

[26]

C. Li and K. Lu, The period function of hyperelliptic Hamiltonians of degree 5 with real critical points, Nonlinearity., 21 (2008), 465-483.  doi: 10.1088/0951-7715/21/3/006.

[27]

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

[28]

M. Sabatini, On the period function of Liénard systems, J. Differ. Equ., 152 (1999), 467-487.  doi: 10.1006/jdeq.1998.3520.

[29]

W. YanZ. Liu and Y. Liang, Existence of solitary waves and periodic waves to a perturbed generalized KdV equation, Math. Model. Anal., 19 (2014), 537-555.  doi: 10.3846/13926292.2014.960016.

[30]

Y. Zhao, The monotonicity of the period function for codimension four quadratic system $Q_{4}$, J. Differ. Equ., 185 (2002), 370-387.  doi: 10.1006/jdeq.2002.4175.

Figure 1.  The phase portraits of the system (2.5).(a) $ n $ is even. (b) $ n $ is odd
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