# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022124
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## Solitary waves of singularly perturbed generalized KdV equation with high order nonlinearity

 1 Department of Mathematics, Shandong University of Science and Technology, Qingdao, Shandong 266590, China 2 International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho, South Africa 3 Samara National Research University, Moskovskoye Shosse 34, Samara, 443086, Russia

* Corresponding author: Lijun Zhang

Received  February 2022 Revised  March 2022 Early access May 2022

The paper is concerned on solitary waves for singularly perturbed generalized KdV equation with high order nonlinear terms. We firstly give the phase portraits of system related to the unperturbed equation under various cases by theory of planar dynamical system. Then by using geometric singular perturbation theory and Melnikov's method, the existence of solitary wave solutions of generalized KdV equations with high order nonlinear terms is established. It is proven that some solitary wave solutions with particular wave speeds will persist under small perturbations.

Citation: Jundong Wang, Lijun Zhang, Elena Shchepakina, Vladimir Sobolev. Solitary waves of singularly perturbed generalized KdV equation with high order nonlinearity. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022124
##### References:
 [1] A. Biswas, 1-soliton solution of the $K(m, n)$ equation with generalized evolution, Phys. Lett. A, 372 (2008), 4601-4602.  doi: 10.1016/j.physleta.2008.05.002. [2] A. Chen, L. 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. [3] A. Chen, L. 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. [4] G. Derks and S. van Gils, On the uniqueness of traveling waves in perturbed Korteveg-de Vries equations, Japan J. Indust. Appl. Math., 10 (1993), 413-430.  doi: 10.1007/BF03167282. [5] 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. [6] Z. Du and Q. Qi, 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. [7] X. Fan and L. Tian, The existence of solitary waves of singularly perturbed mKdV-KS equation, Chaos Solitons Fractals, 26 (2005), 1111-1118.  doi: 10.1016/j.chaos.2005.02.014. [8] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equ., 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9. [9] L. Guo and Y. Zhao, Existence of periodic waves for a perturbed quintic BBM equation, Discrete Cont. Dyn. Sys., 40 (2020), 4689-4703.  doi: 10.3934/dcds.2020198. [10] M. Han, Bifurcation Theory of Limit Cycles, Science press, Beijing, 2017. [11] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Phi. Mag., 39 (1895), 422-443.  doi: 10.1080/14786449508620739. [12] T. Ogawa, Traveling wave solutions to a perturbed Korteweg-de Vries equation, Hiroshima Math. J., 24 (1994), 401-422. [13] P. Rosenau and J. M. Hyman, Compactons: Solitons with finite wavelength, Phys. Rev. Lett., 70 (1993), 564-567. [14] X. Sun, Y. Zeng and P. Yu, Analysis and simulation of periodic and solitary waves in nonlinear dispersive-dissipative solids, Commun. Nonlinear Sci. Numer. Simul., 102 (2021), Paper No. 105921, 13 pp. doi: 10.1016/j.cnsns.2021.105921. [15] Y. Tang, W. Xu, J. Shen and L. Gao, Persistence of solitary wave solutions of singularly perturbed Gardner equation, Chaos Solit. Fract., 37 (2008), 532-538.  doi: 10.1016/j.chaos.2006.09.044. [16] J. Wang, M. Yuen and L. Zhang, Persistence of solitary wave solutions to a singularly perturbed generalized mKdV equation, Appl. Math. Lett., 124 (2022), Paper No. 107668, 7 pp. doi: 10.1016/j.aml.2021.107668. [17] Y. Xu, Z. Du and L. Wei, Geometric singular perturbation method to the existence and asymptotic behavior of traveling waves for a generalized Burgers-KdV equation, Nonlinear Dyn., 83 (2016), 65-73.  doi: 10.1007/s11071-015-2309-5. [18] W. Yan, Z. 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. [19] L. Zhang, M. Han, M. Zhang and C. M. Khalique, A new type of solitary wave solution of the mKdV equation under singular perturbations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050162, 14 pp. doi: 10.1142/S021812742050162X. [20] L. Zhang, J. Wang, E. Shchepakina and V. Sobolev, New type of solitary wave solution with coexisting crest and trough for a perturbed wave equation, Nonlinear Dyn., 106 (2021), 3479-3493. [21] Z. Zhao, Solitary waves of the generalized KdV equation with distributed delays, J. Math. Anal. Appl., 344 (2008), 32-41.  doi: 10.1016/j.jmaa.2008.02.036. [22] Z. Zhao and Y. Xu, Solitary waves for Korteweg-de-Vries equation with small delay, J. Math. Anal. Appl., 368 (2010), 43-53.  doi: 10.1016/j.jmaa.2010.02.014. [23] K. Zhu, Y. Wu, Z. Yu and J. Shen, New solitary wave solutions in a perturbed generalized BBM equation, Nonlinear Dyn., 97 (2019), 2413-2423.

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##### References:
 [1] A. Biswas, 1-soliton solution of the $K(m, n)$ equation with generalized evolution, Phys. Lett. A, 372 (2008), 4601-4602.  doi: 10.1016/j.physleta.2008.05.002. [2] A. Chen, L. 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. [3] A. Chen, L. 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. [4] G. Derks and S. van Gils, On the uniqueness of traveling waves in perturbed Korteveg-de Vries equations, Japan J. Indust. Appl. Math., 10 (1993), 413-430.  doi: 10.1007/BF03167282. [5] 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. [6] Z. Du and Q. Qi, 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. [7] X. Fan and L. Tian, The existence of solitary waves of singularly perturbed mKdV-KS equation, Chaos Solitons Fractals, 26 (2005), 1111-1118.  doi: 10.1016/j.chaos.2005.02.014. [8] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equ., 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9. [9] L. Guo and Y. Zhao, Existence of periodic waves for a perturbed quintic BBM equation, Discrete Cont. Dyn. Sys., 40 (2020), 4689-4703.  doi: 10.3934/dcds.2020198. [10] M. Han, Bifurcation Theory of Limit Cycles, Science press, Beijing, 2017. [11] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Phi. Mag., 39 (1895), 422-443.  doi: 10.1080/14786449508620739. [12] T. Ogawa, Traveling wave solutions to a perturbed Korteweg-de Vries equation, Hiroshima Math. J., 24 (1994), 401-422. [13] P. Rosenau and J. M. Hyman, Compactons: Solitons with finite wavelength, Phys. Rev. Lett., 70 (1993), 564-567. [14] X. Sun, Y. Zeng and P. Yu, Analysis and simulation of periodic and solitary waves in nonlinear dispersive-dissipative solids, Commun. Nonlinear Sci. Numer. Simul., 102 (2021), Paper No. 105921, 13 pp. doi: 10.1016/j.cnsns.2021.105921. [15] Y. Tang, W. Xu, J. Shen and L. Gao, Persistence of solitary wave solutions of singularly perturbed Gardner equation, Chaos Solit. Fract., 37 (2008), 532-538.  doi: 10.1016/j.chaos.2006.09.044. [16] J. Wang, M. Yuen and L. Zhang, Persistence of solitary wave solutions to a singularly perturbed generalized mKdV equation, Appl. Math. Lett., 124 (2022), Paper No. 107668, 7 pp. doi: 10.1016/j.aml.2021.107668. [17] Y. Xu, Z. Du and L. Wei, Geometric singular perturbation method to the existence and asymptotic behavior of traveling waves for a generalized Burgers-KdV equation, Nonlinear Dyn., 83 (2016), 65-73.  doi: 10.1007/s11071-015-2309-5. [18] W. Yan, Z. 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. [19] L. Zhang, M. Han, M. Zhang and C. M. Khalique, A new type of solitary wave solution of the mKdV equation under singular perturbations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050162, 14 pp. doi: 10.1142/S021812742050162X. [20] L. Zhang, J. Wang, E. Shchepakina and V. Sobolev, New type of solitary wave solution with coexisting crest and trough for a perturbed wave equation, Nonlinear Dyn., 106 (2021), 3479-3493. [21] Z. Zhao, Solitary waves of the generalized KdV equation with distributed delays, J. Math. Anal. Appl., 344 (2008), 32-41.  doi: 10.1016/j.jmaa.2008.02.036. [22] Z. Zhao and Y. Xu, Solitary waves for Korteweg-de-Vries equation with small delay, J. Math. Anal. Appl., 368 (2010), 43-53.  doi: 10.1016/j.jmaa.2010.02.014. [23] K. Zhu, Y. Wu, Z. Yu and J. Shen, New solitary wave solutions in a perturbed generalized BBM equation, Nonlinear Dyn., 97 (2019), 2413-2423.
Phase portraits of system (10) for the case when $n>m$, $m$ and $n$ are even
Phase portraits of system (10) in the case $n>m$, $m$ and $n$ are odd
The bifurcation phase portraits of system (10) in the case $n>m$, $m$ is even and $n$ is odd
Phase portraits of system (10) in the case $n>m$, $m$ is odd and $n$ is even
Phase portraits of system (10) in the case $m>n$, $m$ and $n$ are even
Phase portraits of system (10) in the case $m>n$, $m$ and $n$ are odd with $n>1$
Phase portraits of system (10) in the case $m>n$, $m$ is odd and $n$ is even
Phase portraits of system (10) in the case $m>n$, $m$ is even and $n>1$ is odd
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