# American Institute of Mathematical Sciences

February  2020, 3(1): 11-24. doi: 10.3934/mfc.2020002

## Orbital stability of periodic traveling wave solutions to the coupled compound KdV and MKdV equations with two components

 School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, China

* Corresponding author: Xiaoxiao Zheng

Received  December 2019 Published  February 2020

Fund Project: The first author is supported by the Natural Science Foundation of Shandong Province (No. ZR2018BA016), the second author is supported by the National Natural Science Foundation of China (No. 11801306)

In this article, the authors consider the orbital stability of periodic traveling wave solutions for the coupled compound KdV and MKdV equations with two components
 \begin{equation*} \left\{ \begin{aligned} &u_{t}+vv_{x}+\beta u^{2}u_{x}+u_{xxx}-uu_{x} = 0, \ \ \beta>0, \\ &v_{t}+(uv)_{x}+2vv_{x} = 0, \end{aligned} \right. \end{equation*}
Firstly, we show that there exist a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period
 $L$
for the coupled compound KdV and MKdV equations. Then, combining the orbital stability theory presented by Grillakis et al., and detailed spectral analysis given by using Lamé equation and Floquet theory, we show that the dnoidal type periodic wave solution with period
 $L$
is orbitally stable. As the modulus of the Jacobian elliptic function
 $k\rightarrow 1$
, we obtain the orbital stability results of solitary wave solution with zero asymptotic value for the coupled compound KdV and MKdV equations from our work. In addition, we also obtain the stability results for the coupled compound KdV and MKdV equations with the degenerate condition
 $v = 0$
, called the compound KdV and MKdV equation.
Citation: Xiaoxiao Zheng, Hui Wu. Orbital stability of periodic traveling wave solutions to the coupled compound KdV and MKdV equations with two components. Mathematical Foundations of Computing, 2020, 3 (1) : 11-24. doi: 10.3934/mfc.2020002
##### References:
 [1] J. Angulo, Stability of cnoidal waves to Hirota-Satsuma systems, Matemática Contemporânea, 27 (2004), 189–223.  Google Scholar [2] J. Angulo, Stability of dnoidal waves to Hirota-Satsuma system, Differential Integral Equations, 18 (2005), 611-645.   Google Scholar [3] P. Byrd and M. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists 2nd edn, New York: Springer, 1971.  Google Scholar [4] Q. R. Chowdhury and R. Mukherjee, On the complete integrability of the Hirota Satsuma, Journal of Physics A: Mathematical and General, 17 (1977), 231-234.  doi: 10.1088/0305-4470/17/5/002.  Google Scholar [5] S. Q. Dai, Solitary wave at the interface of a two-layer fluid, Applied Mathematics and Mechanics, 3 (1982), 721-731.   Google Scholar [6] S. Q. Dai, G. F. Sigalov and A. V. Diogenov, Approximate analytical solutions for some strong nonlinear problems, Science in China Series A, 33 (1990), 843-853.   Google Scholar [7] C. Guha-Roy, Solitary wave solutions of a system of coupued nonlinear equation, J. Math. Phys., 28 (1987), 2087-2088.  doi: 10.1063/1.527419.  Google Scholar [8] C. Guha-Roy, Exact solutions to a coupled nonlinear equation, Inter. J. Theor. Phys., 27 (1988), 447-450.   Google Scholar [9] C. Guha-Roy, On explicit solutions of a coupled KdV-mKdV equation, Internat. J. Modern Phys. B, 3 (1989), 871-875.  doi: 10.1142/S0217979289000646.  Google Scholar [10] B. L. Guo and L. Chen, Orbital stability of solitary waves of coupled KdV equations, Differential and Integral Equations, 12 (1999), 295-308.   Google Scholar [11] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry I, Journal of Functional Analysis, 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9.  Google Scholar [12] B. L. Guo and S. B. Tan, Global smooth solution for coupled nonlinear wave equations, Mathematical Methods in the Applied Sciences, 14 (1991), 419-425.  doi: 10.1002/mma.1670140606.  Google Scholar [13] W. P. Hong, New types of solitary-wave solutions from the combined KdV-mKdV equation, Nuovo Cimento Della Società Italiana Di Fisica B, 115 (2000), 117-118.   Google Scholar [14] R. Hirota and J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation, Physics Letters A, 85 (1981), 407-408.  doi: 10.1016/0375-9601(81)90423-0.  Google Scholar [15] R. J. Iorio and V. Iorio, Fourier Analysis and Partial Differential Equations, Cambridge Studies in Advanced Mathematics, vol 70, Cambridge: Cambridge University Press, 2001. Google Scholar [16] E. L. Ince, The periodic Lam$\acute{e}$ functions, Proceedings of the Royal Society of Edinburgh, 60 (1940), 47-63.  doi: 10.1017/S0370164600020058.  Google Scholar [17] S. Y. Lou and L. L. Chen, Solitary wave solutions and cnoidal wave solutions to the combined KdV and MKdV equation, Mathematical Methods in the Applied Sciences, 17 (1994), 339-347.  doi: 10.1002/mma.1670170503.  Google Scholar [18] W. Magnus and S. Winkler, Hill's Equation, Tracts in Pure and Appliled Mathematics, vol. 20, Wiley, New York, 1966.  Google Scholar [19] V. Narayanamurti and C. M. Varma, Nonlinear propagation of heat pulses in solids, Physical Review Letters, 25 (1970), 1105-1108.   Google Scholar [20] M. Toda, Waves in nonlinear lattice, Progress of Theoretical Physics Supplement, 45 (1970), 174-200.   Google Scholar [21] W. G. Zhang, G. L. Shi and Y. H. Qin, Orbital stability of solitary waves for the, Nonlinear Analysis: Real World Applications, 12 (2011), 1627-1639.  doi: 10.1016/j.nonrwa.2010.10.017.  Google Scholar [22] X. X. Zheng, Y. D. Shang and X. M. Peng, Orbital stability of periodic traveling wave solutions to the generalized Zakharov equations, Acta Math. Sci., 37 (2017), 998-1018.  doi: 10.1016/S0252-9602(17)30054-1.  Google Scholar

show all references

##### References:
 [1] J. Angulo, Stability of cnoidal waves to Hirota-Satsuma systems, Matemática Contemporânea, 27 (2004), 189–223.  Google Scholar [2] J. Angulo, Stability of dnoidal waves to Hirota-Satsuma system, Differential Integral Equations, 18 (2005), 611-645.   Google Scholar [3] P. Byrd and M. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists 2nd edn, New York: Springer, 1971.  Google Scholar [4] Q. R. Chowdhury and R. Mukherjee, On the complete integrability of the Hirota Satsuma, Journal of Physics A: Mathematical and General, 17 (1977), 231-234.  doi: 10.1088/0305-4470/17/5/002.  Google Scholar [5] S. Q. Dai, Solitary wave at the interface of a two-layer fluid, Applied Mathematics and Mechanics, 3 (1982), 721-731.   Google Scholar [6] S. Q. Dai, G. F. Sigalov and A. V. Diogenov, Approximate analytical solutions for some strong nonlinear problems, Science in China Series A, 33 (1990), 843-853.   Google Scholar [7] C. Guha-Roy, Solitary wave solutions of a system of coupued nonlinear equation, J. Math. Phys., 28 (1987), 2087-2088.  doi: 10.1063/1.527419.  Google Scholar [8] C. Guha-Roy, Exact solutions to a coupled nonlinear equation, Inter. J. Theor. Phys., 27 (1988), 447-450.   Google Scholar [9] C. Guha-Roy, On explicit solutions of a coupled KdV-mKdV equation, Internat. J. Modern Phys. B, 3 (1989), 871-875.  doi: 10.1142/S0217979289000646.  Google Scholar [10] B. L. Guo and L. Chen, Orbital stability of solitary waves of coupled KdV equations, Differential and Integral Equations, 12 (1999), 295-308.   Google Scholar [11] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry I, Journal of Functional Analysis, 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9.  Google Scholar [12] B. L. Guo and S. B. Tan, Global smooth solution for coupled nonlinear wave equations, Mathematical Methods in the Applied Sciences, 14 (1991), 419-425.  doi: 10.1002/mma.1670140606.  Google Scholar [13] W. P. Hong, New types of solitary-wave solutions from the combined KdV-mKdV equation, Nuovo Cimento Della Società Italiana Di Fisica B, 115 (2000), 117-118.   Google Scholar [14] R. Hirota and J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation, Physics Letters A, 85 (1981), 407-408.  doi: 10.1016/0375-9601(81)90423-0.  Google Scholar [15] R. J. Iorio and V. Iorio, Fourier Analysis and Partial Differential Equations, Cambridge Studies in Advanced Mathematics, vol 70, Cambridge: Cambridge University Press, 2001. Google Scholar [16] E. L. Ince, The periodic Lam$\acute{e}$ functions, Proceedings of the Royal Society of Edinburgh, 60 (1940), 47-63.  doi: 10.1017/S0370164600020058.  Google Scholar [17] S. Y. Lou and L. L. Chen, Solitary wave solutions and cnoidal wave solutions to the combined KdV and MKdV equation, Mathematical Methods in the Applied Sciences, 17 (1994), 339-347.  doi: 10.1002/mma.1670170503.  Google Scholar [18] W. Magnus and S. Winkler, Hill's Equation, Tracts in Pure and Appliled Mathematics, vol. 20, Wiley, New York, 1966.  Google Scholar [19] V. Narayanamurti and C. M. Varma, Nonlinear propagation of heat pulses in solids, Physical Review Letters, 25 (1970), 1105-1108.   Google Scholar [20] M. Toda, Waves in nonlinear lattice, Progress of Theoretical Physics Supplement, 45 (1970), 174-200.   Google Scholar [21] W. G. Zhang, G. L. Shi and Y. H. Qin, Orbital stability of solitary waves for the, Nonlinear Analysis: Real World Applications, 12 (2011), 1627-1639.  doi: 10.1016/j.nonrwa.2010.10.017.  Google Scholar [22] X. X. Zheng, Y. D. Shang and X. M. Peng, Orbital stability of periodic traveling wave solutions to the generalized Zakharov equations, Acta Math. Sci., 37 (2017), 998-1018.  doi: 10.1016/S0252-9602(17)30054-1.  Google Scholar
 [1] Xiaojie Hou, Wei Feng. Traveling waves and their stability in a coupled reaction diffusion system. Communications on Pure & Applied Analysis, 2011, 10 (1) : 141-160. doi: 10.3934/cpaa.2011.10.141 [2] Fábio Natali, Ademir Pastor. Orbital stability of periodic waves for the Klein-Gordon-Schrödinger system. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 221-238. doi: 10.3934/dcds.2011.31.221 [3] Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561 [4] Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993 [5] Fengxin Chen. Stability and uniqueness of traveling waves for system of nonlocal evolution equations with bistable nonlinearity. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 659-673. doi: 10.3934/dcds.2009.24.659 [6] Santosh Bhattarai. Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1789-1811. doi: 10.3934/dcds.2016.36.1789 [7] Felipe Linares, M. Panthee. On the Cauchy problem for a coupled system of KdV equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 417-431. doi: 10.3934/cpaa.2004.3.417 [8] Yuqian Zhou, Qian Liu. Reduction and bifurcation of traveling waves of the KdV-Burgers-Kuramoto equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 2057-2071. doi: 10.3934/dcdsb.2016036 [9] Yonggeun Cho, Hichem Hajaiej, Gyeongha Hwang, Tohru Ozawa. On the orbital stability of fractional Schrödinger equations. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1267-1282. doi: 10.3934/cpaa.2014.13.1267 [10] Sevdzhan Hakkaev. Orbital stability of solitary waves of the Schrödinger-Boussinesq equation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1043-1050. doi: 10.3934/cpaa.2007.6.1043 [11] Hua Chen, Ling-Jun Wang. A perturbation approach for the transverse spectral stability of small periodic traveling waves of the ZK equation. Kinetic & Related Models, 2012, 5 (2) : 261-281. doi: 10.3934/krm.2012.5.261 [12] Aiyong Chen, Xinhui Lu. Orbital stability of elliptic periodic peakons for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1703-1735. doi: 10.3934/dcds.2020090 [13] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382 [14] Guangyu Zhao. Multidimensional periodic traveling waves in infinite cylinders. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 1025-1045. doi: 10.3934/dcds.2009.24.1025 [15] Nar Rawal, Wenxian Shen, Aijun Zhang. Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1609-1640. doi: 10.3934/dcds.2015.35.1609 [16] Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029 [17] Rui Huang, Ming Mei, Kaijun Zhang, Qifeng Zhang. Asymptotic stability of non-monotone traveling waves for time-delayed nonlocal dispersion equations. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1331-1353. doi: 10.3934/dcds.2016.36.1331 [18] Yicheng Jiang, Kaijun Zhang. Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations. Kinetic & Related Models, 2018, 11 (5) : 1235-1253. doi: 10.3934/krm.2018048 [19] Yaping Wu, Xiuxia Xing. Stability of traveling waves with critical speeds for $P$-degree Fisher-type equations. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 1123-1139. doi: 10.3934/dcds.2008.20.1123 [20] Judith R. Miller, Huihui Zeng. Multidimensional stability of planar traveling waves for an integrodifference model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 741-751. doi: 10.3934/dcdsb.2013.18.741

Impact Factor: