# 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

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##### 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
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