Lie symmetry analysis and exact solutions for the extended mKdV equation. (English) Zbl 1223.37079

This paper concerns solutions of an extended mKdV equation of the form \[ u_t +a_1 u_{xxx} + a_2 u_x +a_3 u u_x + a_4 u^2 u_x = 0, \] where \(u = u(x,t)\) is an unknown function and the parameters \(a_i\) are real numbers with \(a_1 a_4 \neq 0\). Its exact solutions are obtained by using the method of Lie symmetry analysis as well as the method of dynamical systems, including the bifurcation and traveling wave solutions. The authors determine the conditions of the existence of such solutions and obtain exact analytic solutions by using the power series method.


37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
17B80 Applications of Lie algebras and superalgebras to integrable systems
35A30 Geometric theory, characteristics, transformations in context of PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
Full Text: DOI


[1] Olver, P.J.: Applications of Lie groups to differential equations. In: Grauate Texts in Mathematics, vol. 107. Springer, New York (1993) · Zbl 0785.58003
[2] Tian, C.: Lie Groups and Its Applications to Differential Equations. Science Press, Beijing (2001). (In Chinese)
[3] Chen, D.Y.: Introduction to Solitons. Science Press, Beijing (2006). (In Chinese)
[4] Hirota, R., Satsuma, J.: A variety of nonlinear network equations generated from the Bäcklund transformation for the Tota lattice. Suppl. Prog. Theor. Phys. 59, 64–100 (1976) · Zbl 1079.35536
[5] Nakamura, A.: A direct method of calculating periodic wave solutions to nonlinear evolution equations, II. J. Phys. Soc. Jpn. 48(4), 1365–1370 (1980) · Zbl 1334.35250
[6] Li, Y.S.: Soliton and integrable systems. In: Advanced Series in Nonlinear Science. Shanghai Scientific and Technological Education Publishing House, Shanghai (1999). (In Chinese)
[7] Gu, C.H.: Soliton Theory and Its Applications. Springer/Zhejiang Science and Technology Publishing House, Hangzhou (1995) · Zbl 0834.35003
[8] Liu, H., Li, J., Chen, F.: Exact periodic wave solutions for the hKdV equation. Nonlinear Anal. doi: 10.1016/j.na.2008.03.019 (2008) · Zbl 1162.35312
[9] Li, J.: Exact traveling wave solutions and dynamical behavior for the (n+1)-dimensional multiple sine-Gorden equation. Sci. China Ser. A: Math. 50(2), 153–164 (2007) · Zbl 1214.35005
[10] Li, J.: Bounded travelling wave solutions for the (n+1)-dimensional sine-Gordon equations. Chaos Solitons Fractals 25, 1037–1047 (2005) · Zbl 1070.35068
[11] Wazwaz, A.: The tanh method and a variable separated ODE method for solving double sine-Gordon equation. Phys. Lett. A 350, 41–56 (2006) · Zbl 1195.65210
[12] Liu, H., Li, J., Zhang, Q.: Lie symmetry analysis and exact explicit solutions for general Burgers’ equation. J. Comp. Appl. Math. doi: 10.1016/j.cam.2008.06.009 (2008) · Zbl 1143.39005
[13] Li, J., Liu, Z.: Smooth and non-smooth travelling waves in a nonlinearly dispersive equation. Appl. Math. Model. 25, 41–56 (2000) · Zbl 0985.37072
[14] Li, J., Liu, Z.: Travelling waves for a class of nonlinear dispersive equations. Chin. Ann. Math. 23B, 397–418 (2002) · Zbl 1011.35014
[15] Byrd, P.F., Fridman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists. Springer, Berlin (1970)
[16] Liu, H., Qiu, F.: Analytic solutions of an iterative equation with first order derivative. Ann. Diff. Eqs. 21(3), 337–342 (2005) · Zbl 1090.34600
[17] Liu, H., Li, W.: Discussion on the analytic solutions of the second-order iterative differential equation. Bull. Korean Math. Soc. 43(4), 791–804 (2006) · Zbl 1131.34048
[18] Liu, H., Li, W.: The exact analytic solutions of a nonlinear differential iterative equation. Nonlinear Anal. 69, 2466–2478 (2008) · Zbl 1155.34339
[19] Asmar, N.H.: Partial Differential Equations with Fourier Series and Boundary Value Problems. China Machine Press, Beijing (2005) · Zbl 1348.35001
[20] Rudin, W.: Principles of Mathematical Analysis. China Machine Press, Beijing (2004) · Zbl 0052.05301
[21] Fichtenholz, G.M.: Functional Series. Gordon & Breach, New York/London/Paris (1970) · Zbl 0213.35001
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