## Abstract and Applied Analysis

### Abundant Explicit and Exact Solutions for the Variable Coefficient mKdV Equations

#### Abstract

This paper is concerned with the variable coefficients mKdV (VC-mKdV) equation. First, through some transformation we convert VC-mKdV equation into the constant coefficient mKdV equation. Then, using the first integral method we obtain the exact solutions of VC-mKdV equation, such as rational function solutions, periodic wave solutions of triangle function, bell-shape solitary wave solution, kink-shape solitary wave solution, Jacobi elliptic function solutions, and Weierstrass elliptic function solution. Furthermore, with the aid of Mathematica, the extended hyperbolic functions method is used to establish abundant exact explicit solution of VC-mKdV equation. By the results of the equation, the first integral method and the extended hyperbolic function method are extended from the constant coefficient nonlinear evolution equations to the variable coefficients nonlinear partial differential equation.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 109690, 7 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393449829

Digital Object Identifier
doi:10.1155/2013/109690

Mathematical Reviews number (MathSciNet)
MR3147792

Zentralblatt MATH identifier
1300.35128

#### Citation

Zheng, Xiaoxiao; Shang, Yadong; Huang, Yong. Abundant Explicit and Exact Solutions for the Variable Coefficient mKdV Equations. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 109690, 7 pages. doi:10.1155/2013/109690. https://projecteuclid.org/euclid.aaa/1393449829

#### References

• R. Hirota, “Exact solution of the modified Korteweg-de Vries equation for multiple collisions of solitons,” Journal of the Physical Society of Japan, vol. 33, no. 5, pp. 364–374, 1972.
• K. R. Helfrich, W. K. Melville, and J. W. Miles, “On interfacial solitary waves over slowly varying topography,” Journal of Fluid Mechanics, vol. 149, pp. 305–317, 1984.
• H. Ono, “Soliton fission in anharmonic lattices with reflectionless inhomogeneity,” Journal of the Physical Society of Japan, vol. 61, no. 12, pp. 4336–4343, 1992.
• A. H. Khater, O. H. El-Kalaawy, and D. K. Callebaut, “Bäcklund transformations and exact solutions for Alfvén solitons in a relativistic electron-positron plasma,” Physica Scripta, vol. 58, no. 6, pp. 545–548, 1998.
• R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Waves, Academic Press, London, UK, 1982.
• V. Ziegler, J. Dinkel, C. Setzer, and K. E. Lonngren, “On the propagation of nonlinear solitary waves in a distributed Schottky barrier diode transmission line,” Chaos, Solitons and Fractals, vol. 12, no. 9, pp. 1719–1728, 2001.
• S. K. Liu, Z. T. Fu, S. D. Liu, and Q. Zhao, “Solution by using the Jacobi elliptic function expansion method for variable-coefficient nonlinear equations,” Acta Physica Sinica, vol. 51, no. 9, pp. 1923–1926, 2002.
• Z. Yan, “Symmetry reductions and soliton-like solutions for the variable coefficient MKdV equations,” Communications in Nonlinear Science & Numerical Simulation, vol. 4, no. 4, pp. 284–288, 1999.
• D.-S. Li, Z.-S. Yu, and H.-Q. Zhang, “New soliton-like solutions to variable coefficients MKdV equation,” Communications in Theoretical Physics, vol. 42, no. 5, pp. 649–654, 2004.
• J. L. Zhang, M. L. Wang, and Y. M. Wang, “An extended $F$-expansion method and exact solutions to the KdV and mKdV equations with variable coefficients,” Acta Mathematica Scientia A, vol. 26, no. 3, pp. 353–360, 2006.
• C. Dai, J. Zhu, and J. Zhang, “New exact solutions to the mKdV equation with variable coefficients,” Chaos, Solitons & Fractals, vol. 27, no. 4, pp. 881–886, 2006.
• H. Triki and A.-M. Wazwaz, “Sub-ODE method and soliton solutions for the variable-coefficient mKdV equation,” Applied Mathematics and Computation, vol. 214, no. 2, pp. 370–373, 2009.
• B. Hong, “New Jacobi elliptic functions solutions for the variable-coefficient MKdV equation,” Applied Mathematics and Computation, vol. 215, no. 8, pp. 2908–2913, 2009.
• Y. Zhang, S. Lai, J. Yin, and Y. Wu, “The application of the auxiliary equation technique to a generalized mKdV equation with variable coefficients,” Journal of Computational and Applied Mathematics, vol. 223, no. 1, pp. 75–85, 2009.
• A. H. Salas, “Exact solutions to mKdV equation with variable coefficients,” Applied Mathematics and Computation, vol. 216, no. 10, pp. 2792–2798, 2010.
• S. Guo and Y. Zhou, “Auxiliary equation method for the mKdV equation with variable coefficients,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1476–1483, 2010.
• O. Vaneeva, “Lie symmetries and exact solutions of variable coefficient mKdV equations: an equivalence based approach,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 2, pp. 611–618, 2012.
• X. Hu and Y. Chen, “A direct procedure on the integrability of nonisospectral and variable-coefficient MKdV equation,” Journal of Nonlinear Mathematical Physics, vol. 19, no. 1, pp. 1776–1785, 2012.
• Z. Feng, “The first-integral method to study the Burgers-Korteweg-de Vries equation,” Journal of Physics A, vol. 35, no. 2, pp. 343–349, 2002.
• Z. Feng, “On explicit exact solutions to the compound Burgers-KdV equation,” Physics Letters A, vol. 293, no. 1-2, pp. 57–66, 2002.
• Z. Feng, “Exact solution to an approximate sine-Gordon equation in $(n+1)$-dimensional space,” Physics Letters A, vol. 302, no. 2-3, pp. 64–76, 2002.
• Z. Feng, “The first-integral method to the two-dimensional Burgers-Korteweg-de Vries equation,” Physics Letters A, vol. 302, pp. 57–66, 2002.
• Z. Feng and R. Knobel, “Traveling waves to a Burgers-Korteweg-de Vries-type equation with higher-order nonlinearities,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1435–1450, 2007.
• Z. Feng and G. Chen, “Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth,” Discrete and Continuous Dynamical Systems A, vol. 24, no. 3, pp. 763–780, 2009.
• K. Hosseini, R. Ansari, and P. Gholamin, “Exact solutions of some nonlinear systems of partial differential equations by using the first integral method,” Journal of Mathematical Analysis and Applications, vol. 387, no. 2, pp. 807–814, 2012.
• Y. Huang, Y. Shang, and W. Yuan, “Symbolic computation and the extended hyperbolic function method for constructing exact traveling solutions of nonlinear PDEs,” Journal of Applied Mathematics, vol. 2012, Article ID 716719, 19 pages, 2012.
• J. He and Y. Li, “Designable integrability of the variable coefficient nonlinear Schrödinger equations,” Studies in Applied Mathematics, vol. 126, no. 1, pp. 1–15, 2011. \endinput