/ / Article

## He's Max-Min Approach to a Nonlinear Oscillator with Discontinuous Terms

• Hui-Li Zhang | Fang Xie |
•  Article ID 579731 |
•  Published 30 Jan 2013
• | View Article

## A Note on the Semi-Inverse Method and a Variational Principle for the Generalized KdV-mKdV Equation

• Li Yao | Yun-Jie Yang | Xing-Wei Zhou |
•  Article ID 931643 |
•  Published 27 Mar 2013

Letter to the Editor | Open Access

Volume 2013 |Article ID 931643 | 3 pages | https://doi.org/10.1155/2013/931643

# A Note on the Semi-Inverse Method and a Variational Principle for the Generalized KdV-mKdV Equation

Revised05 Jan 2013
Accepted27 Jan 2013
Published27 Mar 2013

#### Abstract

Ji-Huan He systematically studied the inverse problem of calculus of variations. This note reveals that the semi-inverse method also works for a generalized KdV-mKdV equation with nonlinear terms of any orders.

#### 1. Introduction

In , the semi-inverse method is systematically studied and many examples are given to show how to establish a variational formulation for a nonlinear equation. From the given examples, we found that it is difficult to find a variational principle for nonlinear evolution equations with nonlinear terms of any orders.

For example, consider the following generalized KdV-mKdV equation: where , , , and are constant coefficients, while is a positive number. Equation (1) is an important model in plasma physics and solid state physics.

#### 2. Variational Principle by He’s Semi-Inverse Method

For (1), we introduce a potential function defined as ; we have the following equation: In order to use the semi-inverse method  to establish a Lagrangian for (2), we first check some simple cases: We can easily obtain a variational principle for (2) for , which is Now, according to the semi-inverse method , we construct a trial functional for (2): where is an unknown function of and/or its derivatives.

Making the trial-functional, (5), stationary with respect to results in the following Euler-Lagrange equation: where is called variational differential with respect to , defined as We rewrite (6) in the form Comparison of (8) and (2) leads to the following results: from which we identify the unknown and as follows: We, therefore, obtain the following needed variational principle:

#### 3. Conclusion

This note shows that the semi-inverse method in  works also for the present problem, and it is concluded that the semi-inverse method is a powerful mathematical tool to the construction of a variational formulation for a nonlinear equation; illustrating examples are available in .

The semi-inverse method can be extended to fractional calculus .

#### Acknowledgments

This work was supported by the Chinese Natural Science Foundation Grant no. 10961029 and Kunming University Research Fund (2010SX01).

1. J.-H. He, “Asymptotic methods for solitary solutions and compactons,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012. View at: Publisher Site | Google Scholar
2. J.-H. He, “An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering,” International Journal of Modern Physics B, vol. 22, pp. 3487–3578, 2008. View at: Google Scholar
3. J.-H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141–1199, 2006.
4. J.-H. He, “Variational principles for some nonlinear partial differential equations with variable coefficients,” Chaos, Solitons & Fractals, vol. 19, no. 4, pp. 847–851, 2004.
5. L. Yao, “How to discover a variational principle for a damped vibration problem,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, pp. 171–173, 2010. View at: Google Scholar
6. G.-C. Wu, “Laplace transform overcoming principle drawbacks in application of the variational iteration method to fractional heat equations,” Thermal Science, vol. 16, no. 4, pp. 1257–1261, 2012. View at: Google Scholar
7. X.-W. Zhou and L. Wang, “A variational principle for coupled nonlinear Schrödinger equations with variable coefficients and high nonlinearity,” Computers & Mathematics with Applications, vol. 61, no. 8, pp. 2035–2038, 2011.
8. Z.-L. Tao, “Variational approach to the Benjamin Ono equation,” Nonlinear Analysis. Real World Applications, vol. 10, no. 3, pp. 1939–1941, 2009.
9. Z.-L. Tao, “Solitary solutions of the Boiti-Leon-Manna-Pempinelli equation using He's variational method,” Zeitschrift fur Natuforschung A, vol. 63, no. 10-11, pp. 634–636, 2008. View at: Google Scholar
10. L. Xu, “Variational approach to solitons of nonlinear dispersive K(m, n) equations,” Chaos, Solitons and Fractals, vol. 37, no. 1, pp. 137–143, 2008.
11. D. Baleanu and S. I. Muslih, “Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives,” Physica Scripta, vol. 72, no. 2-3, pp. 119–121, 2005.
12. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, vol. 3, World Scientific, 2012. View at: Publisher Site | MathSciNet
13. G.-C. Wu and D. Baleanu, “Variational iteration method for the Burgers’ flow with fractional derivatives—new Lagrange Mutipliers,” Applied Mathematical Modelling, vol. 37, no. 9, pp. 6183–6190, 2013. View at: Publisher Site | Google Scholar
14. G.-C. Wu, “New trends in variational iteration method,” Communications in Fractional Calculus, vol. 2, no. 2, pp. 59–75, 2011. View at: Google Scholar

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