About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 931643, 3 pages
http://dx.doi.org/10.1155/2013/931643
Letter to the Editor

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

1Department of Mathematics, Kunming University, 2 Puxin Road, Kunming, Yunnan 650214, China
2Institute of Nonlinear Analysis, Kunming University, 2 Puxin Road, Kunming, Yunnan 650214, China

Received 1 January 2013; Revised 5 January 2013; Accepted 27 January 2013

Copyright © 2013 Li Yao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [1], 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 [14] 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 [14], 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 [1] 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 [510].

The semi-inverse method can be extended to fractional calculus [1114].

Acknowledgments

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

References

  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 · View at 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.
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  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.
  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.
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Z.-L. Tao, “Variational approach to the Benjamin Ono equation,” Nonlinear Analysis. Real World Applications, vol. 10, no. 3, pp. 1939–1941, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  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.
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  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 · View at Google Scholar · View at 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 · View at Google Scholar
  14. G.-C. Wu, “New trends in variational iteration method,” Communications in Fractional Calculus, vol. 2, no. 2, pp. 59–75, 2011.