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

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 [1–4] 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 [1–4], 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 [5–10].

The semi-inverse method can be extended to fractional calculus [11–14].

Acknowledgments

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

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Copyright

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.