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].


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