A Note on the Semi-Inverse Method and a Variational Principle for the Generalized KdV-mKdV Equation
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.
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 [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:
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 [5–10].
This work was supported by the Chinese Natural Science Foundation Grant no. 10961029 and Kunming University Research Fund (2010SX01).
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
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
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
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
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