Generalized Variational Principle for Long Water-Wave Equation by He's Semi-Inverse Method
Variational principles for nonlinear partial differential equations have come to play an important role in mathematics and physics. However, it is well known that not every nonlinear partial differential equation admits a variational formula. In this paper, He's semi-inverse method is used to construct a family of variational principles for the long water-wave problem.
2. Variational Formulation
Our aim in this paper is to establish some variational formulations whose stationary conditions satisfy (1.1), (2.5), or (1.2), (2.3), and (2.4). To this end, we will apply He’s semi-inverse method to construct a trial functional:
where is a trial Lagrangian defined as
where is an unknown function of , and/or their derivatives. The advantage of the above trial Lagrangian is that the stationary condition with respect to is one of the governing equations (2.2) or (1.2).
Calculating the above functional equation (2.6) stationary with respect to and , we obtain the followimg Euler-Lagrange equations:
from (2.11), the unknown can be determined as
Finally we obtain the following needed variational formulation:
Proof. Making the above functional equation (2.13) stationary with respect to , , and , we obtain the following Euler-Lagrange equations: Equation (2.14) is equivalent to (1.2), and (2.15) is equivalent to (2.4); in view of (2.4), (2.16) becomes (2.3).
Similary we can also begin with the following trial Lagrangian:
In view of (2.5), we have
From (2.19), the unknown function can be determined as
Therefore, we obtain another needed variational formulation:
We establish a family of variational formulations for the long water-wave problem using He’s semi-inverse method. It is shown that the method is a powerful tool to the search for variational principles for nonlinear physical problems directly from field equations without using Lagrange multiplier. The result obtained in this paper might find some potential applications in future.
The author is deeply grateful to the referee for the valuable remarks on improving the paper.
J.-H. He, “Semi-inverse method of establing generalized variational principles for fluid mechanics with emphasis on turbomachinery aerodynamics,” International Journal of Turbo and Jet Engines, vol. 14, no. 1, pp. 23–28, 1997.View at: Google Scholar
L. Xu, “Variational approach to solitons of nonlinear dispersive equations,” Chaos, Solitons & Fractals, vol. 37, no. 1, pp. 137–143, 2008.View at: Google Scholar
Z.-L. Tao, “Variational principles for some nonlinear wave equations,” Zeitschrift für Naturforschung A, vol. 63, no. 5-6, pp. 237–240, 2008.View at: Google Scholar