Research Article | Open Access
Weimin Zhang, "Generalized Variational Principle for Long Water-Wave Equation by He's Semi-Inverse Method", Mathematical Problems in Engineering, vol. 2009, Article ID 925187, 5 pages, 2009. https://doi.org/10.1155/2009/925187
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, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141–1199, 2006.
- 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.
- J.-H. He, “Variational principle for two-dimensional incompressible inviscid flow,” Physics Letters A, vol. 371, no. 1-2, pp. 39–40, 2007.
- J.-H. He, “Variational approach to (2 + 1)-dimensional dispersive long water equations,” Physics Letters A, vol. 335, no. 2-3, pp. 182–184, 2005.
- J.-H. He, “A generalized variational principle in micromorphic thermoelasticity,” Mechanics Research Communications, vol. 32, no. 1, pp. 93–98, 2005.
- J.-H. He, “Variational principle for non-Newtonian lubrication: Rabinowitsch fluid model,” Applied Mathematics and Computation, vol. 157, no. 1, pp. 281–286, 2004.
- J.-H. He and H.-M. Liu, “Variational approach to diffusion reaction in spherical porous catalyst,” Chemical Engineering and Technology, vol. 27, no. 4, pp. 376–377, 2004.
- J.-H. He, H.-M. Liu, and N. Pan, “Variational model for ionomeric polymer-metal composite,” Polymer, vol. 44, no. 26, pp. 8195–8199, 2003.
- 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.
- J.-H. He, “A family of variational principles for linear micromorphic elasticity,” Computers & Structures, vol. 81, no. 21, pp. 2079–2085, 2003.
- H.-M. Liu, “Generalized variational principles for ion acoustic plasma waves by He's semi-inverse method,” Chaos, Solitons & Fractals, vol. 23, no. 2, pp. 573–576, 2005.
- Y. Wu, “Variational approach to higher-order water-wave equations,” Chaos, Solitons & Fractals, vol. 32, no. 1, pp. 195–198, 2007.
- M. Wang, Y. Zhou, and Z. Li, “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,” Physics Letters A, vol. 216, no. 1–5, pp. 67–75, 1996.
- L. Xu, “Variational principles for coupled nonlinear Schrödinger equations,” Physics Letters A, vol. 359, no. 6, pp. 627–629, 2006.
- L. Xu, “Variational approach to solitons of nonlinear dispersive equations,” Chaos, Solitons & Fractals, vol. 37, no. 1, pp. 137–143, 2008.
- Z.-L. Tao, “Variational principles for some nonlinear wave equations,” Zeitschrift für Naturforschung A, vol. 63, no. 5-6, pp. 237–240, 2008.
- L. Yao, “Variational theory for the shallow water problem,” Journal of Physics: Conference Series, vol. 96, no. 1, Article ID 012033, 3 pages, 2008.
Copyright © 2009 Weimin Zhang. 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.