Generalized Variational Principle for Long Water-Wave Equation by He's Semi-Inverse Method

Abstract

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.

1. Introduction

In this paper we apply He’s semi-inverse method [1–12] to establish a family of variational formulations for the following higher-order long water-wave equations:

𝑢_𝑡−𝑢_𝑥𝑢−𝑣_𝑥+𝛼𝑢_𝑥𝑥𝑣=0,_𝑡−(𝑢𝑣)_𝑥−𝛼𝑣_𝑥𝑥=0. (1.1)(1.2) When 𝛼=1/2, equations (1.1) and (1.2) were investigated in [13], but the generalized variational approach for the discussed problem has not been dealt with.

2. Variational Formulation

We rewrite (1.1) and (1.2) in conservation forms:

𝑢_𝑡1+(−2𝑢²−𝑣+𝛼𝑢_𝑥)_𝑥𝑣=0,_𝑡+(−𝑢𝑣−𝛼𝑣_𝑥)_𝑥=0. (2.1)(2.2) According to (1.1) or (2.1) we can introduce a special function Ψ defined as

Ψ_𝑡1=−2𝑢²−𝑣+𝛼𝑢_𝑥Ψ_𝑥,=−𝑢. (2.3)(2.4) Similarly from (1.2) or (2.2) we can introduce another special function Φ defined as

Φ_𝑡=−𝑢𝑣−𝛼𝑣_𝑥,Φ_𝑥=−𝑣. (2.5) 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:

𝐽(𝑢,𝑣,Ψ)=𝐿𝑑𝑥𝑑𝑡, (2.6) where 𝐿 is a trial Lagrangian defined as

𝐿=𝑣Ψ_𝑡+(−𝑢𝑣−𝛼𝑣_𝑥)Ψ_𝑥+𝐹(𝑢,𝑣), (2.7) where 𝐹 is an unknown function of 𝑢, 𝑣 and/or their derivatives. The advantage of the above trial Lagrangian is that the stationary condition with respect to Ψ is one of the governing equations (2.2) or (1.2).

Calculating the above functional equation (2.6) stationary with respect to 𝑢 and 𝑣, we obtain the followimg Euler-Lagrange equations:

−𝑣Ψ_𝑥+𝛿𝐹Ψ𝛿𝑢=0,_𝑡−𝑢Ψ_𝑥+𝛼Ψ_𝑥𝑥+𝛿𝐹𝛿𝑣=0, (2.8)(2.9) where 𝛿𝐹/𝛿𝑢 is called He’s variational derivative [14–17] with respect to 𝑢, which was first sugested by He in [2], defined as

𝛿𝐹=𝛿𝑢𝜕𝐹−𝜕𝜕𝑢(𝜕𝑡𝜕𝐹𝜕𝑢_𝑡𝜕)−(𝜕𝑥𝜕𝐹𝜕𝑢_𝑥)+⋯. (2.10) We search for such an 𝐹 so that (2.8) is equivalent to (2.3), and (2.9) is equivalent to (2.4). So in view of (2.3) and (2.4), we set

𝛿𝐹𝛿𝑢=𝑣Ψ_𝑥=−𝑢𝑣,𝛿𝐹𝛿𝑣=−Ψ_𝑡+𝑢Ψ_𝑥−𝛼Ψ_𝑥𝑥1=−2𝑢²+𝑣, (2.11) from (2.11), the unknown 𝐹 can be determined as

1𝐹(𝑢,𝑣)=−2𝑢²1𝑣+2𝑣². (2.12) Finally we obtain the following needed variational formulation:

𝐽(𝑢,𝑣,Ψ)={𝑣Ψ_𝑡+(−𝑢𝑣−𝛼𝑣_𝑥)Ψ_𝑥−12𝑢²1𝑣+2𝑣²}. (2.13)

Proof. Making the above functional equation (2.13) stationary with respect to Ψ, 𝑢, and 𝑣, we obtain the following Euler-Lagrange equations: −𝑣_𝑡−(−𝑢𝑣−𝛼𝑣_𝑥)_𝑥=0,−𝑣Ψ_𝑥Ψ−𝑢𝑣=0,_𝑡−𝑢Ψ_𝑥+𝛼Ψ_𝑥𝑥−12𝑢²+𝑣=0. (2.14)(2.15)(2.16) 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:

𝐿₁(𝑢,𝑣,Φ)=𝑢Φ_𝑡1+(−2𝑢²−𝑣+𝛼𝑢_𝑥)Φ_𝑥+𝐺(𝑢,𝑣). (2.17) It is obvious that the stationary condition with respect to Φ is equivalent to (2.1) or (1.1). Now the Euler-Lagrange equations with respect to 𝑢 and 𝑣 are

Φ_𝑡−𝑢Φ_𝑥−𝛼Φ_𝑥𝑥+𝛿𝐺𝛿𝑢=0,−Φ_𝑥+𝛿𝐺𝛿𝑣=0. (2.18) In view of (2.5), we have

𝛿𝐺𝛿𝑢=−Φ_𝑡+𝑢Φ_𝑥+𝛼Φ_𝑥𝑥=0,𝛿𝐺𝛿𝑣=Φ_𝑥=−𝑣. (2.19) From (2.19), the unknown function 𝐺(𝑢,𝑣) can be determined as

1𝐺(𝑢,𝑣)=−2𝑣². (2.20) Therefore, we obtain another needed variational formulation:

𝐽₁(𝑢,𝑣,Φ)={𝑢Φ_𝑡1+(−2𝑢²−𝑣+𝛼𝑢_𝑥)Φ_𝑥−12𝑣²}𝑑𝑥𝑑𝑡. (2.21)

3. Conclusion

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.

Acknowledgments

The author is deeply grateful to the referee for the valuable remarks on improving the paper.