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:

ut-uxu-vx+αuxx=0,vt-(uv)x-αvxx=0.
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:

ut+(-12u2-v+αux)x=0,vt+(-uv-αvx)x=0.
According to (1.1) or (2.1) we can introduce a special function Ψ defined as

Ψt=-12u2-v+αux,Ψx=-u.
Similarly from (1.2) or (2.2) we can introduce another special function Φ defined as

Φt=-uv-αvx,Φx=-v.
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,
where L is a trial Lagrangian defined as

L=vΨt+(-uv-αvx)Ψx+F(u,v),
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:

-vΨx+δFδu=0,Ψt-uΨx+αΨxx+δFδv=0,
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

δFδu=∂F∂u-∂∂t(∂F∂ut)-∂∂x(∂F∂ux)+⋯.
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=vΨx=-uv,δFδv=-Ψt+uΨx-αΨxx=-12u2+v,
from (2.11), the unknown F can be determined as

F(u,v)=-12u2v+12v2.
Finally we obtain the following needed variational formulation:

J(u,v,Ψ)=∬{vΨt+(-uv-αvx)Ψx-12u2v+12v2}.

Proof.

Making the above functional equation (2.13) stationary with respect to Ψ, u, and v, we obtain the following Euler-Lagrange equations:
-vt-(-uv-αvx)x=0,-vΨx-uv=0,Ψt-uΨx+αΨxx-12u2+v=0.
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)=uΦt+(-12u2-v+αux)Φx+G(u,v).
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

Φt-uΦx-αΦxx+δGδu=0,-Φx+δGδv=0.
In view of (2.5), we have

δGδu=-Φt+uΦx+αΦxx=0,δGδv=Φx=-v.
From (2.19), the unknown function G(u,v) can be determined as

G(u,v)=-12v2.
Therefore, we obtain another needed variational formulation:

J1(u,v,Φ)=∬{uΦt+(-12u2-v+αux)Φx-12v2}dxdt.

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.

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