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.

1. Introduction

In [1], 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:
(1)ut+(α+βup+γu2p)ux+uxxx+ηuxxxxx+g(t)u=0,
where α, β, γ, and η are constant coefficients, while p 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 v defined as u=vx; we have the following equation:
(2)vxt+(α+βvxp+γvx2p)vxx+vxxxx+ηvxxxxxx+g(t)vx=0.
In order to use the semi-inverse method [1–4] to establish a Lagrangian for (2), we first check some simple cases:
(3)L=-vxvt2forvxt=0,L=(vxx)22forvxxxx=0,L=-vx36for(vx2)x2=vxvxx=0,L=-vxnn(n-1)forvxn-2vxx=0.
We can easily obtain a variational principle for (2) for g(t)≡0, which is
(4)J(v)=∬{-12vxvt-12αvx2-β(p+2)(p+1)vxp+2-γ(2p+2)(2p+1)vx2p+2+vxx2-η2vxxx2}dxdt,
Now, according to the semi-inverse method [1–4], we construct a trial functional for (2):
(5)J(v)=∬{f(t)[-12vxvt-12αvx2-β(p+2)(p+1)vxp+2-γ(2p+2)(2p+1)vx2p+2+vxx2-η2vxxx2β(p+2)(p+1)vxp+2]+Fβ(p+2)(p+1)vxp+2}dxdt,
where F is an unknown function of u and/or its derivatives.

Making the trial-functional, (5), stationary with respect to v results in the following Euler-Lagrange equation:
(6)12(fvx)t+12(fvt)x+α(fvx)x+β(p+1)(fvxp+1)x+γ(2p+1)(fvx2p+1)x+(fvxx)xx-η(fvxxx)xxx+δFδv=0,
where δF/δv is called variational differential with respect to v, defined as
(7)δFδv=∂F∂v-∂∂t(∂F∂vt)-∂∂x(∂F∂vx)+∂2∂t2(∂F∂vtt)+∂2∂x2(∂F∂vxx)+⋯.
We rewrite (6) in the form
(8)ft2fvx+vxt+αvxx+βvxpvxx+γvx2pvxx+vxxxx+ηvxxxxxx+δFfδv=0.
Comparison of (8) and (2) leads to the following results:
(9)ft2f=g(t),δFfδv=0,
from which we identify the unknown f and F as follows:
(10)f(t)=e2∫g(t)dt,F=0.
We, therefore, obtain the following needed variational principle:
(11)J(v)=∬{e2∫g(t)dt[-12vxvt-αvx2-β(p+2)(p+1)vxp+2-γ(2p+2)(2p+1)vx2p+2+vxx2-η2vxxx2β(p+2)(p+1)vxp+2]β(p+2)(p+1)vxp+2}dxdt.

3. Conclusion

This note shows that the semi-inverse method in [1] 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].

The semi-inverse method can be extended to fractional calculus [11–14].

Acknowledgments

This work was supported by the Chinese Natural Science Foundation Grant no. 10961029 and Kunming University Research Fund (2010SX01).

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