Volume 2013, Issue 1 931643

Letter to the Editor

Open Access

A Note on the Semi-Inverse Method and a Variational Principle for the Generalized KdV-mKdV Equation

Li Yao,

Li Yao

Department of Mathematics, Kunming University, 2 Puxin Road, Kunming, Yunnan 650214, China kmu.edu.cn

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Yun-Jie Yang,

Yun-Jie Yang

Department of Mathematics, Kunming University, 2 Puxin Road, Kunming, Yunnan 650214, China kmu.edu.cn

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Xing-Wei Zhou,

Corresponding Author

Xing-Wei Zhou

[email protected]

Department of Mathematics, Kunming University, 2 Puxin Road, Kunming, Yunnan 650214, China kmu.edu.cn

Institute of Nonlinear Analysis, Kunming University, 2 Puxin Road, Kunming, Yunnan 650214, China kmu.edu.cn

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Li Yao,

Li Yao

Department of Mathematics, Kunming University, 2 Puxin Road, Kunming, Yunnan 650214, China kmu.edu.cn

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Yun-Jie Yang,

Yun-Jie Yang

Department of Mathematics, Kunming University, 2 Puxin Road, Kunming, Yunnan 650214, China kmu.edu.cn

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Xing-Wei Zhou,

Corresponding Author

Xing-Wei Zhou

[email protected]

Department of Mathematics, Kunming University, 2 Puxin Road, Kunming, Yunnan 650214, China kmu.edu.cn

Institute of Nonlinear Analysis, Kunming University, 2 Puxin Road, Kunming, Yunnan 650214, China kmu.edu.cn

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First published: 27 March 2013

https://doi.org/10.1155/2013/931643

Citations: 2

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Abstract

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:

()

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 = v_x; we have the following equation:

()

In order to use the semi-inverse method [1–4] to establish a Lagrangian for (2), we first check some simple cases:

()

We can easily obtain a variational principle for (2) for g(t) ≡ 0, which is

()

Now, according to the semi-inverse method [1–4], we construct a trial functional for (2):

()

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:

()

where δF/δv is called variational differential with respect to v, defined as

()

We rewrite (6) in the form

()

Comparison of (8) and (2) leads to the following results:

()

from which we identify the unknown f and F as follows:

()

We, therefore, obtain the following needed variational principle:

()

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).

References

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A Note on the Semi-Inverse Method and a Variational Principle for the Generalized KdV-mKdV Equation

Abstract

1. Introduction

2. Variational Principle by He’s Semi-Inverse Method

3. Conclusion

Acknowledgments

References

References

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A Note on the Semi-Inverse Method and a Variational Principle for the Generalized KdV-mKdV Equation

This article relates to:

Asymptotic Methods for Solitary Solutions and Compactons

Abstract

1. Introduction

2. Variational Principle by He’s Semi-Inverse Method

3. Conclusion

Acknowledgments

References

References

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