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

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: 𝑢_𝑡+𝛼+𝛽𝑢^𝑝+𝛾𝑢^2𝑝𝑢_𝑥+𝑢_𝑥𝑥𝑥+𝜂𝑢_{𝑥𝑥𝑥𝑥𝑥}+𝑔(𝑡)𝑢=0, (1) 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: 𝑣_𝑥𝑡+𝛼+𝛽𝑣^𝑝_𝑥+𝛾𝑣_𝑥^2𝑝𝑣_𝑥𝑥+𝑣_{𝑥𝑥𝑥𝑥}+𝜂𝑣_{𝑥𝑥𝑥𝑥𝑥𝑥}+𝑔(𝑡)𝑣_𝑥=0. (2) In order to use the semi-inverse method [1–4] to establish a Lagrangian for (2), we first check some simple cases: 𝑣𝐿=−^𝑥𝑣_𝑡2for𝑣_𝑥𝑡𝑣=0,𝐿=^𝑥𝑥22for𝑣_{𝑥𝑥𝑥𝑥}𝑣=0,𝐿=−^3𝑥6𝑣for^2𝑥𝑥2=𝑣_𝑥𝑣_𝑥𝑥𝑣=0,𝐿=−^𝑛𝑥𝑛(𝑛−1)for𝑣_𝑥^𝑛−2𝑣_𝑥𝑥=0. (3) We can easily obtain a variational principle for (2) for 𝑔(𝑡)≡0, which is 𝐽(𝑣)=−12𝑣_𝑥𝑣_𝑡−12𝛼𝑣²_𝑥−𝛽𝑣𝑝+2𝑝+1^𝑥𝑝+2−𝛾𝑣2𝑝+22𝑝+1^𝑥2𝑝+2+𝑣²_𝑥𝑥−𝜂2𝑣²_𝑥𝑥𝑥𝑑𝑥𝑑𝑡, (4) Now, according to the semi-inverse method [1–4], we construct a trial functional for (2): 𝐽(𝑣)=𝑓(𝑡−1)2𝑣_𝑥𝑣_𝑡−12𝛼𝑣²_𝑥−𝛽𝑣𝑝+2𝑝+1^𝑥𝑝+2−𝛾𝑣2𝑝+22𝑝+1^𝑥2𝑝+2+𝑣²_𝑥𝑥−𝜂2𝑣²_𝑥𝑥𝑥+𝐹𝑑𝑥𝑑𝑡, (5) 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: 12𝑓𝑣_𝑥𝑡+12𝑓𝑣_𝑡𝑥+𝛼𝑓𝑣_𝑥𝑥+𝛽𝑝+1𝑓𝑣_𝑥^𝑝+1_𝑥+𝛾2𝑝+1𝑓𝑣_𝑥^2𝑝+1_𝑥+𝑓𝑣_{𝑥𝑥𝑥𝑥}−𝜂𝑓𝑣_{𝑥𝑥𝑥𝑥𝑥𝑥}+𝛿𝐹𝛿𝑣=0, (6) where 𝛿𝐹/𝛿𝑣 is called variational differential with respect to 𝑣, defined as 𝛿𝐹=𝛿𝑣𝜕𝐹−𝜕𝜕𝑣𝜕𝑡𝜕𝐹𝜕𝑣_𝑡−𝜕𝜕𝑥𝜕𝐹𝜕𝑣_𝑥+𝜕²𝜕𝑡²𝜕𝐹𝜕𝑣_𝑡𝑡+𝜕²𝜕𝑥²𝜕𝐹𝜕𝑣_𝑥𝑥+⋯. (7) We rewrite (6) in the form 𝑓_𝑡𝑣2𝑓^𝑥+𝑣_𝑥𝑡+𝛼𝑣_𝑥𝑥+𝛽𝑣^𝑝_𝑥𝑣_𝑥𝑥+𝛾𝑣_𝑥^2𝑝𝑣_𝑥𝑥+𝑣_{𝑥𝑥𝑥𝑥}+𝜂𝑣_{𝑥𝑥𝑥𝑥𝑥𝑥}+𝛿𝐹𝑓𝛿𝑣=0. (8) Comparison of (8) and (2) leads to the following results: 𝑓_𝑡(𝑡),2𝑓=𝑔𝛿𝐹𝑓𝛿𝑣=0, (9) from which we identify the unknown 𝑓 and 𝐹 as follows: 𝑓(𝑡)=𝑒^{2∫𝑔(𝑡)𝑑𝑡},𝐹=0. (10) We, therefore, obtain the following needed variational principle: 𝐽(𝑣)=𝑒^{2∫𝑔(𝑡)𝑑𝑡}−12𝑣_𝑥𝑣_𝑡−𝛼𝑣²_𝑥−𝛽𝑣𝑝+2𝑝+1^𝑥𝑝+2−𝛾𝑣2𝑝+22𝑝+1^𝑥2𝑝+2+𝑣²_𝑥𝑥−𝜂2𝑣²_𝑥𝑥𝑥𝑑𝑥𝑑𝑡. (11)

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).