# Benjamin–Bona–Mahony Equation with Variable Coefficients: Conservation Laws

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fundamental Notations and Relations

^{[2]}given by

**Definition 1**. The vector field X, of the form Equation (5), is called a Noether operator corresponding to a second-order Lagrangian L of Equation (10) if

^{1}(t, x, z) and B

^{2}(t, x, z).

**Theorem 1.**(

**Noether**[17]) If X, as given in Equation (5), is a Noether point symmetry generator corresponding to a second-order Lagrangian L of Equation (10), then the vector T = (T

^{1}, T

^{2}) with components

_{t}τ − z

_{x}ξ is the Lie characteristic function.

## 3. Conservation Laws of Equation (4)

_{x}. Thus, Equation (4) becomes a fourth-order equation, namely

**Case 1.**α(t) and β(t) arbitrary but not of the form contained in Cases 2–4.

_{i}T

^{i}|

_{(4)}= 0, as some excessive terms emerge that require some further analysis. By making a slight adjustment to these terms, it can be shown that these terms can be absorbed into the divergence condition. For,

**Case 2**. α(t) = γ, β(t) = λ, where γ and λ are non-zero constants.

_{1}, X

_{2}given by the operators Equations (30) and (31) and X

_{3}given by

_{3}yields

_{1}, X

_{2}and X

_{3}are

**Case 3.**α(t) = γe

^{σt}, β(t) = λe

^{σt}, where γ, σ and λ are nonzero constants.

_{1}, X

_{2}given by the operators Equations (30) and (31) and X

_{3}given by

_{3}, gives

_{1}, X

_{2}and X

_{3}are, respectively,

**Case 4.**α(t) = γ/a(t), β(t) = λ/a(t), where γ, λ are constants, with γ, λ ≠ 0 and a(t) an arbitrary function of t.

_{1}, X

_{2}given by Equations (30) and (31) and X

_{3}given by

_{1}, X

_{2}and X

_{3}, in this case, are

**Remark 1.**Remark: It should be noted that since the Lagrangian Equation (16) is invariant under the spatial translation symmetry, this will give rise to the linear momentum conservation laws. Moreover, if α(t) = γ, β(t) = λ, where γ and λ are non-zero constants, then the corresponding Lagrangian Equation (16) is also invariant under the time translation symmetry and thus the linear momentum and energy are both conserved.

## 4. Exact Solution of Equation (4) for a Special Case Using Conservation Laws

**Definition 2.**Suppose that X is a symmetry of Equation (10) and T a conserved vector of Equation (10). Then if X and T satisfy

^{t}= v

_{x}, T

^{x}= −v

_{t}. Then using the similarity variables r, s, w with the generator $X=\frac{\partial}{\partial s}$, we have T

^{r}= v

_{s}, T

^{s}= −v

_{r}, and the conservation law is then rewritten as

**Theorem 2.**An n-th order partial differential equation with two independent variables, which admits a symmetry X that is associated with a conserved vector T, is reduced to an ordinary differential equation of order n − 1; namely, T

^{r}= k, where T

^{r}is defined by Equation (72) for solutions invariant under X.

_{1}and X

_{2}. We now define the combination of X

_{1}and X

_{2}by X = ρX

_{1}+X

_{2}. Thus, the canonical coordinates of X are given by

_{r}= p(u), we have ... Then Equation (78) reduces to the first order ordinary differential equation

_{1}+ X

_{2}.

## 5. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Muatjetjeja, B.; Khalique, C.M.
Benjamin–Bona–Mahony Equation with Variable Coefficients: Conservation Laws. *Symmetry* **2014**, *6*, 1026-1036.
https://doi.org/10.3390/sym6041026

**AMA Style**

Muatjetjeja B, Khalique CM.
Benjamin–Bona–Mahony Equation with Variable Coefficients: Conservation Laws. *Symmetry*. 2014; 6(4):1026-1036.
https://doi.org/10.3390/sym6041026

**Chicago/Turabian Style**

Muatjetjeja, Ben, and Chaudry Masood Khalique.
2014. "Benjamin–Bona–Mahony Equation with Variable Coefficients: Conservation Laws" *Symmetry* 6, no. 4: 1026-1036.
https://doi.org/10.3390/sym6041026