In [28, 29] Bluman *et al*. introduced a method to find a new class of symmetries for a PDE. They are called potential symmetries and can be obtained for any differential equation which can be written as a conservation law.

It means that given scalar PDE of second order
$$F(x,t,u,{u}_{x},{u}_{t},{u}_{xx},{u}_{xt},{u}_{tt})=0,$$(10)
where the subscripts denote the partial derivatives of *u*, it can be written as a conservation law
$$\frac{D}{Dt}{F}_{1}(x,t,u,{u}_{x},{u}_{t})-\frac{D}{Dx}{F}_{2}(x,t,u,{u}_{x},{u}_{t})=0,$$(11)
for some functions *F*_{1} and *F*_{2} of the indicated arguments and where $\frac{D}{Dx}\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{n}\mathrm{d}\phantom{\rule{thinmathspace}{0ex}}\frac{D}{Dt}$ are the total derivative operators defined by
$$\begin{array}{rc}\frac{D}{Dx}& =\frac{\mathrm{\partial}}{\mathrm{\partial}x}+{u}_{x}\frac{\mathrm{\partial}}{\mathrm{\partial}u}+{u}_{xx}\frac{\mathrm{\partial}}{\mathrm{\partial}{u}_{x}}+{u}_{xt}\frac{\mathrm{\partial}}{\mathrm{\partial}{u}_{t}}+\dots ,\\ \\ \frac{D}{Dt}& =\frac{\mathrm{\partial}}{\mathrm{\partial}t}+{u}_{t}\frac{\mathrm{\partial}}{\mathrm{\partial}u}+{u}_{xt}\frac{\mathrm{\partial}}{\mathrm{\partial}{u}_{x}}+{u}_{tt}\frac{\mathrm{\partial}}{\mathrm{\partial}{u}_{t}}+\dots .\end{array}$$

Then if we introduce a potential variable *v* = *v*(*x*, *t*) for the PDE written in the conserved form (6), we obtain a potential system (system approach) that we call *S*$$\left\{\begin{array}{}{v}_{x}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{F}_{1}(x,t,u,{u}_{x},{u}_{t}),\\ {v}_{t}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{F}_{2}(x,t,u,{u}_{x},{u}_{t}).\end{array}\right.$$(12)

Furthermore, for many physical equations one can eliminate *u* from the potential system (12) and form an auxiliary integrated or potential equation (integrated equation approach)
$$G(x,t,v,{v}_{x},{v}_{t},{v}_{xx},{v}_{xt},{v}_{tt})=0,$$(13)
for some function *G* of the indicated arguments.

On the other hand, any Lie group of transformations for (12)
$$\begin{array}{ll}{X}_{S}=& \xi (x,t,u,v){\mathrm{\partial}}_{x}+\tau (x,t,u,v){\mathrm{\partial}}_{t}+\psi (x,t,u,v){\mathrm{\partial}}_{u}\\ & +\phi (x,t,u,v){\mathrm{\partial}}_{v}.\end{array}$$
induces a nonlocal symmetry, *potential symmetry*, for the given PDE (10) when at least one of the infinitesimals which correspond to the variables *x* and *u* depends explicitly on the potential *v*. So, we obtain potential symmetries if and only if the following condition is satisfied
$${\xi}_{v}^{2}+{\tau}_{v}^{2}+{\varphi}_{v}^{2}\ne 0.$$(14)
In order to find potential symmetries of (1) and from the conservation law (11), we consider the equation in conserved form and the associated potential system is given by
$$\left\{\begin{array}{l}{v}_{x}=u,\\ {v}_{t}=\mu {u}_{x}+\delta {u}_{xx}-\lambda {u}_{xxx}-f(u).\end{array}\right.$$(15)

In the present work, we present the point symmetries of (15) and we study which symmetries induce potential symmetries of equation (1). These symmetries are such that the condition (14) is satisfied. If the above relation does not hold, then the point symmetries of (15) project into point symmetries of (1).

A Lie point symmetry admitted by *S*(*x*, *t*, *u*, *v*) is a symmetry characterized by an infinitesimal transformation of the form
$$\begin{array}{l}{x}^{\ast}=x+\u03f5\xi (x,t,u,v)+\mathcal{O}({\u03f5}^{2})\\ {t}^{\ast}=t+\u03f5\tau (x,t,u,v)+\mathcal{O}({\u03f5}^{2})\\ {u}^{\ast}=u+\u03f5\eta (x,t,u,v)+\mathcal{O}({\u03f5}^{2})\\ {v}^{\ast}=v+\u03f5\phi (x,t,u,v)+\mathcal{O}({\u03f5}^{2})\end{array}$$(16)
admitted by system (15). System (15) admits Lie symmetries if and only if
$$\begin{array}{l}{\text{pr}}^{(1)}X({v}_{x}-u)=0,\\ \\ {\text{pr}}^{(3)}X({v}_{t}-\mu {u}_{x}-\delta {u}_{xx}+\lambda {u}_{xxx}+f(u))=0,\end{array}$$
where pr^{(3)}*V* is the third extended generator of
$$\begin{array}{ll}{X}_{S}=& \xi (x,t,u,v){\mathrm{\partial}}_{x}+\tau (x,t,u,v){\mathrm{\partial}}_{t}+\psi (x,t,u,v){\mathrm{\partial}}_{u}\\ & +\phi (x,t,u,v){\mathrm{\partial}}_{v}.\end{array}$$

In other words, we require that the infinitesimal generator leaves invariant the set of solutions of (15). This yields to an overdetermined system of thirteen equations for the infinitesimals *ξ*(*x*, *t*, *u*, *v*), *τ*(*x*, *t*, *u*, *v*), *ψ*(*x*, *t*, *u*, *v*) and *ϕ*(*x*, *t*, *u*, *v*).

From this system we have obtained that
$$\begin{array}{lll}\xi =\xi (x,t),& \tau =\tau (t),& \psi =\alpha (x,t,v)u+\beta (x,t,v),\\ \phi =\phi (x,t,v)& & \end{array}$$
where *ξ*, *τ*, *α*, *β* and *ϕ* must satisfy the following equations
$$\begin{array}{rr}a{\alpha}_{v}& =0,\\ {\phi}_{v}-\alpha -{\tau}_{t}+3{\xi}_{x}& =0,\\ 4a{\alpha}_{v}u+a{\beta}_{v}+3a{\alpha}_{x}-3a{\xi}_{xx}-b{\xi}_{x}& =0,\\ {\phi}_{v}u-\alpha u-{\xi}_{x}u+{\phi}_{x}-\beta & =0,\\ 6a{\alpha}_{vv}{u}^{2}+3a{\beta}_{vv}u+9a{\alpha}_{vx}u-3b{\alpha}_{v}u& \\ +3a{\beta}_{vx}-b{\beta}_{v}+3a{\alpha}_{xx}-2b{\alpha}_{x}& \\ -a{\xi}_{xxx}+b{\xi}_{xx}-2c{\xi}_{x}& =0,\\ a{\alpha}_{vvv}{u}^{4}+a{\beta}_{vvv}{u}^{3}+3a{\alpha}_{vvx}{u}^{3}& \\ -b{\alpha}_{vv}{u}^{3}+3a{\beta}_{vvx}{u}^{2}-b{\beta}_{vv}{u}^{2}+3a{\alpha}_{vxx}{u}^{2}& \\ -2b{\alpha}_{vx}{u}^{2}-c{\alpha}_{v}{u}^{2}+3a{\beta}_{vxx}u-2b{\beta}_{vx}u& \\ -c{\beta}_{v}u+a{\alpha}_{xxx}u-b{\alpha}_{xx}u-c{\alpha}_{x}u& \\ +{f}_{u}\alpha u-{\xi}_{t}u+{\phi}_{t}+a{\beta}_{xxx}-b{\beta}_{xx}& \\ -c{\beta}_{x}+{f}_{u}\beta -f\alpha +3{\xi}_{x}f& =0\end{array}$$(17)

From system (17) we have considered the following cases:

–

The parameters *a*, *b*, *c* are arbitrary constants, with *c* ≠ 0, and *f* is an arbitrary function. From system (17) we have obtained the infinitesimals:
$$\begin{array}{lllllll}\xi ={k}_{1},& \phantom{\rule{0.2cm}{0ex}}& \tau ={k}_{2},& \phantom{\rule{0.2cm}{0ex}}& \phi ={k}_{3},& \phantom{\rule{0.2cm}{0ex}}& \psi =0.\end{array}$$
However, it is not a potential symmetry of the equation (1) because the condition (14) is not satisfied.

–

If *f* = *k* (Ln (*u*) − 1) *u* + *m u* + *n*, from system (17) we have obtained the infinitesimals:
$$\begin{array}{l}\xi ={k}_{1}t+{k}_{2},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\tau ={k}_{3},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\psi =-{k}_{1}\phantom{\rule{thinmathspace}{0ex}}u,\\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phi ={k}_{1}\phantom{\rule{thinmathspace}{0ex}}v+t\phantom{\rule{thinmathspace}{0ex}}\left(\left(1-k\right)\phantom{\rule{thinmathspace}{0ex}}{k}_{1}\phantom{\rule{thinmathspace}{0ex}}u+{k}_{1}\phantom{\rule{thinmathspace}{0ex}}n\right)+r.\end{array}$$
And again, it is not a potential symmetry of the equation (1) because the condition (14) is not satisfied.

Consequently, we have concluded that the equation (1) does not admit potential symmetries.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.