<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In this paper, we consider a generalized Gardner equation from the point of view of classical and nonclassical symmetries in partial differential equations. We perform a complete analysis of the symmetry reductions by using the similarity variables and the similarity solutions which allow us to reduce our equation into an ordinary differential equation. Moreover, we prove that the nonclassical method applied to the equation leads to new symmetries, which cannot be obtained by using the Lie classical method. Finally, we calculate exact travelling wave solutions of the equation by using the simplest equation method.</p></abstract>

In this paper, we consider a generalized Gardner equation from the point of view of classical and nonclassical symmetries in partial differential equations. We perform a complete analysis of the symmetry reductions by using the similarity variables and the similarity solutions which allow us to reduce our equation into an ordinary differential equation. Moreover, we prove that the nonclassical method applied to the equation leads to new symmetries, which cannot be obtained by using the Lie classical method. Finally, we calculate exact travelling wave solutions of the equation by using the simplest equation method.

On the classical and nonclassical symmetries of a generalized Gardner equation

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"On the classical and nonclassical symmetries of a generalized Gardner equation"}

In this paper, we consider a generalized Gardner equation from the point of view of classical and nonclassical symmetries in partial differential equations. We...

Subcase	Optimal system of subalgebras	Similarity variables	Similarity solutions
arbitrary	< λv₁ + μv₂ >	z= μ x — λ t	u = h(z)
1.1. a)	< λv₁ + μv₂ >	z = μ x — λ t	u = h(z)
1.1. b)	< λv₁ + μv₃ >	z = t	$u = h (z) + \frac{μ \exp (- e t) x}{λ - \frac{k μ}{e} \exp (- e t)}$ $\begin{array}{} \displaystyle u = h(z) + \frac{{\mu \exp ( - et)x}}{{\lambda - \frac{{k\mu }}{e}\exp ( - et)}} \end{array}$
2.1. a)	< λv₁ + μv₂ >	z = μ x — λ t	u = h(z)
2.1. b)	< λv₁ + μv₅ >	z = t	$u = \frac{h (z) + μ x}{λ + μ k t}$ $\begin{array}{} \displaystyle u= \displaystyle \frac{h(z)+\mu x}{\lambda+ \mu k t} \end{array}$
2.1. c)	< v₄ >	$z = (x - d t) t^{- \frac{1}{3}} + \frac{f k t^{\frac{5}{3}}}{2}$ $\begin{array}{} \displaystyle z=(x-d t) t^{-\frac{1}{3}} + \frac{f k t^{\frac{5}{3}}}{2} \end{array}$	$u = t^{- \frac{2}{3}} h (z) - f t$ $\begin{array}{} \displaystyle u= t^{-\frac{2}{3}}h(z)- f t \end{array}$
2.2. a)	< λv₁ + μv₂ >	z = μ x — λ t	u = h(z)
2.2. b)	< v₆ >	$z = (x - d t) t^{- \frac{1}{3}}$ $\begin{array}{} \displaystyle z= (x-d t) t^{-\frac{1}{3}} \end{array}$	$u = t^{- \frac{1}{3 n}} h (z)$ $\begin{array}{} \displaystyle u=t^{-\frac{1}{3n}}h(z) \end{array}$
2.3. a)	< λv₁ + μv₂ >	z = μ x — λ t	u = h(z)
2.3. b)	< v₇ >	$z = (x - d t) t^{- \frac{1}{3}}$ $\begin{array}{} \displaystyle z= (x-d t) t^{-\frac{1}{3}} \end{array}$	$u = t^{- \frac{2}{3 n}} h (z)$ $\begin{array}{} \displaystyle u=t^{-\frac{2}{3n}}h(z) \end{array}$
2.4. a)	< λv₁ + μv₂ >	z = μ x — λ t	u = h(z)
2.4. b)	< v₈ >	$z = x t^{- \frac{1}{3}} + \frac{(a^{2} - 4 b d) t^{\frac{2}{3}}}{4 b}$ $\begin{array}{} \displaystyle z=\displaystyle x t^{-\frac{1}{3}}+ \frac{(a^2 - 4 b d)t^{\frac{2}{3}}}{4 b } \end{array}$	$u = t^{- \frac{1}{3}} h (z) - \frac{a}{2 b}$ $\begin{array}{} \displaystyle u=\displaystyle t^{-\frac{1}{3}} h(z)- \frac{a}{2 b } \end{array}$

Subcase	ODEs
arbitrary	cμ³h‴ + bμh²ⁿh′ + aμhⁿh′ + (dμ - λ)h′ + eh + f = 0
1.1. a)	cμ³h‴ + kμhh′ + (dμ - λ )h′ + eh + f = 0
1.1. b)	λ e exp(ez) (h′ + f) - μ (e²x + kh′ - ehk + fk - de) = 0
2.1. a)	cμ³h‴ + kμhh′ + (dμ - λ)h′ + f = 0
2.1. b)	h′ + μd + (λ + μkz) f = 0
2.1. c)	3 c h‴ + 3k h h′-h′ z - 2 h = 0
2.2. a)	cμ³h‴ + bμh²ⁿh′ + (dμ - λ)h′ = 0
2.2. b)	3 c n h‴ + 3 k n h²ⁿh′- n h′ z - h = 0
2.3. a)	cμ³h‴ + μahⁿh′ + (dμ - λ)h′ = 0
2.3. b)	3 c n h‴ + 3 a n hⁿ h′ - nh′ z - 2 h = 0
2.4. a)	cμ ³h‴ + bμh²h′ + aμhh′ + (dμ - λ )h′ = 0
2.4. b)	3 c h‴ + 3 b h² h′ - h′ z - h = 0

On the classical and nonclassical symmetries of a generalized Gardner equation

Published Online: May 11, 2016

Page range: 263 - 272

Received: Feb 05, 2016

Accepted: May 11, 2016

DOI: https://doi.org/10.21042/AMNS.2016.1.00021

Keywords
Partial differential equations, Symmetries, Exact solutions

© 2016 R. de la Rosa, M.S. Bruzón published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Similarity solutions and similarity variables of equation (1)

Reduced equations

On the classical and nonclassical symmetries of a generalized Gardner equation

Published Online: May 11, 2016

Page range: 263 - 272

Received: Feb 05, 2016

Accepted: May 11, 2016

DOI: https://doi.org/10.21042/AMNS.2016.1.00021

KeywordsPartial differential equations, Symmetries, Exact solutions

© 2016 R. de la Rosa, M.S. Bruzón published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Similarity solutions and similarity variables of equation (1)

Reduced equations

Keywords
Partial differential equations, Symmetries, Exact solutions