On the classical and nonclassical symmetries of a generalized Gardner equation

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 >	$z = x t^{- \frac{1}{3}} + \frac{(a^{2} - 4 b d) t^{\frac{2}{3}}}{4 b}$ $\begin{array}{} \displaystyle z=\displaystyle x t^{-\frac{1}{3}}+ \frac{(a^2 - 4 b d)t^{\frac{2}{3}}}{4 b } \end{array}$	$u = t^{- \frac{1}{3}} h (z) - \frac{a}{2 b}$ $\begin{array}{} \displaystyle u=\displaystyle t^{-\frac{1}{3}} h(z)- \frac{a}{2 b } \end{array}$

Reduced equations

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

Language:: English

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On the classical and nonclassical symmetries of a generalized Gardner equation

R. de la Rosa

M.S. Bruzón

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