Symmetry Reductions for a Generalized Fifth Order KdV Equation

Adjoint table for the Lie algebra {X1,X2,X32} $\{X_1, X_2, X_3^2\}$ _

Ad	X1	X2	X32 $X_3^2$
X1	X1	X2	X32−εX1 $X_3^2-\epsilon X_1$
X2	X1	X2	X32−ε((4c−d2a)X1+5X2) $X_3^2-\epsilon \left( \left( 4 c -\tfrac{d^2}{a}\right) X_1 +5 X_2 \right)$
X32 $X_3^2$	eεX1	Y2	X32 $X_3^2$

Commutator table for the Lie algebra {X1,X2,X32} $\{X_1, X_2, X_3^2 \}$ _

	X1	X2	X32 $X_3^2$
X1	0	0	X1
X2	0	0	(4c−d2a)X1+5X2 $\left( 4 c -\tfrac{d^2}{a} \right) X_1+5 X_2$
X32 $X_3^2$	−X1	−(4c−d2a)X1−5X2 $-\left( 4 c -\tfrac{d^2}{a} \right) X_1-5 X_2$	0

Adjoint table for the Lie algebra {X1,X2,X33} $\{X_1, X_2, X_3^3\}$ _

Ad	X1	X2	X33 $X_3^3$
X1	X1	X2	X33 $X_3^3$
X2	X1	X2	X33−aεX1 $X_3^3-a \epsilon X_1$
X33 $X_3^3$	X1	X2 + aε X1	X33 $X_3^3$

Commutator table for the Lie algebra {X1,X2,X31} $\{X_1, X_2, X_3^1 \}$ _

	X1	X2	X31 $X_3^1$
X1	0	0	X1
X2	0	0	4cX1 + 5X2
X31 $X_3^1$	−X1	−4cX1−5X2	0

Commutator table for the Lie algebra {X1,X2,X31,X33 |a=0 } $\{X_1, X_2, X_3^1, X_3^3 \left|_{a=0} \right_ \}$ _

	X1	X2	X31 $X_3^1$	X33 |a=0 $\bf \text X^3_3|_{a=0}$
X1	0	0	X1	0
X2	0	0	4cX1 + 5X2	0
X31 $X_3^1$	−X1	−4cX1−5X2	0	2X33 |a=0 $2\text X^3_3|_{a=0}$
X33 |a=0 $\text X^3_3|_{a=0}$	0	0	−2X33 |a=0 $-2\text X^3_3|_{a=0}$	0

Adjoint table for the Lie algebra {X1,X2,X31,X33 |a=0 } $\{X_1, X_2, X_3^1, X_3^3 \left|_{a=0} \right \}$ _

Ad	X1	X2	X31 $X_3^1$	X4
X1	X1	X2	X31−εX1 $X_3^1-\epsilon X_1$	X33 |a=0 $X_3^3|_{a=0}$
X2	X1	X2	X31−ε(4cX1+5X2) $X_3^1-\epsilon \left( 4 c X_1 +5 X_2 \right)$	X33 |a=0 $X_3^3|_{a=0}$
X31 $X_3^1$	eεX1	Y1	X31 $X_3^1$	e−2εX33 |a=0 $e^{-2 \epsilon} X_3^3|_{a=0}$
X33 |a=0 $X_3^3|_{a=0}$	X1	X2	X31+2εX33 |a=0 $X_3^1+2\epsilon X_3^3|_{a=0}$	X33 |a=0 $X_3^3|_{a=0}$

Adjoint table for the Lie algebra {X1,X2,X31} $\{X_1, X_2, X_3^1\}$ _

Ad	X1	X2	X31 $X_3^1$
X1	X1	X2	X31−εX1 $X_3^1-\epsilon X_1$
X2	X1	X2	X31−ε(4cX1+5X2) $X_3^1-\epsilon \left( 4 c X_1 +5 X_2 \right)$
X31 $X_3^1$	eεX1	Y1	X31 $X_3^1$

Commutator table for the Lie algebra {X1,X2,X33} $\{X_1, X_2, X_3^3 \}$ _

	X1	X2	X33 $X_3^3$
X1	0	0	0
X2	0	0	aX1
X33 $X_3^3$	0	−aX1	0