Group Analysis of Generalized Fifth-Order Korteweg–de Vries Equations with Time-Dependent Coefficients

Abstract

We perform enhanced Lie symmetry analysis of generalized fifth-order Korteweg-de Vries equations with time-dependent coefficients. The corresponding similarity reductions are classified and some exact solutions are constructed.

Group analysis of general Burgers–Korteweg–de Vries equations

“…, A r tt , B tt , C tt , A 0 t and C t . This leads to the systemX 1 A 0 = · · · =X 1 A r = 0,X 1 B = 0,X 1 C = 0,X 1 Y 1 = 0,X 1 t = 0 and (1/X 1 ) Y 2 Y 2 = 0, whose general solution is of the form (17). The expressions for the transformed arbitrary elements A 0 , .…”

“…Indeed, let us fix any equationLθ from the classK 3 . The set Tθ of all admissible transformations with source atθ is parameterized by the arbitrary smooth functions T , X 0 , U 1 and U 00 of t and the arbitrary constants ε 0 , ε 1 , ε ′ 0 and ε ′ 1 with T t U 1 = 0, δ := ε 0 ε ′ 1 − ε ′ 0 ε 1 = 0 and δU 1 X 1 > 0, where X 1 is defined by (17) for the fixed value of the arbitrary element Y 2 , Y 2 = Y 2 (t). Each admissible transformation from Tθ is generated by the equivalence transformation from M with the same values of T , ε 0 , ε 1 , ε ′ 0 and ε ′ 1 , and the values of X 00 , X 01 and V defined by…”

“…Proposition 7 and Section 9. Variable-coefficient generalizations of the Kawahara equation were studied in a similar way in [17,18]. The group classification of Galilei-invariant equations of the form u t + uu x = F (u r ) was carried out in [5,8].…”