On the Lie algebras, generalized symmetries and darboux transformations of the fifth-order evolution equations in shallow water

Abstract

By considering the one-dimensional model for describing long, small amplitude waves in shallow water, a generalized fifth-order evolution equation named the Olver water wave (OWW) equation is investigated by virtue of some new pseudo-potential systems. By introducing the corresponding pseudo-potential systems, the authors systematically construct some generalized symmetries that consider some new smooth functions··· ,n β=1,2,··· ,N depending on a finite number of partial derivatives of the nonlocal variables v β and a restriction i,α,β ∂ξ i ∂v β 2 + ∂η α ∂v β 2 = 0, i.e., i,α,β ∂G α ∂v β 2 = 0. Furthermore, the authors investigate some structures associated with the Olver water wave (AOWW) equations including Lie algebra and Darboux transformation. The results are also extended to AOWW equations such as Lax, Sawada-Kotera, Kaup-Kupershmidt, Itô and Caudrey-Dodd-Gibbon-Sawada-Kotera equations, et al. Finally, the symmetries are applied to investigate the initial value problems and Darboux transformations.

Solitary Wave and Quasi-Periodic Wave Solutions to a (3+1)-Dimensional Generalized Calogero-Bogoyavlenskii-Schiff Equation

“…It is also integrable by the one-dimensional inverse scattering transform and Painléve test [28][29][30][31]. Recently, more and more people are interested in studying some generalized nonlinear evolution equations [32][33][34][35][36][37][38], resulting from their more widely applications in many physical fields [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. To our knowledge, Riemann theta function periodic wave solutions for Eq.…”

Space-time fractional Rosenou-Haynam equation: Lie symmetry analysis, explicit solutions and conservation laws

“…FDEs have been studied nowadays to describe several physical aspects and procedure in natural conditions [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. However, fractional partial differential equations (FPDEs) having only time derivative have been analyzed via the Lie symmetry method [20][21][22][23][24][25][26]. Recently, the Lie method has been developed to the systems of time FPDEs [27].…”

On some new analytical solutions for new coupled Konno–Oono equation by the external trial equation method

“…Authors of [5] obtained trawling wave solution for coupled Konno-Oono (CKO) equation using the modified exponential function method. A special case of system (1.1) to be considered transformed into new Konno-Oono equation system which is a coupled integrable dispersionless equations given as form of: )-expansion method [13][14][15], the G G ¢ -expansion method [16,17], the generalized G G ¢ -expansion method [18], the Bernoulli sub-equation function method [19][20][21][22], the sine-Gordon expansion method [23][24][25], the Ricatti equation expansion [26], the formal linearization method [27], the Lie symmetry [28][29][30], the Bäcklund transformations [31][32][33], the Darboux transformation [34], the Fokas method [35][36][37], the Hirota bilinear method [38][39][40][41]43], and so on.…”