New similarity solutions for the generalized variable-coefficients KdV equation by using symmetry group method

Abstract

In this paper, a generalized variable-coefficient KdV equation (vcKdV) arising in fluid mechanics, plasma physics and ocean dynamics is investigated by using symmetry group analysis. Two basic generators are determined, and for every generator in the optimal system the admissible forms of the coefficients and the corresponding reduced ordinary differential equation are obtained. The search for solutions to those reduced ordinary differential equations yields new exact solutions for the generalized vcKdV equation.

New rogon waves for the nonautonomous variable coefficients Schrö dinger equation

“…Recently, many new methods have been constructed to obtain new solutions for nonlinear partial differential equations like symmetry groups, tanh method, trial equation method, sin-Gordon equation method, etc. (Chen et al 2019(Chen et al , 2020Hua et al 2019;Liu et al 2019;Mao et al 2019;Xia et al 2020;Xu et al 2020) (El-Sayed et al 2015, 2014, 2021, 2018aEl-Shiekh and Rehab 2018b;El-Shiekh 2017, 2015, 2013El-Shiekh and Gaballah 2020b;Moatimid et al 2013;Moussa et al 2012;El-Shiekh 2010, 2011;Moatimid and El Shikh 2008;Moussa and El Shikh 2006;Chen et al 2021;He et al 2021;Lü and Ma 2016;Xia et al 2020;Yin et al 2020).…”

“…In the following we are going to apply one of the similarity techniques, the direct similarity reduction method (El-Shiekh 2019, 2017, 2015, 2013, 2018aEl-Shiekh and Gaballah 2020b;El-Shiekh 2008, 2011) where ′ denotes the derivative with respect to . Collect the U coefficient and its derivatives, also, the real and imaginary parts together, assuming that e i( (z,t)) ≠ 0, we have…”

Similarity analytical solutions for the Schrӧdinger equation with the Riesz fractional derivative in quantum mechanics

“…Apart from that, the symmetry properties were discussed for the fractional diffusion equation by Gazizov et al 18 Furthermore, the scaling transformation was studied by Kandasamy et al 19 for the fluid viscosity on the process of mass and heat transfer. Apart from that, El‐Sheikh 20 studied regarding the similarity solutions for KdV equation via symmetry method. In a different paper, the Burgers equation has been treated in terms of freezing solutions by Matthes 21 .…”

Novel solitons and periodic wave solutions for Davey–Stewartson system with variable coefficients

“…Therefore, many new methodologies, for example, tanh method, direct algebric method, Darboux transformation, Painlevé test, integral methods, Bä cklund transformation, trail equation method, etc [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26], are constructed for obtaining solitary and soliton wave solutions to nonlinear partial differential equations (NPDEs), but symmetry methods are still the most important techniques in construction and exploitation of nature laws for NPDEs [27][28][29][30][31][32][33]. Many modifications and generalizations are made for symmetry to deal with different types of NPDEs.…”