Exact vector multipole and vortex solitons in the media with spatially modulated cubic–quintic nonlinearity

Resonant Soliton and Soliton-Cnoidal Wave Solutions for a (3+1)-Dimensional Korteweg-de Vries-Like Equation

“…That is to say, if the function satisfies the consistent conditions (8) and (9), then (4) with (5)-(7) is an auto-Bäcklund transformation between the solutions , V, and 0 , V 0 , 0 of (2) and (3). According to the residual symmetry theorem, the residuals 1 , V 1 and 1 are just the nonlocal symmetry with respect to the solutions 0 , V 0 and 0 of (5)- (7) and one should transform them to a local Lie point symmetry for studying this nonlocal symmetry [25].…”

“…In scientific and engineering fields, nonlinear evolution equations have been studied in wide applications, such as in the nonlinear optics [1][2][3][4][5][6][7], plasma physics [8,9], fluid mechanics [10,11], textile engineering [12], and wave propagation phenomena [13][14][15][16]. Explicitly, for finding solutions, which including solitons, cnoidal waves, Painlevé waves, Airy waves, Bessel waves, etc., people often take the symmetry reduction approach with nonlocal symmetries with the aid of Darboux transformation, Bäklund transformation, and residual symmetry [17][18][19][20][21].…”

Soliton, breather, lump and their interaction solutions of the ($2+1$)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation

“…A set of systematic methods have been used in the literature to obtain reliable treatments of nonlinear evolution equations. So far, researchers have established several methods to find the exact solutions, including the inverse scattering transform [1], the Bäcklund transformation [2][3][4][5], the Darboux transformation [6][7][8][9][10][11][12][13][14], the Riemann-Hilbert approach [15][16][17] and Hirota's bilinear method [18][19][20][21][22][23][24][25][26][27][28], Jacobian elliptic function method and modified tanh-function method [29][30][31][32][33]. Each of these approaches has its features, Hirota's bilinear method is widely popular due to its simplicity and directness.…”

Application of the Exp (−φ(ζ))-expansion method for solitary wave solutions

“…Therefore, different techniques for finding exact solutions are used for a diversified field of partial differential equations like, homogenious balance technique for example (Wang, 1995;Zayed et al, 2004), Hirota's bilinear approach (Hirota, 1973;Hirota & Satsuma, 1981), Auxiliary equivalence technique (Sirendaoreji, 2004), Trial task technique (Inc & Evans, 2004), Jacobi elliptic task system (Ali, 2011), Tanhfunction technique (Abdou, 2007), and method of sine-cosine (Wazwaz, 2004), truncated Painleve expansion technique (Weiss et al, 1983), variational iteration method (VIM) (Abbasbandy, 2007), Expfunction technique (He & Wu, 2006;, ðG 0 =GÞ-expansion approach (Wang et al, 2008;Zayed, 2010;Zayed & Gepreel, 2009;Shehata, 2010;Zhang et al, 2010), Exact soliton solution (Khan & Akbar, 2013;Liping et al, 2009;Zhang et al, 2013). Several theoretical and experimental works for solitons includes (Wang et al, 2017;Dai et al, 2018;Wang et al, 2016;Wang et al, 2018;Ding et al, 2017). On exact solution, some novel results and computational methods involved to travelling-wave transformation, see the references (Yang, 2016;Yang et al, 2016Saad et al, 2017;Feng et al, 2017).…”