A new rational sine-Gordon expansion method and its application to nonlinear wave equations arising in mathematical physics

The new extended rational SGEEM for construction of optical solitons to the (2+1)–dimensional Kundu–Mukherjee–Naskar model

“…Recently, Yamgoue et al [44] introduced the rational sine-Gordon expansion method. The authors introduced the following trial solution which was generated from the sine-Gordon equation [45,46]:…”

Gramian solutions and solitonic interactions of a (2+1)-dimensional Broer–Kaup–Kupershmidt system for the shallow water

“…Nonlinear and dispersive long gravity waves traveling along two horizontal directions in the shallow water of uniform depth have been said to be governed by a (2 + 1)-dimensional BKK system as follows (Ying and Lou, 2001; Ruan and Chen, 1998; Tang et al , 2020; Rizvi et al , 2020; Wen, 2011; Kassem and Rashed, 2019; Yamgoué et al , 2019; Jiang et al , 2017; Lan et al , 2017): where u = u ( x , y , t ) denotes the height of the water surface above a horizontal bottom, v = v ( x , y , t ) is the horizontal velocity of the water wave, and the subscripts represent the partial derivatives with respect to the scaled space variables x , y and time variable t . When y = x , System (1) has been reduced to a (1 + 1)-dimensional BKK system, which is used to describe the propagation of the long waves in shallow water (Ying and Lou, 2001).…”

“…Painlevé property and infinitely many symmetries for System (1) have been discussed (Ruan and Chen, 1998). Binary Bell polynomial approach (Tang et al , 2020), Hirota bilinear method (Tang et al , 2020; Rizvi et al , 2020; Wen, 2011), hidden symmetries of the Lie optimal system (Kassem and Rashed, 2019) and rational sine-Gordon expansion method (Yamgoué et al , 2019) have been applied to System (1). Types of the solutions of System (1), such as the N -soliton solutions ( N is a positive integer) (Wen, 2011; Kassem and Rashed, 2019), cuspon (Kassem and Rashed, 2019), breather (Jiang et al , 2017), lump and rogue wave solutions (Rizvi et al , 2020), have been obtained.…”

“…Nonlinear and dispersive long gravity waves traveling along two horizontal directions in the shallow water of uniform depth have been said to be governed by a (2 þ 1)-dimensional BKK system as follows (Ying and Lou, 2001;Ruan and Chen, 1998;Tang et al, 2020;Rizvi et al, 2020;Wen, 2011;Kassem and Rashed, 2019;Yamgoué et al, 2019;Jiang et al, 2017;Lan et al, 2017):…”

Solitons of nonlinear dispersive wave steered from Navier-Bernoulli hypothesis and Love's hypothesis in the cylindrical elastic rod with compressible Murnaghan's materials

“…The extended sinh-Gordon method is applied for the (2 + 1)− dimensional hyperbolic Schr ödinger equation and cubic-quintic Schr ödinger equations for the optical solutions in [11]. The generalized modified Zakharov-Kuznetsov equation and Broer-Kaup-Kupershmidt equation is solved by rational sine-Gordon method in [12]. The Kerr law and quadratic-cubic law nonlinearities of nonlinear Schr ödinger equation for optical solitons using sine-Gordon method in [13].…”