Extended generalized Darboux transformation to hybrid rogue wave and breather solutions for a nonlinear Schrödinger equation

Application of the Multiple Exp-Function, Cross-Kink, Periodic-Kink, Solitary Wave Methods, and Stability Analysis for the CDG Equation

“…Appending Equation (10) into Equation (7) and ordering the numerator of the rational function to zero, we can achieve 2 Advances in Mathematical Physics a series of nonlinear algebraic equations about the variables α i , δ i , Δ rs,ij and Ω rs,ij . Solving the solutions for these nonlinear algebraic equations and putting these solutions into Equation (10), the multiple soliton solutions to Equation ( 7) can be obtained in the below form as…”

“…It is known that these exact solutions of nonlinear evolution equations (NLEEs), especially the soliton solutions [1][2][3], can be given by using a variety of different methods [4,5], such as Jacobi elliptic function expansion method [6], inverse scattering transformation (IST) [7,8], Darboux transformation (DT) [9], extended generalized DT [10], Lax pair (LP) [11], Lie symmetry analysis [12], Hirota bilinear method [13], and others [14,15]. The Hirota bilinear method is an efficient tool to construct exact solutions of NLEEs, and there exists plenty of completely integrable equations which are studied in this way.…”

New various multisoliton kink‐type solutions of the (1 + 1)‐dimensional Mikhailov–Novikov–Wang equation

“…Over the past few years, many methods are being developed to expose new traveling solitary wave solutions for the nonlinear partial differential equation (PDE) representing the different areas of science and engineering. [1][2][3][4] Some of the analytical methods such as extended (G ′ /G)-expansion method, Darboux transformation, Pfaffian technique, sech-tanh method, sine-cosine method, Painlevé analysis, 5 Hirota bilinear method, [6][7][8][9][10][11][12][13][14][15] extended generalized Darboux transformation method, 16,17 Bäcklund transformation, and simplified Hirota's method [18][19][20][21][22][23][24] are used to solve different models involving nonlinear PDE. There is no specified method to solve all types nonlinear PDE.…”

Mixed Rational Lump-Solitary Wave Solutions to an Extended (2+1)-Dimensional KdV Equation

“…More recently, a lot of rational lump solutions, hybrid solutions consisting of lump waves and kink waves, loop-like kink breather solutions, and the lump interacting with the line soliton solutions have been constructed via the Hirota bilinear method . Although these results can also be derived by Darboux transformation [47][48][49][50], modified extended mapping method [1], and direct algebraic method [2], the bilinear method is still a powerful tool for solving integrable systems. It is worth mentioning that Seadawy et al obtained some new exact solutions of many integrable systems by using various methods, such as extend simple equation method and the exp ðϕðξÞÞ expansion method [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16].…”