Optical soliton solutions of the pulse propagation generalized equation in parabolic-law media with space-modulated coefficients

Symmetry reductions of the ( 3 + 1 ) $(3+1)$ -dimensional modified Zakharov–Kuznetsov equation

“…One of the main functions for finding symmetry reductions is to use them to seek exact solutions. There are many effective direct methods that can be used to solve the obtained reduced equations such as the tanh method [28], the homogeneous balance method [29], the Horota bilinear method [30], the Darboux transformation method [31], and so on (see [32][33][34][35][36][37][38][39] for reference). Here, we use the traveling wave transformation to transform reduced equations (11) and (12) to ODEs for obtaining exact solutions.…”

“…By solving the above equation, one can obtain the following solutions which are expressed in the form of trigonometric functions, hyperbolic functions, and Weierstrass function: of the mZK equation with k = 1. The evolutions of solution (34) are given in Fig. 1, Fig.…”

New analytical internal long wave solutions described by the Benjamin Ono equation in deep water

“…Under using symbolic computation systems, the studying of the exact solutions of the NLEEs has attracted the attention of research community. A variety of approaches have been investigated and applied to the NLEEs, including the unified method (UM) [1][2][3][4][5] and its generalized form [6][7][8][9][10], the extended Jacobi elliptic function expansion method [11,12], the Bernoulli sub-equation function method [13,14], the sine-Gordon expansion method [15,16] and the Ricatti equation expansion [17,18].…”

Bilinearization and new soliton solutions of Whitham-Broer-Kaup equations with time-dependent coefficients

“…Researchers often investigate solutions of such nonlinear PDEs (see for examples [17][18][19][20][21]) to gain more insight into these physical phenomena for further applications. Since the celebrated Korteweg-de Vries (KdV) equation was exactly solved by Gardner et al [13], finding exact solutions of nonlinear PDEs has gradually developed into one of the most important and significant directions and many effective methods have been proposed such as the inverse scattering method [1,61,66,69], Hirota's bilinear method [15], Bäcklund transformation [33], Darboux transformation [31,47,64], Painlevé expansion [46,59,60], homogeneous balance method [44], subsidiary equation method [12,22,67,68], first integral method [4], residual power series method [23], and the exp-function method [14,55,56].…”