Exact traveling wave solutions of the Boussinesq–Burgers equation

Exact soliton solutions of the modified KdV–KP equation and the Burgers–KP equation by using the first integral method

“…In order to better understand these nonlinear phenomena, many mathematicians and physical scientists make efforts to seek more exact solutions of them. Several powerful methods have been proposed to obtain exact solutions of nonlinear evolution equations, such as multiple exp-function method [1], tanh-sech method [2][3][4][5], extended tanh method [6][7][8][9], hyperbolic function method [10], sine-cosine method [11][12][13], Jacobi elliptic function expansion method [14], F-expansion method [15], transformed rational function method [16], homogeneous balance method [17][18][19][20][21][22] and so on.…”

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

“…we can give uniform formulae of n-soliton solutions of (1.1) and (1.2) as follows: 25) where the summation µ=0,1 refers to all possible combinations of each µ i = 0, 1 for i = 1, 2, · · · , n, and Z 1 (µ), Z 2 (µ) and Z 3 (µ) denote that when we select all the possible combinations µ j (j = 1, 2, · · · , 2n) the following conditions hold, respectively:…”

“…When γ 1 (t) = γ 4 (t) = 2, γ 2 (t) = γ 6 (t) = −1/2 and γ 3 (t) = γ 5 (t) = 0, equations (1.1) and (1.2) degenerate into the Boussinesq-Burgers (BB) equations [25]:…”

“…With the help of Riccati equation and its some special solutions, Khalfallah [25] obtained hyperbolic function solutions and rational solutions of (1.9) and (1.10). By using the compatibility method, Yan and Zhou [53] obtained many explicit solutions of the Boussinesq-Burgers (1.13) and (1.14), which include solutions expressed by error function, Bessel function, exponential function and Airy function.…”

Combined ZK-mZK equation for Rossby solitary waves with complete Coriolis force and its conservation laws as well as exact solutions

“…Many efficient techniques, such as homogeneous balance technique [47], symmetry theory [48], Jacobi elliptic function method [49], homotopy analysis transform method [50], homotopy perturbation transform method [51], Darboux transformation [52][53][54], Bilinear method [55,56], and so on [57], have been proposed to seek solitary waves solutions. In addition, some numerical methods, i.e., modified binomial and Monte Carlo methods [58], high accurate NRK, and MWENO [59][60][61], have also been used to solve partial differential equations.…”