Auto-Bäcklund transformation and analytic solutions for general variable-coefficient KdV equation

On the Variable-coefficient Burgers-Hlavaty Equation

“…in Ref. [6,7], When substituting (3) into (1), we make the coefficients of like powers of 0 to vanish, so as to get…”

A New Auto-Bäcklund Transformation of the KdV Equation with General Variable Coefficients and Its Application

“…Equation (1) is well known as a model equation describing the propagation of weakly nonlinear and weakly dispersive waves in inhomogeneous media. In recent decades, much progress has been made in the studies of (1) obtaining its auto-BT and exact solutions [2][3][4][5][6]. In [3], by using the truncate Painlevé expansion [7], Hong and Jung obtained an auto-BT which includes 5 Painlevé-Bäcklund equations to be solved.…”

Some Soliton Solutions of Non Linear Partial Differential Equations by Tan-Cot Method

“…In order to understand the non-linear phenomena of equations (1.1) − (1.4) in a better way it is important to seek their more exact solutions. A variety of useful methods notably Algebraic method [14], Exp-function method [15], Adomian modified method [16], Inverse scattering method [17], Tanh function method [18], Variational method [19], Similarity transformation methods using Lie group theory [20], Homotopy perturbation method [21], Jacobi elliptic function expansion method [22], Hirota method [23], Backlu nd transformations method [24,25], F-expansion method [26], Differential transformations method [27], Darboux transformations [28], Balance method [29], Sine-cosine method [30] and Tan-Cot method [31] were applied to investigate the solutions of non-linear partial differential equations.…”