Darboux transformation and explicit solutions for some ()-dimensional equations

Analytic Solutions of the Thomas Equation by Generalised Tanh and Travelling Wave Hypothesis Methods

“…Therefore, many effective methods such as Darboux transformation [5,15], Bäcklund transformation [13,8], the F-expansion method [7,14], Exponent function method [6,4], sine -cosine method [16,17], Jacobi elliptic function method [3,2], the tanh-function method [9,17], etc, are introduced to obtain different types of exact solutions for nonlinear equations.…”

Exact two waves solutions with variable amplitude to the KdV equation

“…We know that, Darboux transformation method is one of the powerful and direct method for finding exact multiple line-soliton solutions, it is scarcely used for solving other type of multiple waves solutions because of the matter of calculation [8,9,10,11,12]. Generating new solutions from the non-trivial solution amounts to solving the variable coefficient partial differential system by this method, a major difficult is that for a given partial differential equations there is to date no completely systematic method of solving it, work has usually relied on some special techniques.…”

Doubly and Triply Periodic Waves Solutions for the KdV Equation

“…It is always useful to seek more and various multi-periodic solutions for recovering interactions among some simple periodic waves in a nonlinear medium. We know that the Darboux transformation method is the main method to construct exact multi-soliton solutions, and this method is scarcely used for solving multiperiodic solutions [8][9][10]. In the paper, not only explicit doubly periodic solutions are available, but also a group of explicit triply periodic solutions is obtained by means of the Darboux transformation method.…”