The first-integral method applied to the Eckhaus equation

Explicit solutions for couple higher order nonlinear Schrödinger equation by (G′G)-expansion method

“…(1) and separating real and imaginary parts, we get (10) (11) It is possible to integrate Eq. (11) and the resulting integrated equation can be written as (12) The necessary and sufficient condition for a non-constant function satisfying both Eq. (10) and (12) is that the coefficients of Eq.…”

“…Various powerful methods have been presented to find the exact solutions of nonlinear partial differential equations, for example, several important techniques have been developed such as the tanh-method [1,2], sine-cosine method [3,4], tanh-coth method [5], exp-function method [6], homogeneous-balance method [7,8], Jacobi-elliptic function method [9,10], first-integral method [11,12] and so on, all the methods have some limitations in their applications. In fact, there is no unified method that can be used to handle all types of nonlinear partial differential equations (NLPDE).…”

Exact solution of sixth-order thin-film equation by GʹG-expansion method

“…The powerful and efficient methods to find analytic solutions of nonlinear partial differential equations NLPDEs by using various method has become the main aim for many authors. Many powerful methods have been created and developed to obtain analytic solutions of NLPDEs, such as the the tanh-method [1], sinecosine method [2], tanh-coth method [3], exp-function method [4], homogeneous-balance method [5], Jacobielliptic function method [6], first-integral method [7] and so on. One of the most effective direct method to build traveling wave solution of NLPDEs is the G Gexpansion method, which was first proposed by Wang et al [8].…”

The homogeneous balance of undetermined coefficients method and its application

“…In [34], the Eckhaus equation is linearized by making some transformations of dependent and independent variables. Exact traveling wave solutions of the Eckhaus equation can be obtained by the .G 0 =G/-expansion method [10] and the first integral method [4].…”

“…Seeking the exact solutions of nonlinear PDEs has long been an interesting topic in the nonlinear mathematical physics. With the development of soliton theory, various methods for obtaining the exact solutions of nonlinear PDEs have been presented, such as the inverse scattering method [1], the Bäcklund and Darboux transformation method [2], the homotopy perturbation method [3], the first integral method [4], the variational iteration method [5], the Riccati-Bernoulli sub-ODE method [6], the Jacobi elliptic function method [7], the tanhsech method [8], the .G 0 =G/-expansion method [9,10], the Hirota's method [11], the homogeneous balance method (HBM) [12,13], the differential transform method (DTM) [14][15][16][17] and so on.…”