Link between solitary waves and projective Riccati equations

The Painlevé Approach to Nonlinear Ordinary Differential Equations

“…In particular, with µ 0 = 0 one thus obtains immediately the class of solutions polynomial in the two variables tanh and sech [33], thus augmenting the class indicated at the end of previous section. Evidently, if the DE has only one family, no dependence on sech can be found.…”

“…the second order linear equation with constant coefficients ψ xx − k 2 4 ψ = 0 (7.30) Ψ 2 = C 1 e k 2 x + C 2 e − k 2 x = C 0 cosh k 2 (x − x 0 ), (7.31) ψ 1 (x) = Ψ 2 (x + a), ψ 2 (x) = Ψ 2 (x − a), a arbitrary, (7.32) Substituting (7.29) into (7.17) and eliminating any derivative of (ψ 1 , ψ 2 ) of order higher than or equal to two in x results into a polynomial in the two variables ψ 1,x /ψ 1 , ψ 2,x /ψ 2 . Before identifying it to the null polynomial, one must take account of the first integral µ 0 , the ratio of two constant Wronskians 33) which splits the polynomial of two variables into the sum of two polynomials in one variable : are just two different representations [34] of a solution of (7.17) depending on two arbitrary constants (µ 0 , x 0 ) : with this simple assumption, we have obtained the general solution u −2 = v = 1 2β…”

Solving Benjamin-Bona-Mahony equation by using the sn-ns method and the tanh-coth method

“…θ(ξ) = ξ. Using the transformation (9) and putting ordinary derivative of v(ξ) instead of the partial derivatives of u(x, t), (8) In this method, we seek the traveling wave solutions to (10) in the form…”

“…And then, many methods have been established and used for obtaining exact solutions to NPLDEs. Some of the powerful methods are the generalized tanh method [1], the tanh-coth method (extended tanh method) [2], the tanh-sech method [3,4], sine-cosine method [5]- [6], the exp-function method [7], the projective Ricatti equations method [8], the generalized projective Ricatti equations method [9], the (G'/G)-expansion method [10], and the sn-ns method [11].…”

Periodic solutions to ZK-MEW equation in the form Lame equation

“…Recently, many authors presented various powerful method to deal with this interest subject, such as Backlund transformation [1], Darboux transformation [2], the extended tanh-function method [3], the F-expansion method [4], projective Riccati equations method [5], the Jacobian elliptic functions expansion method [6] and so on [7][8][9][10]. Recently , vast research results [11][12][13][14][15] have been obtained in this field.…”