Linearity inside nonlinearity: exact solutions to the complex Ginzburg-Landau equation

The Painlevé Approach to Nonlinear Ordinary Differential Equations

“…When a DE admits two families with opposite principal parts, such as ( 7.17), it is natural to seek particular solutions described by two singular manifolds [34] v = 1 2β [∂ x Log ψ 1 − ∂ x Log ψ 2 + v 0 ], (7.29) in which (ψ 1 , ψ 2 ) is a basis of the two-dimensional space of solutions of some ODE whose general solution is entire, e.g. 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 .…”

“…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…”

Solitary Waves of Nonlinear Nonintegrable Equations

“…because of the elementary identities [9] tanh z − 1 tanh z = −2i sech 2z + i π 2 , tanh z + 1 tanh z = 2 tanh 2z + i π 2 .…”

“…Despite the multivalued dominant behaviour of the complex amplitude A of CGL3 and SH, one can define two variables with a single valued dominant behaviour. In this complex modulus representation [9] …”

“…Therefore, if one wants to extend the three above solutions, it is advisable to eliminate ϕ between the system of two real equations equivalent to (9),…”