Interdependence Between the Laurent-Series and Elliptic Solutions of Nonintegrable Systems

Elliptic solutions of the quintic complex one-dimensional Ginzburg–Landau equation

“…into (36) with g r = 0 and g i = 0 proves that they are exact solutions. The coefficients of the Laurent-series solutions do not include arbitrary parameters, so the obtained solutions are unique single-valued solutions and system (36) has no elliptic solution at g r = 0 and g i = 0.…”

“…The first package of computer algebra procedures, which realize the third and the fourth steps of the algorithm, has been written in AMP [33] by Conte. One can also use our Maple and REDUCE packages of procedures, which are accessible via the Internet [34] and are described in [35,36]. So, one passes the first four steps of algorithm without any difficulties.…”

“…To find elliptic solutions of system (36) we use the Conte-Musette method. The function ψ(ξ ) is a solution of both systems (36) and (37).…”

“…To find elliptic solutions of system (36) we use the Conte-Musette method. The function ψ(ξ ) is a solution of both systems (36) and (37). Therefore it has only two different Laurentseries expansions, whereas the functions M(ξ ) should have four different Laurent-series expansions, so it is easier to find the first-order differential equation for ψ(ξ ) than M(ξ ).…”