Extended Mixed AKNS-Lund-Regge Model and Its Self-similarity Reduction

A symmetric system of mixed Painlevé III–V equations and its integrable origin

“…A model that mixes the second and the third Painlevé equations and possesses Painlevé property was obtained from a mixture of mKdV-Sine Gordon integrable models [12]. Likewise a mixture of Lund-Regge and AKNS models was to shown to reduce to the ordinary differential equation of Painlevé type [19].…”

On the mKdV–Liouville hierarchy and its self-similarity reduction

“…such that the first equation of the hierarchy is 0 (t) y xxxx − 3 2 y 2 x y xx + y xt + β(t)e y + δ(t)e −y = 0 (2) It was shown in [7] that the self-similarity reduction of (2) yields Kudryashov's equation [8]. Such equation passes the necessary condition for absence of movable branches points, called Painlevé property, and it can reduce to two Painlevé equation by appropriate choices of the parameters.…”

“…Such equation passes the necessary condition for absence of movable branches points, called Painlevé property, and it can reduce to two Painlevé equation by appropriate choices of the parameters. The Painlevé equations are second-order nonlinear ODEs (ordinary differential equations) which define new transcendental functions [9] and it has motivated many studies in higher order ODEs [7,8,10,11,12,13,14,15,16,17,18,19,20]. Due the connection with two Painlevé equations, Kudryashov's equation was conjectured as a possibility of defining a new transcendental function [8,21].…”

“…Later, this mixed model showed to be suitable for describing few-optical-cycles pulse in transparent media [6]. The generalized mKdV-sinh-Gordon hierarchy [7] is an mixed model which include the mixed mKdV-sinh-Gordon (δ(t) = −β(t)) and the mixed mKdV-Liouville hierarchies (δ(t) = 0), namely…”