Asymptotic solutions of a fourth—order analogue for the Painlevé equations

[摘要] Asymptotic solutions of a fourth-order analogue for the Painlevé equations that is self-similar reduction of the modified Sawada-Kotera and Kaup-Kupershmidt equation is considered. The Boutroux variables of two types have been found which allows us to find asymptotic solutions of the equation in the neighbourhood of the infinity. It was shown that asymptotic of self-similar solution for the modified Sawada-Kotera and Kaup-Kupershmidt equations can be determined as solutions of autonomous differential equations. Asymptotic solutions expressed by elementary functions have been found too. Besides asymptotic solutions expressed by logarithmic derivative of two elliptic Weierstrass functions have been found. Connection between obtained asymptotic solutions and asymptotic solutions of the Sawada-Kotera and Kaup-Kupershmidt equations has been discussed.

[发布机构] National Research Nuclear University MEPhl, Kashirskoe sh. 31, Moscow; 115409, Russia^1

[效力级别] 数学

[关键词] Asymptotic solutions;Autonomous differential equation;Elementary function;Logarithmic derivatives;Neighbourhood;Self-similar;Self-similar solution;Weierstrass functions