Two hierarchies of ordinary differential equations and their properties
The author considers the two systems \(u_t+ \theta_1G_n(u)= 0\), and \(a_t+ \theta_2H_n(a)= 0\), \(n= 1,2,3,\dots\), with \(\theta_1= \theta_2= D^3+ 2uD+ u_x\), \(D= {\partial\over\partial x}\). The operators \(G_n\) and \(H_n\) are obtained by the recursion relations \(G_{n+2}= J_1(u)\theta_1(u) G_n\), and \(H_{n+2}= J_2(a) \theta_2(a) H_n\), when \(G_0= 1\), \(G_1= u_{xx} +{1\over 4} u^2\), \(H_0= 1\), \(H_1= a_{xx}+ 4a^2\) and the operators \(J_1= D^3+{1\over 2} D^2uD^{-1}+{1\over 2}D^{- 1} uD^2+{1\over 8}(u^2 D^{-1}+ D^{-1} u^2)\), and \(J_2= D^3+ 3(uD+ Du)+ 2(D^2 uD^{-1}+ D^{-1} uD^2)+ 8(u^2 D^{-1}+ D^{-1} u^2)\). This article is one of many by the author on this subject. These two systems are modified and when the Painlevé test is applied to these equations another system is obtained which is comparable to the system studied in an earlier paper. The author shows that this system is comparable with a desired system and has similar properties.
scientific article; zbMATH DE number 1412362