Meromorphic solutions of some forth-order differential equations
The object of the research pointed in the introduction is nonlinear differential forth-order equations with permanent coefficients. The sets of resonances appropriate to studied equations are adduced. The aim of the research is to define meromorphic property of general solutions of studied equations as well as other equations concerned with the studied ones; to find two-parameter rational decisions of studied equations and to get conditions in the context of which rational decisions can be found from general solutions of present equations. The so-called Painleve test for nonlinear higher order equations is adduced in the work of American mathematicians (M. Ablovitz, A. Ramani, H. Segur). According to this test in order to possess the Painleve property an equation should have whole and different resonances. Besides it is claimed that one resonance should be equal to number -1, all other resonances should not possess negative number as real part (when considering systems of differential equations, equations corresponding to resonances can have multiple roots). Later in other works the additional investigations were held in order to define if the Painleve property is present when there are negative ones not equal to number -1 among resonances. However, if an equation possesses the Painleve property, then rational decision should correspond to negative resonance not equal to number -1. In the main part meromorphic property of general solutions of forth-order equations, pointed in the introduction, as well as other equations concerned with the studied ones, is established. The rational decisions for studied equations with negative resonances are formulated. The conditions in the context of which these rational decisions can be found from general solutions are gained as well. The results of the research can be applied to the analytical theory of the nonlinear differential equations.