Studies on singular solutions of nonlinear differential equations.

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As a result of our studies on two types of new singular solutions, named black hole solutions and white hole solutions, it is shown that the black hole solutions may exist for a class of second order nonlinear differential equations whose differential operators have a certain singularity, and that the white hole solutions may exist for a class of second order nonlinear differential equations whose differential operators do not admit any singularity. The result exhibits a remarkable duality existing in the nonlinear structures of the differential equations having black hole solutions and white hole solutions. Furthermore, successful attempts have been made to extend the result mentioned above to a class of two-dimensional systems of second order nonlinear differential equations, thereby characterizing the existence of singular solutions of both black hole type and white hole type for such differential systems. The problem of coexistence of black hole solutions (or white hole solutions) and other types of singular solutions (such as blowing-up solutions an extinct solutions) has also been studied.It is an important question to have detailed information about the structure of the totality of solutions, both proper and singular, of the given nonlinear differential equations with or without singularity. Given a differential equation, the two cases occur : either it has no singular solution at all or it possesses a proper solution and a singular solution simultaneously. Half-linear equations are an example of the latter.An in-depth investigation of qualitative properties of poper solutions of these two types of nonlinear equations has also been one of the subject in our research project.