Studies on singular solutions of nonlinear differential equations.

通过对两类新的奇异解--黑洞解和白洞解的研究，证明了一类具有一定奇性的二阶非线性微分方程解的黑洞解可能存在，一类二阶非线性微分方程解的白洞解可能存在，而这类二阶非线性微分方程解的奇性不允许有任何奇性。结果表明，具有黑洞解和白洞解的微分方程解的非线性结构具有显著的对偶性。此外，还成功地将上述结果推广到一类二阶非线性微分方程组，从而刻画了这类微分方程组的黑洞型和白洞型奇异解的存在性。还研究了黑洞解(或白洞解)与其他类型的奇异解(如爆破解和绝迹解)的共存问题。对于给定的具有或不具有奇性的非线性微分方程解的整体结构的详细信息是一个重要的问题。给定一个微分方程，就会出现两种情况：要么根本没有奇异解，要么同时存在真性解和奇异解。半线性方程就是后者的一个例子。深入研究这两类非线性方程的波普解的定性性质也是我们的研究课题之一。

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.