Analytical Studies of Nonlinear Problems in Gravitational Field Theory

本课题旨在发展非线性数学物理中的解析方法，并将其应用于引力场理论中的非线性方程。得到了以下结果：1。分析方法的发展。(1)将约化微扰法应用于多维波传播，证明了一些非线性演化方程可以约化为Kadomtsev-Petviashvili方程和Zabolotskaya-Khokhlov方程。(2)研究了不可积微扰对孤子系统的影响。发现在弱外力作用下可以激发出一种不同于孤子的新的平稳非线性波，并通过色散（或耗散）扰动将束缚2孤子分裂成单个孤子。(3)通过截断奇异流形展开，将一些不可积方程转化为Hirota的双线性形式，并推导出其显式解。引力场方程的应用。(1)应用特征方法研究了弦诱导重力理论中非线性波的传播特性。并推导出了与史瓦西解具有不同热力学性质的新黑洞解。(2)利用柱形爱因斯坦方程的孤子解，证明了引力波非线性相互作用导致的引力法拉第旋转是有效发生的。(3)基于Schwarzschild解的摄动理论，研究了两个黑洞的正面碰撞问题。构造初始数据，计算引力场随时间的演化。明确地指出了脉冲波的发射和黑洞面积的增大。不可积引力场方程解析方法的进一步发展有待于进一步的研究。

This research project aimed to develop analytical methods in nonlinear mathematical physics and to apply them to nonlinear equations in gravitational field theory. The following results were obtained: 1.Developments of analytical methods.(1)By applying the reductive perturbation method to multi-dimensional wave propagation,it was shown that some nonlinear evolutional equations can be reduced to the Kadomtsev-Petviashvili aquation and the Zabolotskaya-Khokhlov equation.(2)Effects of nonintegrable perturbations on soliton systems were studied. It was found that a new stationary nonlinear wave which is different from the soliton can be excited by a weak external force,and a bound 2-soliton can split into individual solitons through a dispersive (or dissipative) perturbation.(3)Some nonintegrable equations were transformed into Hirota's bilinear form by a truncation of the singular manifold expansion,and the explicit solutions were derived.2.Applications to gravitational field equations.(1)By applying the method of characteristics,a propagation character of nonlinear waves in a string-induced gravity theory was studied. The new black-hole solution which has a thermodynamic property different from the Schwarzschild solution was also derived.(2)By using soliton solutions of the cylindlical Einstein equations,it was shown that the gravitational Faraday rotation due to a nonlinear interaction of gravitational waves occurs effectively. (3)Based on the perturbation theory for the Schwarzschild solution,the ploblem of a head-on collision of two black holes was studied. The initial data were constructed,and the subsequent time-evolution of gravitational field was calculated.The emission of pulse-like waves and the increase increase of the area of black holes were explicitly shown. Further developments of analytical methods in nonintegrable gravitational field equations remain to future investigations.