Analytical Studies of Nonlinear Problems in Gravitational Field Theory

本研究项目旨在发展非线性数学物理中的分析方法，并将其应用于引力场理论中的非线性方程。1.分析方法的发展。(1)将约化摄动法应用于多维波的传播，证明了某些非线性发展方程可归结为Kadomtsev-Petviashvili方程和Zbolotskaya-Khokhlov方程。(2)研究了不可积扰动对孤子系统的影响。结果表明，弱外力可以激发出与孤子不同的新的定常非线性波，束缚的2-孤子可以通过色散(或耗散)扰动分裂成单个孤子。(3)通过截断奇异流形展开，将一些不可积方程变换为Hirota双线性形式，并得到显式解。2.在引力场方程中的应用。(1)利用特征线法研究了弦诱导引力理论中非线性波的传播特性。(2)利用柱面爱因斯坦方程的孤子解，证明了由于引力波的非线性相互作用而产生的引力法拉第旋转是有效的。(3)基于Schwarzschild解的微扰理论，研究了两个黑洞的迎头碰撞问题。构造了初始数据，计算了引力场的后续时间演化，清晰地显示了脉冲波的发射和黑洞面积的增加。不可积引力场方程分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.