Analytical Studies of Nonlinear Problems in Gravitational Field Theory

该研究项目旨在发展非线性数学物理学的分析方法，并将其应用于引力场理论中的非线性方程。主要研究结果如下：1.分析方法的发展。(1)By将约化摄动方法应用于多维波动传播问题，证明了某些非线性发展方程可以约化为Kadomtsev-Petviashvili方程和Zabolotskaya-Khokhlov方程。（2）研究了不可积扰动对孤子系统的影响。研究发现，在弱外力作用下，可以激发出一种不同于孤子的新的定态非线性波，并且通过色散（或耗散）微扰，束缚2-孤子可以分裂成单个孤子。（3）通过截断奇异流形展开，将不可积方程转化为Hirota双线性形式，并得到了显式解。2.引力场方程的应用。(1)By摘要应用特征线方法研究了弦致引力理论中非线性波的传播特性。并导出了一个新的黑洞解，它具有不同于Schwarzschild解的热力学性质。(2)By利用圆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引起的引力法拉第旋转的有效性。（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，计算了引力场随时间的演化，明确显示了脉冲波的发射和黑洞面积的增加.不可积引力场方程解析方法的进一步发展有待于进一步研究。

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.