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Analytical Studies of Nonlinear Problems in Gravitational Field Theory

引力场论非线性问题的解析研究

基本信息

批准号：
62540279
负责人：
TOMIMATSU Akira
金额：
$ 1.22万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1987
资助国家：
日本
起止时间：
1987 至 1988
项目状态：
已结题

项目摘要

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.

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。