Analysis of pure nonlinear lattice with x^4 potential -A method of computational physics including computer algebra-

为了研究复杂系统中可能存在的共同特征，如1/<有界积分>涨落，考虑并分析了非线性格的最简单模型：一个单项x^4位的纯非线性格。其数值解随着时间的推移变得混沌，解的功率谱具有1/<有界积分>分布。粒子的轨迹在2N （N为粒子数）相空间中具有相等的停留概率，本研究项目旨在利用分析技术阐明晶格的动力学。晶格模型近似为准线性波动方程gamma_u=(gamma^2gamma_x)_x。对于一类特殊的初始条件，方程有解析解。在时间t_<v min>后，解在某个区域M内一般是多值的。区域M随着时间的推移而扩展。格的解析解和数值解在t<t_<v min>的整个区域内一致，在t<t_<v min>的单值区域内一致。推广到M的能量积分在包括M在内的整个区域对所有t都是守恒的。晶格解在M中会变得混沌。一维空间碰撞粒子系统的数值分析表明，速度随时间的分布趋向于1/上铁分布。

In order to investigate the common feature which may exist in the complex systems, such as 1/<bounded integral> fluctuation, the simplest model of nonlinear lattice is considered and analyzed : a purely nonlinear lattice with a single term of x^4 potential. Its solution obtained by numerical technique became chaotic as time elapses and the power spectrum of the solution has 1/<bounded integral> distribution. The trajectories of particles has equal sojourn probability in the 2N (N is the number of particles) phase spaceThis research project aims to clarify the dynamics of the lattice by using also the analytical technique. The model of the lattice is approximated by a quasi-linear wave equation, gamma_u=(gamma^2gamma_x)_x. The equation has analytical solutions for a special class of initial conditions. The solutions become generally multivalued in some region M after a time t_<v min>. The region M spreads with time. The analytical solutions and numerical solutions to the lattice agree in the whole region fot t<t_<v min> and they agree in the single valued region for t>t_<v min>. The energy integral extended to be applicable also to M is shown to be conserved in the whole region including M for all t. The solution to the lattice will become chaotic in M.The numerical analysis of the system of colliding particles in one dimensional space clarified that the velocity distribution tends to 1/upsiron distribution with time.