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

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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.