Analysis of pure nonlinear lattice with x^4 potential A method of computational physics including computer algebra
Project/Area Number  07640538 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
物理学一般

Research Institution  Tokyo Engineering University 
Principal Investigator 
SHIMOJI Sadao Tokyo Engineering University Department of Engineering Science Professor, 工学部, 教授 (50216123)

CoInvestigator(Kenkyūbuntansha) 
KAWAI Toshio Keio University Department of Physics Professor, 理工学部, 教授 (30146724)

Project Fiscal Year 
1995 – 1996

Project Status 
Completed(Fiscal Year 1996)

Budget Amount *help 
¥2,000,000 (Direct Cost : ¥2,000,000)
Fiscal Year 1996 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1995 : ¥1,400,000 (Direct Cost : ¥1,400,000)

Keywords  nonlinear lattice / quasilinear wave equation / multivalued solution / chaotic behavior / 純非線形格子 / 非線形波動方程式 / 多価解 / カオス的挙動 / MongeAmpere方程式 / 対称変換 / 超長期の運動 
Research Abstract 
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 space This 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 quasilinear 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.

Report
(4results)
Research Output
(7results)