Low-mode truncation methods in the sine-Gordon equation

Thesis/Dissertation · Tue Jan 01 04:00:00 EST 1991

OSTI ID:5261595

Xiong, Chuyu

In this dissertation, the author studies the chaotic and coherent motions (i.e., low-dimensional chaotic attractor) in some near integrable partial differential equations, particularly the sine-Gordon equation and the nonlinear Schroedinger equation. In order to study the motions, he uses low mode truncation methods to reduce these partial differential equations to some truncated models (low-dimensional ordinary differential equations). By applying many methods available to low-dimensional ordinary differential equations, he can understand the low-dimensional chaotic attractor of PDE's much better. However, there are two important questions one needs to answer: (1) How many modes is good enough for the low mode truncated models to capture the dynamics uniformly (2) Is the chaotic attractor in a low mode truncated model close to the chaotic attractor in the original PDE And how close is He has developed two groups of powerful methods to help to answer these two questions. They are the computation methods of continuation and local bifurcation, and local Lyapunov exponents and Lyapunov exponents. Using these methods, he concludes that the 2N-nls ODE is a good model for the sine-Gordon equation and the nonlinear Schroedinger equation provided one chooses a good' basis and uses enough' modes (where enough' depends on the parameters of the system but is small for the parameter studied here). Therefore, one can use 2N-nls ODE to study the chaos of PDE's in more depth.

• Research Organization: Ohio State Univ., Columbus, OH (United States)
• OSTI ID: 5261595
• Country of Publication: United States
• Language: English

Related Subjects

662110* -- General Theory of Particles & Fields-- Theory of Fields & Strings-- (1992-)
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
99 GENERAL AND MISCELLANEOUS
990200 -- Mathematics & Computers
ATTRACTORS
CALCULATION METHODS
DIFFERENTIAL EQUATIONS
EQUATIONS
FIELD EQUATIONS
LYAPUNOV METHOD
NONLINEAR PROBLEMS
PARTIAL DIFFERENTIAL EQUATIONS
SCHROEDINGER EQUATION
SINE-GORDON EQUATION
STOCHASTIC PROCESSES
WAVE EQUATIONS