I study spatio-temporal complicated behaviors in the nonlinear Schrödinger (NLS) equation and its near-integrable generalized form, the perturbed NLS equation, under the periodic boundary condition. The main goal of this thesis is to study low-dimensional behaviors of the systems....[ Read more ]

I study spatio-temporal complicated behaviors in the nonlinear Schrödinger (NLS) equation and its near-integrable generalized form, the perturbed NLS equation, under the periodic boundary condition. The main goal of this thesis is to study low-dimensional behaviors of the systems.

Aiming at this goal, I develop a perturbation theory for the NLS equations. In this theory, I obtain an order-function representation for the NLS equations. This representation consists of a very simple finite-dimensional dynamical system.

I show that the order-function representation is a simple but very powerful tool in between the studies on the NLS equations. I reveal many interesting relations NLS equations and their order-function representations: the modulation instability in the NLS equations is related to Hamiltonian bifurcation in the order-function representations; the spatially dependent and independent fixed point solutions of the NLS equations are related to the extrema of a potential given in the representations; the homoclinic structure corresponds to the double-well potential obtained in the representations; pattern types are related to unconnected regions in the phase space. In particular, I figure out that the mechanism of pattern competition is homoclinic crossing in terms of the order-function representation. And I reveal a new type of instability in the NLS equations. I called it the drifting instability, which is related to the ignored coordinate instability in the order-function representation.

Some open problems and direction for future studies are also indicated.