Much of my research in this field was discussed in a 45 minute invited presentation given at the international conference on "Singular Behavior and Nonlinear Dynamics" in Greece. Also this work resulted in an invitation to discuss possible research directions in nonlinear studies at the Los Alamos National Laboratory in 1990.
A part of nerve fibers and brains, as well as DNA consist of liquid crystals. It has been pointed there is an impressive analogy between never propagation and the director wave in liquid crystal.
The results in dynamics of antiferroelectric liquid crystals provide a unified and quantitative description for experiments. Kinetics of antiferroelectric liquid crystals at the critical temperature reveals a novel experimental phenomenon of free growth in a one dimension. The work has been recommended by physical review focus of APS. The results have an impact to liquid crystals display and sensor designing. The result reveals a biophysical significance of antiferroelectric liquid crystals.
The research in this field has been directly stimulated by industrial applications to polymer. This research includes studies of the kinetics of light-induced phase separation and periodic structures in polymeric-liquid crystal systems
A generalized free energy functional for polymer liquid crystals has been constructed which can be used to model the distinct effects of the long chain nature of polymers. Exact solutions and nonlinear analysis of the model equation with a forth-order derivative and revealed physics give a convincing and unified explanation for the formation and evolution of the universal phases.
Case II diffusion in polymer networks is an outstanding nonlinear problem with great practical importance for many fields, including medical physics and polymer engineering. The work is the incorporation of swelling effects and interfacial energy in the kinetic equations. The results should give a clear explanation for case II diffusion.
Hydrogen-bonded chain systems
Many hydrogen-bonded chain systems, such as water, DNA, various biopolymers, and bio-membranes, constitute the fundamental substances of life. Research in this area continues to be very challenging, with the potential for an enormous impact on both physics and biology. A basic experimental fact of these systems is that mass, energy, and charge is transported by "proton hopping" along a specific path. The research in this area includes the development model for proton conduction in a hydrogen-bonded chain system by a critical examination of the coupled dynamics of proton and heavy ion motion. Exact coupled soliton solutions have been explicitly obtained. The discovery of both topological and non-topological solitons in this type of system has revealed a new mechanism for charge and mass transfer.
Nonlinear dynamics in complex fluids
Complex fluids, especially liquid crystals, are particularly suitable for nonlinear studies. Various nonlinear phenomena such as solitary waves, pattern formation, selection mechanism and chaos in these materials have been systematically studied . The research is directly stimulated by the challenging problems in this field.
The work has included obtaining the exact solitary-wave solutions of the generalized Fisher equation used in the model of domain wall motion in liquid crystals. The method used to obtain this solution involves creating an ansatz based on both a symmetry analysis and an energy analysis of the physical system. The exact solutions have since been used to describe many phenomena in physics, engineering, and biology in the literature. This method has been subsequently expanded to obtain the exact solutions of the coupled equations used in field theory, plasmas and polymers.
The work has enabled complete and exact solutions to a class of nonlinear diffusion equations related to the selection problem in nonlinear dissipative systems to be found. Based on these exact solutions, a selection hypothesis that describes the spatial-temporal evolution of unstable dynamic systems has been proposed. These results provide an analytical basis for the marginal stability principle in unstable dynamic systems.