Pattern selection and instability in nonlinear wave equation: An aspect of soliton and chaos

Pattern selection problems are found in a variety of phenomena. Fluid dynamic systems and nonlinear diffusion phenomena give examples of pattern formation problems in dissipative systems. In some cases the dissipation reduces the effective dimension of the system, which leads to several strikingly universal behaviors which were initially found in simple model systems with a few degrees of freedom. Nonlinear wave equations describe systems without dissipation in which the situation is more complicated. Many completely intergrating systems are known in nonlinear wave equations, where neither ergodicity nor chaos is expected. With addition of small perturbation to completely integrable systems, the growth of instability and the role of coherent structures in the pattern selection problem is observed. Two of these aspects are discussed.