Chaos, solitons and fractals in the nonlinear Dirac equation

By means of the asymptotic perturbation (AP) method, analytical investigation of a nonlinear Dirac equation shows the existence of interacting coherent excitations such as the dromions, lumps, ring soliton solutions and breathers as well as instanton solutions. The interaction between the localized solutions are completely elastic, because they pass through each other and preserve their shapes and velocities, the only change being a phase shift. Finally, one may obtain approximate lower-dimensional chaotic patterns such as chaotic-chaotic patterns, periodic-chaotic patterns, chaotic line soliton patterns and chaotic dromion patterns, due to the possibility of selecting appropriately some arbitrary functions. In a similar way, fractal dromion and lump patterns as well as stochastic fractal excitations can appear in the solution.