Application of catastrophe theory to molecules and solitons

We present the general elements of catastrophe theory and apply that theory to the investigation of the stable configurations of molecuels. For each molecule we introduce the concept of a grand catastrophe. With its aid we classify the stable configurations of the molecular structure. We study in particular simple molecules, containing only a few atoms. The relationship between symmetry breaking (Jahn-Teller effect) and transversality (a concept of catastrophe theory) is demonstrated. We define smooth mapping for different types of nonlinear equations having soliton solutions. By classifying the singularities of this smooth mapping we are able to investigate the stability or instability of the solitons considered. The theory is explicitly applied for studying solitons associated with the generalized Schrödinger equations, the generalized Kadomtsev-Petviashvili equations, equations with exponential and logarithmic nonlinearities. We also study solitons generated in random potentials, magnetic solitons, and solitons in nonlinear field theory.