On the Breaking of Progressive Waves in Shallow Water Part 2. Catastrophe Potential for Breaking Model

As a fundamental study on the elucidation of wave motions in the coast, one of analytical methods proposing the dynamical systems that represent the history of waves is presented as follows: The wave equation, expressed in the surface displacement only, and its Lagrangian function for the variational principle are obtained from the results of application of the perturbation method, where the method is applied to basic equations of hydrodynamics in the same way as the non-linear wave theory in shallow water. Motions of gravity waves are physical phenomena on the potential field. In this case, Hamiltonian function is identical to the integral surface (energy surface) of basic equations, when Euler-Lagrange's dynamical systems are transformed into Hamiltonian systems, and the independent variables do not appear explicitly. But quasi-linearised differential equations, based on the conventional perturbation method, are impossible to express the breaking of waves. Therefore, new concepts are necessary in order to have reasonable solutions analyzing the problem where proper dynamical systems are to be clarified the breaking. For the phenomena with discontinuities, R.Thom established the catastrophe theory, which deals with singular points of mappings and its structural stabilities, and has classified the singularities for certain classes of functions. This theory gives us clear understanding of the qualitative aspect of discontinuity in natural processes. The main theoretical significance of Thom's classification is to allow us to determine the stable equilibria of gradient systems subjected to small number of constraints, and to describe how these equilibria change as the constraints vary. On the other hand, since Hamiltonian function is the manifold constructed by two physical variables and basic equations are related to dynamical systems on this manifold, the topology of this differentiable manifold becomes important. Consequently, Hamiltonian function should be taken as Taylor's approximation to the exact energy surface at a local position near the stationary point of the variational principle, where topologically important properties are preserved. From the analytical studies on the catastrophe theory and the properties of Hamiltonian function mentioned above, it becomes clear that Hamiltonian function corresponds to the catastrophe potential for the breaking model. The numerical calculation using the first approximation to the breaking of the solitary waves is shown in which classical hydrodynamics and R.Thom's catastorophe theory are combined