INTEGRABLE SYSTEMS · Integrable systems in the theory of waves in fluids

Integrable systems in the theory of waves in fluids

In many real world phenomena nonlinear effects play an essential role. This has motivated an enormous amount of research into the study of nonlinearity, especially in the context of nonlinear waves. The interplay between nonlinearity and other characteristics of a model can be studied using a relatively simple model equation. The Korteweg-de Vries (KdV) equation is probably the best known model equation describing the competition between nonlinear and dissipative effects in water waves. In the recent years the Camassa-Holm (CH) equation has generated a tremendous deal of interest from top-class mathematicians with different areas of expertise: spectral problems, integrable systems, analysis, differential geometry etc. CH is an integrable nonlinear dispersive wave equation that has been used as a model of the propagation of unidirectional shallow water waves over a flat bottom. The CH equation has been introduced as a model in the theory of water waves by Roberto Camassa and Darryl Holm in 1993. Within the project the following studies have been completed: 1. The relations between smooth and peaked soliton solutions are reviewed for the Camassa-Holm (CH) shallow water wave equation in one spatial dimension. The canonical Hamiltonian formulation of the CH equation in action-angle variables is expressed for solitons by using the scattering data for its associated isospectral eigenvalue problem, rephrased as a Riemann-Hilbert problem. The dispersionless limit of the CH equation and its resulting peakon solutions are examined by using an asymptotic expansion in the dispersion parameter. 2. The Lax pair formulation of the two-component Camassa-Holm equation (CH2) is generalised to produce an integrable multi-component family, CH (n, k), of equations with n components and |k| velocities. All of the members of the CH (n, k) family show fluid-dynamics properties with coherent solitons following particle characteristics. We determine their Lie-Poisson Hamiltonian structures and give numerical examples of their soliton solution behaviour. We concentrate on the CH (2, k) family with one or two velocities, including the CH (2, -1) equation in the Dym position of the CH2 hierarchy. A brief discussion of the CH (3, 1) system reveals the underlying graded Lie-algebraic structure of the Hamiltonian formulation for CH (n, k) when n > 3.

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