The so-called integrable evolution equations possess several remarkable properties. In particular, their initial value problem can be solved using a nonlinear version of the Fourier transform method, called inverse scattering (spectral) method. For evolution equations in one and two spatial dimensions this method involves the Riemann-Hilbert and the d-bar formalisms, respectively.
An important advantage of these formulations is that they can be used for the explicit evaluation of the long time behaviour of the solution. Among the most important integrable evolution equations in one space dimension is the Camassa-Holm (CH) equation, which is a certain generalization of the celebrated Korteweg-de Vries equation.
The main objective of this project is to evaluate the long time behaviour of the solution of the CH equation using the Riemann-Hilbert formalism, to implement the inverse scattering method to equations analogous with the CH equation which are generalizations of the nonlinear Schroedinger and of the sine-Gordon equations, and to extend these results to integrable generalizations of the above equations in two and three spatial variables.
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