| Abstract: |
| The Kaup-Boussinesq equation is a well known weakly nonlinear equation arising as an approximate model for shallow water flows. The equation is known to be integrable, in the sense that it may be written as a compatibility condition imposed on a pair of linear spectral problems. A particularly interesting aspect of the Lax pair associated with this fluid model is its energy dependence, whereby the potential of the spectral problem depends on the spectral parameter. In this talk we will present a recently developed pair of energy-dependent conjugate spectral problems associated with the Kaup-Boussinesq equation and briefly outline how this pair spectral problems is used to construct a topological solution of the system, via the Inverse Scattering Transform method. Following this, we will present a recent extension of this work, in which soliton solutions of the Kaup-Boussinesq equation have been constructed via the Inverse Scattering Transform. |
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