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Full-Text Articles in Applied Mathematics
Zakharov-Shabat System With Constant Boundary Conditions. Reflectionless Potentials And End Point Singularities, Tihomir Valchev, Rossen Ivanov, Vladimir Gerdjikov
Zakharov-Shabat System With Constant Boundary Conditions. Reflectionless Potentials And End Point Singularities, Tihomir Valchev, Rossen Ivanov, Vladimir Gerdjikov
Conference papers
We consider scalar defocusing nonlinear Schroedinger equation with constant boundary conditions. We aim here to provide a self contained pedagogical exposition of the most important facts regarding integrability of that classical evolution equation. It comprises the following topics: direct and inverse scattering problem and the dressing method.
Two Component Integrable Systems Modelling Shallow Water Waves: The Constant Vorticity Case, Rossen Ivanov
Two Component Integrable Systems Modelling Shallow Water Waves: The Constant Vorticity Case, Rossen Ivanov
Articles
In this contribution we describe the role of several two-component integrable systems in the classical problem of shallow water waves. The starting point in our derivation is the Euler equation for an incompressible fluid, the equation of mass conservation, the simplest bottom and surface conditions and the constant vorticity condition. The approximate model equations are generated by introduction of suitable scalings and by truncating asymptotic expansions of the quantities to appropriate order. The so obtained equations can be related to three different integrable systems: a two component generalization of the Camassa-Holm equation, the Zakharov-Ito system and the Kaup-Boussinesq system. The …
On An Integrable Two-Component Camassa-Holm Shallow Water System, Adrian Constantin, Rossen Ivanov
On An Integrable Two-Component Camassa-Holm Shallow Water System, Adrian Constantin, Rossen Ivanov
Articles
The interest in the Camassa-Holm equation inspired the search for various generalizations of this equation with interesting properties and applications. In this letter we deal with such a twocomponent integrable system of coupled equations. First we derive the system in the context of shallow water theory. Then we show that while small initial data develop into global solutions, for some initial data wave breaking occurs. We also discuss the solitary wave solutions. Finally, we present an explicit construction for the peakon solutions in the short wave limit of system.