Rossen I. Ivanov

Inverse scattering transform for the Camassa?Holm equation

An inverse scattering method is developed for the Camassa–Holm equation. As an illustration of our approach the solutions corresponding to the reflectionless potentials are constructed in terms of the scattering data. The main difference with respect to the standard inverse scattering transform lies in the fact that we have a weighted spectral problem. We therefore have to develop different asymptotic expansions.

Water waves and integrability

Eulers equations describe the motion of inviscid fluid. In the case of shallow water, when a perturbative asymptotic expansion of Eulers equations is considered (to a certain order of smallness of the scale parameters), relations to certain integrable equations emerge. Some recent results concerning the use of integrable equation in modelling the motion of shallow water waves are reviewed in this contribution.

Inverse scattering transform for the Degasperis–Procesi equation

We develop the inverse scattering transform (IST) method for the Degasperis–Procesi equation. The spectral problem is an Zakharov–Shabat problem with constant boundary conditions and finite reduction group. The basic aspects of the IST, such as the construction of fundamental analytic solutions, the formulation of a Riemann–Hilbert problem, and the implementation of the dressing method, are presented.

Extended Camassa-Holm Hierarchy and Conserved Quantities

An extension of the Camassa-Holm hierarchy is constructed in this paper. The conserved quantities of the hierarchy are studied and a recurrent formula for the integrals of motion is derived. - PACS numbers: 02.30.Ik; 05.45.Yv; 45.20.Jj; 02.30.Jr.

On the Integrability of a Class of Nonlinear Dispersive Wave Equations

Abstract We investigate the integrability of a class of 1+1 dimensional models describing non-linear dispersive waves in continuous media, e.g. cylindrical compressible hyperelastic rods, shallow water waves, etc. The only completely integrable cases coincide with the Camassa-Holm and Degasperis-Procesi equations.

Two-component CH system: inverse scattering, peakons and geometry

An inverse scattering transform method corresponding to a Riemann–Hilbert problem is formulated for CH2, the two-component generalization of the Camassa–Holm (CH) equation. As an illustration of the method, the multi-soliton solutions corresponding to the reflectionless potentials are constructed in terms of the scattering data for CH2.

N-wave interactions related to simple Lie algebras. ℤ2-reductions and soliton solutions

The reductions of the integrable N-wave type equations solvable by the inverse scattering method with the generalized Zakharov-Shabat systems L and related to some simple Lie algebra are analysed. The Zakharov-Shabat dressing method is extended to the case when is an orthogonal algebra. Several types of one-soliton solutions of the corresponding N-wave equations and their reductions are studied. We show that one can relate a (semi-)simple subalgebra of to each soliton solution. We illustrate our results by four-wave equations related to so(5) which find applications in Stokes-anti-Stokes wave generation.

Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples

The Lax pair formulation of the two-component Camassa-Holm equation (CH2) is generalized to produce an integrable multi-component family, CH(n,k), of equations with

On the dressing method for the generalised Zakharov–Shabat system

n

Empirical Balanced Truncation of Nonlinear Systems

components and