A novel nonlinear evolution equation integrable by the inverse scattering method

A Backlund transformation for an evolution equation (ut+u ux)x+u=0 transformed into new coordinates is derived. An inverse scattering problem is formulated. The inverse scattering method has a third order eigenvalue problem. A procedure for finding the exact N-soliton solution of the Vakhnenko equation via the inverse scattering method is described

Special singularity function for continuous part of the spectral data in the associated eigenvalue problem for nonlinear equations

The procedure for finding the solutions of the Vakhnenko-Parkes equation by means of the inverse scattering method is described. The continuous spectrum is taken into account in the associated eigenvalue problem. The suggested special form of the singularity function for continuous part of the spectral data gives rise to the multimode solutions. The sufficient conditions are proved in order that these solutions become real functions. The interaction of the N periodic waves is studied. The procedure is illustrated by considering a number of example

Solutions associated with discrete and continuous spectrums in the inverse scattering method for the Vakhnenko-Parkes equation

In this paper the inverse scattering method is applied to the Vakhnenko-Parkes equation. We describe a procedure for using the inverse scattering transform to find the solutions that are associated with both the bound state spectrum and continuous spectrum of the spectral problem. The suggested special form of the singularity function gives rise to the multi-mode periodic solutions. Sufficient conditions are obtained in order that the solutions become real functions. The interaction of the solitons and multi-mode periodic waves is studied. The procedure is illustrated by considering a number of examples

The connection of the Degasperis-Procesi equation with the Vakhnenko equation

Travelling-wave solutions of the Degasperis-Procesi equation (DPE) are investigated. The solutions are characterized by two parameters. Hump-like, loop-like and coshoidal periodicwave solutions are found; hump-like, loop-like and peakon solitary-wave solutions are obtained as well. Hone and Wang showed a connection between the DPE and the Vakhnenko equation (VE). Comparing the solutions of the DPE and the VE, we observe that, for both equations at interaction of waves, there are three kinds of phaseshift that depend on the ratio of wave amplitudes. In particular, there is a case when two interacted waves have phaseshifts in the positive direction

Solitons in coupled Ablowitz-Ladik chains

A model of two coupled Ablowitz-Ladik (AL) lattices is introduced. While the system as a whole is not integrable, it admits reduction to the integrable AL model for symmetric states. Stability and evolution of symmetric solitons are studied in detail analytically (by means of a variational approximation) and numerically. It is found that there exists a finite interval of positive values of the coupling constant in which the symmetric soliton is stable, provided that its mass is below a threshold value. Evolution of the unstable symmetric soliton is further studied by means of direct simulations. It is found that the unstable soliton breaks up and decays into radiation, or splits into two counter-propagating asymmetric solitons, or evolves into an asymmetric pulse, depending on the coupling coefficient and the mass of the initial soliton.Comment: To appear in Phys. Lett.

Hamiltonian Structures for the Ostrovsky-Vakhnenko Equation

We obtain a bi-Hamiltonian formulation for the Ostrovsky-Vakhnenko equation using its higher order symmetry and a new transformation to the Caudrey-Dodd-Gibbon-Sawada-Kotera equation. Central to this derivation is the relation between Hamiltonian structures when dependent and independent variables are transformed.Comment: 13 page