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 introduct...

In this paper we aim to derive solutions for the SU($\mathcal{N}$) self-dual Yang-Mills (SDYM) equation with arbitrary $\mathcal{N}$. A set of noncommutative relations are introduced construct a matrix that can be reduced SDYM equation. It is shown these generated from two different Sylvester equations, which correspond Cauchy schemes (matrix) Kadomtsev-Petviashvili hierarchy and Ablowitz-Kaup-...

It is well known that fractional differential equations appeared more and more frequently in different research areas, such as fluid mechanics, viscoelasticity, biology, physics, engineering and other areas of science [1-30]. Considerable attention have been spent in recent years to develop techniques to look for solutions of nonlinear fractional partial differential equations (NFPDEs). Consequ...

Zero curvature formulations are given for the “dual hierarchies” of standard soliton equation hierarchies, recently introduced by Olver and Rosenau, including the physically interesting Fuchssteiner-Fokas-CamassaHolm hierarchy. PACS Numbers: 03.40.Kf, 47.20.Ky, 52.35.Sb Through the years since the discovery of the notion of “integrability” in PDEs, quite a number of integrable PDEs have been di...

The supercomplexification is a special method of N = 2 supersymmetrization the integrable equations in which bosonic sector can be reduced to complex version these equations. ...

C. de Graaf, ∗ C. Sousa, and R. Broer Department of Physical and Inorganic Chemistry, Universitat Rovira i Virgili Plaça Imperial Tàrraco 1, 43005 Tarragona, Spain Departament de Qúımica F́ısica and Parc Cient́ıfic, Universitat de Barcelona C/ Mart́ı i Franquès 1, 08028 Barcelona, Spain Theoretical Chemistry, Materials Science Centre, Rijksuniversiteit Groningen Nijenborgh 4, 9747 AG Groningen, Th...

The theory of soliton gas had been previously developed for unidirectional integrable dispersive hydrodynamics in which the properties are determined by overtaking elastic pairwise interactions between solitons. In this paper, we extend to gases bidirectional Eulerian systems where both head-on and collisions solitons take place. We distinguish two qualitatively different types gases: isotropic...

Zakharov-Shabat–Ablowitz-Kaup-Newel-Segur (ZS–AKNS) representation for Stokes-anti-Stokes stimulated Raman scattering (SRS) is proposed. Periodical waves, solitons and self-similarity solutions are derived. Transient and bright threshold solitons are discussed. December 29, 1994 Sofia

The concept of the nonholonomic deformation formulated recently for the AKNS family is extended to the Kaup-Newell class. Applying this construction we discover a novel mixed integrable hierarchy related to the deformed derivative nonlinear Schrödinger (DNLS) equation and found the exact soliton solutions exhibiting unusual accelerating motion for both its field and the perturbing functions. Ex...