On N-wave type systems and their gauge equivalent

The class of nonlinear evolution equations - gauge equivalent to the N-wave equations related to the simple Lie algebra g are derived and analyzed. They are written in terms of the functions S(x,t) satisfying r= rank g nonlinear constraints. The corresponding Lax pairs and the time evolution of the scattering data are found. The Zakharov-Shabat dressing method is appropriately modified to construct their soliton solutions.Comment: 5 pages, LaTeX 2e, revised versio

Nonlinear integrable systems related to arbitrary space-time dependence of the spectral transform

We propose a general algebraic analytic scheme for the spectral transform of solutions of nonlinear evolution equations. This allows us to give the general integrable evolution corresponding to an arbitrary time and space dependence of the spectral transform (in general nonlinear and with non-analytic dispersion relations). The main theorem is that the compatibility conditions gives always a true nonlinear evolution because it can always be written as an identity between polynomials in the spectral variable $k$. This general result is then used to obtain first a method to generate a new class of solutions to the nonlinear Schroedinger equation, and second to construct the spectral transform theory for solving initial-boundary value problems for resonant wave-coupling processes (like self-induced transparency in two-level media, or stimulated Brillouin scattering of plasma waves or else stimulated Raman scattering in nonlinear optics etc...).Comment: 27 pages, Latex file, Submitted to Journ Math Phy

Fordy-Kulish models and spinor Bose-Einstein condensates

A three-component nonlinear Schrodinger-type model which describes spinor Bose-Einstein condensate (BEC) is considered. This model is integrable by the inverse scattering method and using Zakharov-Shabat dressing method we obtain three types of soliton solutions. The multi-component nonlinear Schrodinger type models related to symmetric spaces C.I Sp(4)/U(2) is studied.Comment: 8 pages, LaTeX, jnmp styl

Reductions and Conservation Laws of a Generalized Third-Order PDE via Multi-Reduction Method

In this work, we consider a family of nonlinear third-order evolution equations, where two arbitrary functions depending on the dependent variable appear. Evolution equations of this type model several real-world phenomena, such as diffusion, convection, or dispersion processes, only to cite a few. By using the multiplier method, we compute conservation laws. Looking for traveling waves solutions, all the the conservation laws that are invariant under translation symmetries are directly obtained. Moreover, each of them will be inherited by the corresponding traveling wave ODEs, and a set of first integrals are obtained, allowing to reduce the nonlinear third-order evolution equations under consideration into a first-order autonomous equation

On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be interpreted as ZCRs. In [arXiv:1303.3575], for any (1+1)-dimensional scalar evolution equation $E$, we defined a family of Lie algebras $F(E)$ which are responsible for all ZCRs of $E$ in the following sense. Representations of the algebras $F(E)$ classify all ZCRs of the equation $E$ up to local gauge transformations. In [arXiv:1804.04652] we showed that, using these algebras, one obtains necessary conditions for existence of a B\"acklund transformation between two given equations. The algebras $F(E)$ are defined in terms of generators and relations. In this paper we show that, using the algebras $F(E)$, one obtains some necessary conditions for integrability of (1+1)-dimensional scalar evolution PDEs, where integrability is understood in the sense of soliton theory. Using these conditions, we prove non-integrability for some scalar evolution PDEs of order $5$. Also, we prove a result announced in [arXiv:1303.3575] on the structure of the algebras $F(E)$ for certain classes of equations of orders $3$, $5$, $7$, which include KdV, mKdV, Kaup-Kupershmidt, Sawada-Kotera type equations. Among the obtained algebras for equations considered in this paper and in [arXiv:1804.04652], one finds infinite-dimensional Lie algebras of certain polynomial matrix-valued functions on affine algebraic curves of genus $1$ and $0$. In this approach, ZCRs may depend on partial derivatives of arbitrary order, which may be higher than the order of the equation $E$. The algebras $F(E)$ generalize Wahlquist-Estabrook prolongation algebras, which are responsible for a much smaller class of ZCRs.Comment: 29 pages; v2: consideration of zero-curvature representations with values in infinite-dimensional Lie algebras added. arXiv admin note: text overlap with arXiv:1303.3575, arXiv:1804.04652, arXiv:1703.0721