Related papers: On new types of integrable 4-wave interactions
A class of elliptic curves with associated Lax matrices is considered. A family of dynamical systems on e(3) parametrized by polynomial a with above Lax matrices are constructed. Five cases from the family are selected by the condition of…
We consider integrability structures of the generalized Hunter--Saxton equation. In particular, we obtain the Lax representation with nonremovable spectral parameter, find local recursion operators for symmetries and cosymmetries, generate…
Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general…
In this paper, we carry out the algebraic study of integrable differential-difference equations whose field variables take values in an associative (but not commutative) algebra. We adapt the Hamiltonian formalism to nonabelian difference…
Two types of integrable coupled nonlinear Schrodinger (NLS) equations are derived by using Zakharov-Shabat (ZS) dressing method.The Lax pairs for the coupled NLS equations are also investigated using the ZS dressing method. These give new…
We start with a Riemann-Hilbert problem (RHP) related to a BD.I-type symmetric spaces $SO(2r+1)/S(O(2r-2s +1)\otimes O(2s))$, $s\geq 1$. We consider two Riemann-Hilbert problems: the first formulated on the real axis $\mathbb{R}$ in the…
We describe a family of 1+1 classical integrable space-discrete models of the Landau-Lifshitz type through the usage of ansatz for $U$-$V$ (Lax) pair with spectral parameter satisfying the semi-discrete Zakharov-Shabat equation. The ansatz…
Four new integrable evolutions equations with operator Lax pairs are found for an octonion variable. The method uses a scaling ansatz to set up a general polynomial form for the evolution equation and the Lax pair, using KdV and mKdV…
We obtain new integral representations, expressed as contour integrals in the complex Fourier plane, for the solution of fully nonhomogeneous interface problems for the linearized Cahn-Hilliard equation with arbitrary initial data on the…
We translate effectively our earlier quantum constructions to the classical language and using Yang-Baxterisation of the Faddeev-Reshetikhin-Takhtajan algebra are able to construct Lax operators and associated $r$-matrices of classical…
We study a new class of Fourier integral operators defined in R^N. Their symbols are allowed to satisfy a differential inequality with certain multi-parameter characteristic. We prove these operators of order -(N-1)/2 bounded from the…
We consider a matrix Riemann-Hilbert problem for the sextic nonlinear Schr\"{o}dinger equation with a non-zero boundary conditions at infinity. Before analyzing the spectrum problem, we introduce a Riemann surface and uniformization…
We present a new approach to construction of recursion operators for multidimensional integrable systems which have a Lax-type representation in terms of a pair of commuting vector fields. It is illustrated by the examples of the…
Simple deformations, with a parameter $\epsilon$, of classical $R$-matrices which follow from decomposition of appropriate Lie algebras, are considered. As a result nonstandard Lax representations for some well known integrable systems are…
The \hbar-dependent KP hierarchy is a formulation of the KP hierarchy that depends on the Planck constant \hbar and reduces to the dispersionless KP hierarchy as \hbar -> 0. A recursive construction of its solutions on the basis of a…
Short-wave perturbations in a relaxing medium, governed by a special reduction of the Ostrovsky evolution equation, and later derived by Whitham, are studied using the gradient-holonomic integrability algorithm. The bi-Hamiltonicity and…
We study the Lax integrability of a nonlinear system of two coupled second-order evolution equations introduced by Ibragimov and Shabat. For this system we find a zero-curvature representation with an essential parameter, construct an…
The multi-component nonlinear Schrodinger equation related to C.I=Sp(2p)/U(p) and D.III=SO(2p)/U(p)-type symmetric spaces with non-vanishing boundary conditions is solvable with the inverse scattering method (ISM). As Lax operator L we use…
We present a representation of skew-orthogonal polynomials of symplectic type ($\beta=4$) in terms of a matrix Riemann-Hilbert problem, for weights of the form ${\rm e}^{-V(z)}$ where $V$ is a polynomial of even degree and positive leading…
This work proposes a new way for handling obstacles to asymptotic integrability in perturbed nonlinear PDEs within the method of Normal Forms - NF - for the case of multi-wave solutions. Instead of including the whole obstacle in the NF,…