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Integrable models of resonant interaction of two or more waves in 1+1 dimensions are known to be of applicative interest in several areas. Here we consider a system of three coupled wave equations which includes as special cases the vector…
A new description of the universal Whitham hierarchy in terms of a factorization problem in the Lie group of canonical transformations is provided. This scheme allows us to give a natural description of dressing transformations, string…
We study a simple nonlinear model defined on the cubic lattice. We propose a bilinearization scheme for the field equations and demonstrate that the resulting system is closely related to the well-studied integrable models, such as the…
Our purpose in this paper is to study when a planar differential system polynomial in one variable linearizes in the sense that it has an inverse integrating factor which can be constructed by means of the solutions of linear differential…
We present a hermitian matrix chain representation of the general solution of the Hirota bilinear difference equation of three variables. In the large N limit this matrix model provides some explicit particular solutions of continuous…
A class of two-dimensional systems of second-order ordinary differential equations is identified in which a system requires fewer Lie point symmetries than required to solve it. The procedure distinguishes among those which are…
We develop a systematic approach to deriving rational solutions and obtaining classification of their parameters for dressing chains of even N periodicity or equivalently $A^{(1)}_{N-1}$ invariant Painlev\'e equations. This construction…
In this paper, we mainly analyze the long-time asymptotics of high-order soliton for the Hirota equation. Two different Riemann-Hilbert representations of Darboux matrix with high-order soliton are given to establish the relationships…
Families of solutions to the field equations of the covariant BRST invariant effective action of the membrane theory are constructed. The equations are discussed in a double dimensional reduction, they lead to a nonlinear equation for a one…
The auxiliary linear problems are presented for all discretization levels of the Hirota-Ohta system. The structure of these linear problems coincides essentially with the structure of Nonlinear Schr\"odinger hierarchy. The squared…
A new approach for obtaining the transformations of solutions of nonlinear ordinary differential equations representable as the compatibility condition of the overdetermined linear systems is proposed. The corresponding transformations of…
We propose the Hirota bilinearization of the Fokas-Lenells derivative nonlinear Schrodinger equation with a non-vanishing background. The bilinear method is applied using an auxilary function to obtain the dark one soliton solution, dark…
A systematic framework is presented for the construction of hierarchies of soliton equations. This is realised by considering scalar linear integral equations and their representations in terms of infinite matrices, which give rise to all…
We further develop the method of dressing the boundary for the focusing nonlinear Schr\"odinger equation (NLS) on the half-line to include the new boundary condition presented by Zambon. Additionally, the foundation to compare the solutions…
A typical system of k difference (or differential) equations can be compressed, or folded into a difference (or ordinary differential) equation of order k. Such foldings appear in control theory as the canonical forms of the controllability…
Mixed type (bright-dark) soliton solutions of the integrable N-coupled nonlinear Schr{\"o}dinger (CNLS) equations with mixed signs of focusing and defocusing type nonlinearity coefficients are obtained by using Hirota's bilinearization…
Hirota's bilinear approach is a very effective method to construct solutions for soliton systems. In terms of this method, the nonlinear equations can be transformed into linear equations, and can be solved by using perturbation method. In…
A review of selected topics in Hirota's bilinear difference equation (HBDE) is given. This famous 3-dimensional difference equation is known to provide a canonical integrable discretization for most important types of soliton equations.…
A generalized derivative nonlinear Schr\"odinger equation, \ii q_t + q_{xx} + 2\ii \gamma |q|^2 q_x + 2\ii (\gamma-1)q^2 q^*_x + (\gamma-1)(\gamma-2)|q|^4 q = 0 , is studied by means of Hirota's bilinear formalism. Soliton solutions are…
We consider certain boundary conditions supporting soliton solutions in the generalized non-linear Schr\"{o}dinger equation (AKNS$_r$)\,($r=1,2$). Using the dressing transformation (DT) method and the related tau functions we study the…