Related papers: Hirota bilinear forms of the AKNS($N$) systems

In this work we continue to study negative AKNS($N$) that is AKNS($-N$) system for $N=3,4$. We obtain all possible local and nonlocal reductions of these equations. We construct the Hirota bilinear forms of these equations and find…

We study two members of the multi-component AKNS hierarchy. These are multi-NLS and multi-MKdV systems. We derive the Hirota bilinear forms of these equations and obtain soliton solutions. We find all possible local and nonlocal reductions…

We first construct a $(2+1)$-dimensional negative AKNS hierarchy and then we give all possible local and (discrete) nonlocal reductions of these equations. We find Hirota bilinear forms of the negative AKNS hierarchy and give one- and…

Superpositions of hierarchies of integrable equations are also integrable. The superposed equations, such as the Hirota equations in the AKNS hierarchy, cannot be considered as new integrable equations. Furthermore if one applies the Hirota…

We introduce a class of reductions of the two-component KP hierarchy, which includes the Hirota-Ohta system hierarchy. The description of the reduced hierarchies is based on the Hirota bilinear identity and an extra bilinear relation…

In this paper, we give a procedure for discretizing recursion operators by utilizing unified bilinear forms within integrable hierarchies. To illustrate this approach, we present unified bilinear forms for both the AKNS hierarchy and the…

Starting from a multi-component AKNS system, we obtain new shifted nonlocal nonlinear Schr\"{o}dinger equations. We find 13 different shifted nonlocal nonlinear Schr\"{o}dinger equations with two-place nonlocalities and 10 shifted nonlocal…

Bilinearization of a given nonlinear partial differential equation is very important not only to find soliton solutions but also to obtain other solutions such as the complexitons, positons, negatons, and lump solutions. In this work we…

We consider multilinear generalization of the Hirota derivative, which serves as a building block for integrable solitonic hierarchies. 2 special integrable mutlilinear equations are shown to be splittable into pairs of bilinear operators,…

The N=2 supersymmetric KdV equations are studied within the framework of Hirota's bilinear method. For two such equations, namely $N=2, a=4$ and $N=2, a=1$ supersymmetric KdV equations, we obtain the corresponding bilinear formulations.…

This article presents a novel application of the Hirota bilinear formalism to the $N=2$ supersymmetric KdV and Burgers equations. This new approach avoids splitting N=2 equations into two $N=1$ equations. We use the super Bell polynomials…

We give the correct prescriptions for the terms involving the inverse of the derivative of the delta function, in the Hamiltonian structures of the AKNS and DNLS systems, in order for the Jacobi identities to hold. We also establish that…

An interpretation of Hirota bilinear relations for classical $\tau$ functions is given in terms of intertwining operators. Noncommutative example of $U_q(sl_2)$ is presented.

We present a bilinear Hirota representation of the N=2 supersymmetric extension of the Korteweg-de Vries equation. This representation is deduced using binary Bell polynomials, hierarchies and fermionic limits. We, also, propose a new…

We derive a set of bilinear functional equations of Hirota type for the partition functions of the $sl(2)$ related integrable statistical models defined on a random lattice. These equations are obtained as deformations of the Hirota…

In this note, we compute Hirota bilinear forms arising from both homogeneous and principal realization of vertex representations of 2-toroidal Lie algebras of type $A_l, D_l, E_l$.

We consider the resolution of the N=2 supersymmetric KdV equation with a=-2 (SKdV_{a=-2}) from two approaches, the group invariant method (or symmetry reduction) and the Hirota formalism. A bilinear form of the SKdV_{a=-2} equation is…

We establish the binary nonlinearization approach of the spectral problem of the super AKNS system, and then use it to obtain the super finite-dimensional integrable Hamiltonian system in supersymmetry manifold $\mathbb{R}^{4N|2N}$. The…

In this paper, we are going to solve nonlinear nonlocal reverse-time six-component six-order AKNS system. We used reverse-time reduction to reduce the coupled system to an integrable six-order NLS-type equation. Starting from the spectral…

We introduce multilinear operators, that generalize Hirota's bilinear $D$ operator, based on the principle of gauge invariance of the $\tau$ functions. We show that these operators can be constructed systematically using the bilinear $D$'s…