In [36], we found explicit Lax pairs for the soft edge of beta ensembles with even integer values of β. Using this general result, the case β = 6 is further considered here. This is the smallest even β, when the corresponding Lax pair and its relation to Painlevé II (PII) have not been known before, unlike cases β = 2 and 4. It turns out that again everything can be expressed in terms of the Ha...

The Hirota–Miwa equation (also known as the discrete KP equation, or the octahedron recurrence) is a bilinear partial difference equation in three independent variables. It is integrable in the sense that it arises as the compatibility condition of a linear system (Lax pair). The Hirota–Miwa equation has infinitely many reductions of plane wave type (including a quadratic exponential gauge tran...

We study the elliptic C n and BC n Ruijsenaars-Schneider models which is elliptic generalization of system given in [1]. The Lax pairs for these models are constructed by Hamiltonian reduction technology. We show that the spectral curves can be parameter-ized by the involutive integrals of motion for these models. Taking nonrelativistic limit and scaling limit, we verify that they lead to the s...

The discrete non-linear Schrödinger (NLS) model in the presence of an integrable defect is examined. The problem is viewed from a purely algebraic point of view, starting from the fundamental algebraic relations that rule the model. The first charges in involution are explicitly constructed, as well as the corresponding Lax pairs. These lead to sets of difference equations, which include partic...

Point-like Liouville integrable dynamical defects are introduced in the context of the Landau-Lifshitz and Principal Chiral (Faddeev-Reshetikhin) models. Based primarily on the underlying quadratic algebra we identify the first local integrals of motion, the associated Lax pairs as well as the relevant sewing conditions around the defect point. The involution of the integrals of motion is shown...

In this paper, we propose a recursive Darboux transformation in generalized form of focusing vector Nonlinear Schrödinger Equation (NLSE) known as the Manakov System. We apply to Lax-pairs system view generating Nth-order generalization rogue wave solutions with rule iteration. discuss from first- three-order generalizations while illustrating these features some 3D, 2D graphical depictions. il...

In this letter, taking the well known (2+1)-dimensional soliton systems, Davey-Stewartson (DS) model and the asymmetric Nizhnik-Novikov-Veselov (ANNV) model, as two special examples, we show that some types of lower dimensional chaotic behaviors may be found in higher dimensional soliton systems. Especially, we derive two famous chaotic system described by ordinary differential equation (ODE) s...

We consider N=2 supersymmetric extensions of the Camassa-Holm and HunterSaxton equations. We show that they admit geometric interpretations as Euler equations on the superconformal algebra of contact vector fields on the 1|2dimensional supercircle. We use the bi-Hamiltonian formulation to derive Lax pairs. Moreover, we present some simple examples of explicit solutions. As a byproduct of our an...

We study the elliptic C n and BC n Ruijsenaars-Schneider models which is elliptic generalization of system given in [1]. The Lax pairs for these models are constructed by Hamiltonian reduction technology. We show that the spectral curves can be parameter-ized by the involutive integrals of motion for these models. Taking nonrelativistic limit and scaling limit, we verify that they lead to the s...

We continue a recently developed systematic approach to the Bousinesq (Bsq) hierarchy and its algebro-geometric solutions. Our formalism includes a recursive construction of Lax pairs and establishes associated Burchnall-Chaundy curves, Baker-Akhiezer functions and Dubrovin-type equations for analogs of Dirichlet and Neumann divisors. The principal aim of this paper is a detailed theta function...