Lax pairs are useful in studying nonlinear partial differential equations, although finding them is often difficult. A standard approach for finding them was developed by Wahlquist and Estabrook [1]. It was designed to apply to for equations with two independent variables and generally produces incomplete Lie algebras (called “prolongation structures”), which can be written as relations among c...

We suggest a generalized Lax pair on a Hermitian symmetric space to generate a new coupled higher-order nonlinear Schrödinger equation of a dual type which contains both bright and dark soliton equations depending on parameters in the Lax pair. Through the generalized ways of reduction and the scaling transformation for the coupled higher-order nonlinear Schrödinger equation, two integrable typ...

Using a contraction procedure, we obtain Toda theories and their structures, from affine Toda theories and their corresponding structures. By structures, we mean the equation of motion, the classical Lax pair, the boundary term for half line theories, and the quantum transfer matrix. The Lax pair and the transfer matrix so obtained, depend nontrivially on the spectral parameter. A Toda field th...

Based on the Gerstenhaber Theory it is clarified how operadic dynamics may be introduced. Operadic observables satisfy the Gerstenhaber algebra identities and their time evolution is governed by the Gerstenhaber-Lax or Gerstenhaber-Heisenberg equation. In this way one can describe classical and quantum time evolution of operations. The notion of an operadic Lax pair is introduced as well.

We study the C2 Ruijsenaars-Schneider(RS) model with interaction potential of trigonometric type. The Lax pairs for the model with and without spectral parameter are constructed. Also given are the involutive Hamiltonians for the system. Taking nonrelativistic limit, we obtain the Lax pair of C2 Calogero-Moser model.

We provide a Lax pair for the surfaces of Voss and Guichard, we show that such particular considered by Gambier are characterized third Painleve function.

We consider an integrable generalization of the nonlinear Schrödinger (NLS) equation that was recently derived by one of the authors using bi-Hamiltonian methods. This equation is related to the NLS equation in the same way that the Camassa Holm equation is related to the KdV equation. In this paper we: (a) Use the bi-Hamiltonian structure to write down the first few conservation laws. (b) Deri...

(Received 25 June 1998, revised version received 05 January 1999) The Lax pair for a derivative nonlinear Schrödinger equation proposed by Chen-Lee-Liu is generalized into matrix form. This gives new types of integrable coupled derivative nonlinear Schrödinger equations. By virtue of a gauge transformation, a new multi-component extension of a derivative nonlinear Schrödinger equation proposed ...

We propose a new integrable N = 2 supersymmetric Toda lattice hierarchy which may be relevant for constructing a supersymmetric one–matrix model. We define its first two Hamil-tonian structures, the recursion operator and Lax–pair representation. We provide partial evidence for the existence of an infinite-dimensional N = 2 superalgebra of its flows. We study its bosonic limit and introduce new...

Calogero-Moser models can be generalised for all of the finite reflection groups. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H3, H4, and the dihedral group I2(m), besides the well-known ones based on crystallographic root systems, namely those associated with Lie algebras. Universal Lax pair operators for all of the...