With the aid of spectral gradient method Fuchssteiner, compatible pair Hamiltonian operators for coupled NLS hierarchy is rediscovered. This result enables us to construct a hierarchy, which contains vector generalization Fokas-Lenells system. The system shown be bi-Hamiltonian and possess Lax pair.

In this paper, we study the possible second order Lax operators for all the possible (1+1)-dimensional models with Schwarz variants and some special types of high dimensional models. It is shown that for every (1+1)-dimensional model and some special types of high dimensional models which possess Schwarz variants may have a second order Lax pair. The explicit Lax pairs for (1+1)-dimensional Kor...

We obtain 3 × 3 matrix Lax pairs for systems of ODEs that are solvable in terms of the fourth, fifth and sixth Painlevé equations by considering similarity reductions of the scattering Lax pair for the (2+1)-dimensional three-wave resonant interaction system. These results allow us to construct new 3× 3 Lax representations for the fourth and fifth Painlevé equations, together with the previousl...

In this article, we prove that we can introduce a small ~ parameter in the six Painlevé equations through their corresponding Lax pairs and Hamiltonian formulations. Moreover, we prove that these ~-deformed Lax pairs satisfy the Topological Type property proposed by Bergère, Borot and Eynard for any generic choice of the monodromy parameters. Consequently we show that one can reconstruct the fo...

We derive a Riemann--Hilbert representation for the solution of an integrable nonlinear evolution equation with $3 \times 3$ Lax pair. use derived to obtain formulas long-time asymptotics.

We study nonlocal symmetries of Plebanski’s second heavenly equation in an infinite-dimensional covering associated to a Lax pair with non-removable spectral parameter. show that all local sym...

Using the symmetry reductions of the self-dual Yang-Mills (SDYM) equations in (2+2) dimensions, we introduce new integrable equations which are nonautonomous versions of the chiral model in (2 + 1) dimensions, generalized nonlinear Schrödinger, Korteweg-de Vries, Toda lattice, Garnier and Euler-Arnold equations. The Lax pairs for all of these equations are derived by the symmetry reductions of ...

We derive the long-time asymptotics for solution of initial-boundary value problem coupled nonlinear Schrodinger equation whose Lax pair involves 3 × matrix in present paper. Based on a no...

As an application of Hirota bilinear method, perturbation expansion truncated at different levels is used to obtain exact soliton solutions to (2+1)-dimensional nonlinear evolution equation in much simpler way in comparison to other existing methods. We have derived bilinear form of nonlinear evolution equation and using this bilinear form, bilinear Backlund transformations and construction of ...