In this paper, the Lakshmanan-Porsezian-Daniel (LPD) equation is considered. This integrable and admits Lax pair. The LPD generalization of nonlinear Schrodinger (NLS) described by Ablowitz-Kaup-Newell-Segur (AKNS) system. Using sine-cosine method hyperbolic tangent a variety new exact solutions are obtained. These methods effective tools for searching partial differential equations in mathemat...

We consider the 4D Martínez Alonso–Shabat equation E uty=uzuxy−uyuxz (also referred to as universal hierarchy equation) and using its known Lax pair construct two infinite-dimensional differential coverings over E. In these coverings, we give a complete description of Lie algebras nonlocal symmetries. particular, our results generalize ones obtained in Morozov Sergyeyev (2014) contain construct...

In this paper, the two-dimensional generalized nonlinear Schrödinger equations are introduced with Lax pair. The existence of pair defines integrability for partial differential equation, so integrable. Related to development was understanding that certain coherent structures called solitons play a basic role in phenomena as fluid mechanics, optics relativity, and lattice dynamics. Via Hirota b...

We study the Hankel determinant generated by Laguerre weight with jump discontinuities at $t_k, k=1,\cdots,m$. By employing ladder operator approach to establish Riccati equations, we show that $\sigma_n(t_1,\cdots,t_m)$, logarithmic derivative of $n$-dimensional determinant, satisfies a generalization $\sigma$-from Painlev\'{e} V equation. Through investigating Riemann-Hilbert problem for asso...

We review the subject of four dimensional anti-self-dual conformal structures with signature (+ +−−). Both local and global questions are discussed. Most of the material is well known in the literature and we present it in a way which underlines the connection with integrable systems. Some of the results e.g. the Lax pair characterisation of the scalar–flat Kähler condition and a twistor constr...

The Jacobs-Rebbi equation arises in many contexts where vortical motion in two-dimensional ideal media is investigated. Alternatively, it can be derived in the Abelian Higgs field theory. It is considered non-integrable and numerical solutions have been found, consisting of localised, robust vortices. We show in this work that the equation is integrable and provide the Lax pair. The exact solut...

We wish to explore a link between the Lax integrability of the q-Painlevé equations and the symmetries of the q-Painlevé equations. We shall demonstrate that the connection preserving deformations that give rise to the q-Painlevé equations may be thought of as elements of the groups of Schlesinger transformations of their associated linear problems. These groups admit a very natural lattice str...