Related papers: On the Schlesinger transformations of the sixth Pa…
We find all non-abelian generalizations of $\text{P}_1 - \text{P}_6$ Painlev\'e systems such that the corresponding autonomous system obtained by freezing the independent variable is integrable. All these systems have isomonodromic Lax…
In a recent paper we presented a truncation-type method of deriving Backlund transformations for ordinary differential equations. This method is based on a consideration of truncation as a mapping that preserves the locations of a natural…
We consider the asymptotic behaviour of the second discrete Painlev\'{e} equation in the limit as the independent variable becomes large. Using asymptotic power series, we find solutions that are asymptotically pole-free within some region…
The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is strong evidence that all integrable…
We show the relation of the non-stationary difference equation proposed by one of the authors and the quantized discrete Painlev\'e VI equation. The five-dimensional Seiberg-Witten curve associated with the difference equation has a…
Discrete Painlev\'e equations are nonlinear, nonautonomous difference equations of second-order. They have coefficients that are explicit functions of the independent variable $n$ and there are three different types of equations according…
In this paper, we present new results regarding the orbital stability of solitary standing waves for the general fourth-order Schr\"odinger equation with mixed dispersion. The existence of solitary waves can be determined both as minimizers…
We present a large family of {\it{exact}} solitary wave solutions of the one dimensional Gross-Pitaevskii equation, with time-varying scattering length and gain/loss, in both expulsive and regular parabolic confinement regimes. The…
A starting point of this paper is a classification of quadratic polynomial transformations of the monodromy manifold for the 2x2 isomonodromic Fuchsian systems associated to the Painleve VI equation. Up to birational automorphisms of the…
This letter introduces the novel concept of Painlev\'e solitons -- waves arising from the interaction between Painlev\'e waves and solitons in integrable systems. Painlev\'e solitons may also be viewed as solitons propagating against a…
We study dynamics of solutions in the initial value space of the sixth Painlev\'e equation as the independent variable approaches zero. Our main results describe the repeller set, show that the number of poles and zeroes of general…
We derive the integrability conditions of nonautonomous nonlinear Schr$\rm\ddot o$dinger equations using the Lax Pair and Similarity Transformation methods. We present a comparative analysis of these integrability conditions with those of…
The problem of formulating self-consistent local and global stability exponents is shown to require global separation of variables. Posing the separation of variable problem, we see that many such separations are possible, but only one is…
Using the Painlev\'{e} analysis, we investigate the integrability properties of a system of two coupled nonlinear Schr\"{o}dinger equations that describe the propagation of orthogonally polarized optical waves in an isotropic medium.…
This paper extends deterministic notions of Strong Stability Preservation (SSP) to the stochastic setting, enabling nonlinearly stable numerical solutions to stochastic differential equations (SDEs) and stochastic partial differential…
The Painlev\'e equations are here connected to other classes of equations with the Painlev\'e Property (Ince's equations) by the same degeneracy procedure that connects the Painlev\'e equations (coalescence). These Ince's equations here are…
Given any two sequences of complex numbers, we establish simple relations between their binomial convolution and the binomial convolution of their individual binomial transforms. We employ these relations to derive new identities involving…
We introduce certain B\"acklund transformations for rational solutions of the Painlev\'e VI equation. These transformations act ona family of Painlev\'e VI tau functions. They are obtained from reducing the Hirota bilinear equations that…
In this paper we \emph{explicitly} compute the transformation that maps the generic second order differential equation $y''= f(x, y, y')$ to the Painlev\'e first equation $y''=6y^2+x$ (resp. the Painlev\'e second equation ${y''=2 y^{3}+yx+…
A Miura chain is a (closed) sequence of differential (or difference) equations that are related by Miura or B\"acklund transformations. We describe such a chain for the sixth Painlev\'e equation (\pvi), containing, apart from \pvi itself, a…