arXiv subjects

N. Joshi

Publications and source records attributed to N. Joshi.

29 records · Page 2Linked to original sources

arXiv2015

Stokes Phenomena in Discrete Painlevé I

In this study, we consider the asymptotic behaviour of the first discrete Painlev\{e} equation in the limit as the independent variable becomes large. Using an asymptotic series expansion, we identify two types of solutions which are pole-free within some sector of the complex plane containing the positive real axis. Using exponential asymptotic techniques, we determine the Stokes Phenomena effects present within these solutions, and hence the regions in which the asymptotic series expression is valid. From a careful analysis of the switching behaviour across Stokes lines, we find that the first type of solution is uniquely defined, while the second type contains two free parameters, and that the region of validity may be extended for appropriate choice of these parameters.

math-ph↗

arXiv2013

A Lax pair for the complete QRT mapping

QRT maps are translations on smooth biquadratic curves, also known as elliptic curves. Special cases of QRT maps are known to arise as compatibility conditions for an associated system of linear equations, known as a Lax pair. Here, we provide $2\times 2$ Lax pairs explicitly for the whole 18-parameter family of QRT maps.

nlin.SI↗

arXiv2011

Okamoto's space for the first Painlevé equation in Boutroux coordinates

We study the completeness and connectedness of asymptotic behaviours of solutions of the first Painlevé equation $\op{d}^2y/\op{d}x^2=6\, y^2+ x$, in the limit $x\to\infty$, $x\in\C$. This problem arises in various physical contexts including the critical behaviour near gradient catastrophe for the focusing nonlinear Schrödinger equation. We prove that the complex limit set of solutions is non-empty, compact and invariant under the flow of the limiting autonomous Hamiltonian system, that the infinity set of the vector field is a repellor for the dynamics and obtain new proofs for solutions near the equilibrium points of the autonomous flow. The results rely on a realization of Okamoto's space, i.e., the space of initial values compactified and regularized by embedding in $\C\Proj 2$ through an explicit construction of nine blow-ups.

math.CA↗

arXiv2009

On non-QRT Mappings of the Plane

We construct 9-parameter and 13-parameter dynamical systems of the plane which map bi-quadratic curves to other bi-quadratic curves and return to the original curve after two iterations. These generalize the QRT maps which map each such curve to itself. The new families of maps include those that were found as reductions of integrable lattices.

nlin.SI↗

arXiv2008

On the Linearization of the First and Second Painleve' Equations

We found Fuchs--Garnier pairs in 3X3 matrices for the first and second Painleve' equations which are linear in the spectral parameter. As an application of our pairs for the second Painleve' equation we use the generalized Laplace transform to derive an invertible integral transformation relating two its Fuchs--Garnier pairs in 2X2 matrices with different singularity structures, namely, the pair due to Jimbo and Miwa and the one found by Harnad, Tracy, and Widom. Together with the certain other transformations it allows us to relate all known 2X2 matrix Fuchs--Garnier pairs for the second Painleve' equation with the original Garnier pair.

math.CA↗

arXiv2007

On the Linearization of the Painleve' III-VI Equations and Reductions of the Three-Wave Resonant System

We extend similarity reductions of the coupled (2+1)-dimensional three-wave resonant interaction system to its Lax pair. Thus we obtain new 3x3 matrix Fuchs--Garnier pairs for the third and fifth Painleve' equations, together with the previously known Fuchs--Garnier pair for the fourth and sixth Painleve' equations. These Fuchs--Garnier pairs have an important feature: they are linear with respect to the spectral parameter. Therefore we can apply the Laplace transform to study these pairs. In this way we found reductions of all pairs to the standard 2x2 matrix Fuchs--Garnier pairs obtained by M. Jimbo and T. Miwa. As an application of the 3x3 matrix pairs, we found an integral auto-transformation for the standard Fuchs--Garnier pair for the fifth Painleve' equation. It generates an Okamoto-like Bäcklund transformation for the fifth Painleve' equation. Another application is an integral transformation relating two different 2x2 matrix Fuchs--Garnier pairs for the third Painleve' equation.

math.CA↗

arXiv2004

The Dirichlet Boundary Value Problem for Real Solutions of the first Painlevé Equation on Segments on Non-Positive Semi-Axis

We develop a qualitative theory for real solutions of the equation $y''=6y^2 -x$. In this work a restriction $x\leq0$ is assumed. An important ingredient of our theory is the introduction of several new transcendental functions of one, two, and three variables that describe different properties of the solutions. In particular, the results obtained allow us to completely analyse the Dirichlet boundary value problem $y(a)=y^0$, $y(b)=y_0$ for $a<b\leq0$.

math.CA↗

arXiv2002

Existence and Uniqueness of Tri-tronquée Solutions of the second Painlevé hierarchy

The first five classical Painlevé equations are known to have solutions described by divergent asymptotic power series near infinity. Here we prove that such solutions also exist for the infinite hierarchy of equations associated with the second Painlevé equation. Moreover we prove that these are unique in certain sectors near infinity.

math.CA↗

arXiv2000

Truncation-type methods and Backlund transformations for ordinary differential equations: the third and fifth Painleve equations

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 subset of the movable poles that the equation possesses. Here we apply this approach to the third and fifth Painleve equations. For the third Painleve equation we are able to obtain all fundamental Backlund transformations for the case where the parameters satisfy $γδ\neq0$. For the fifth Painleve equation our approach yields what appears to be all known Backlund transformations.

nlin.SI↗

arXiv1999

On the discrete and continuous Miura Chain associated with the Sixth Painlevé Equation

A Miura chain is a (closed) sequence of differential (or difference) equations that are related by Miura or Bäcklund transformations. We describe such a chain for the sixth Painlevé equation (\pvi), containing, apart from \pvi itself, a Schwarzian version as well as a second-order second-degree ordinary differential equation (ODE). As a byproduct we derive an auto-Bäcklund transformation, relating two copies of \pvi with different parameters. We also establish the analogous ordinary difference equations in the discrete counterpart of the chain. Such difference equations govern iterations of solutions of \pvi under Bäcklund transformations. Both discrete and continuous equations constitute a larger system which include partial difference equations, differential-difference equations and partial differential equations, all associated with the lattice Korteweg-de Vries equation subject to similarity constraints.

solv-int↗

arXiv1998

A Bilinear Approach to Discrete Miura Transformations

We present a systematic approach to the construction of Miura transformations for discrete Painlevé equations. Our method is based on the bilinear formalism and we start with the expression of the nonlinear discrete equation in terms of $τ$-functions. Elimination of $τ$-functions from the resulting system leads to another nonlinear equation, which is a ``modified'' version of the original equation. The procedure therefore yields Miura transformations. In this letter, we illustrate this approach by reproducing previously known Miura transformations and constructing new ones.

solv-int↗