arXiv subjects

Nalini Joshi

Publications and source records attributed to Nalini Joshi.

66 records · Page 4Linked to original sources

arXiv2003

Exact Localized Solutions of Quintic Discrete Nonlinear Schrödinger Equation

We study a new quintic discrete nonlinear Schrödinger (QDNLS) equation which reduces naturally to an interesting symmetric difference equation of the form $ϕ_{n+1}+ϕ_{n-1}=F(ϕ_n)$. Integrability of the symmetric mapping is checked by singularity confinement criteria and growth properties. Some new exact localized solutions for integrable cases are presented for certain sets of parameters. Although these exact localized solutions represent only a small subset of the large variety of possible solutions admitted by the QDNLS equation, those solutions presented here are the first example of exact localized solutions of the QDNLS equation. We also find chaotic behavior for certain parameters of nonintegrable case.

nlin.PS↗

arXiv1999

On a Schwarzian PDE associated with the KdV Hierarchy

We present a novel integrable non-autonomous partial differential equation of the Schwarzian type, i.e. invariant under Möbius transformations, that is related to the Korteweg-de Vries hierarchy. In fact, this PDE can be considered as the generating equation for the entire hierarchy of Schwarzian KdV equations. We present its Lax pair, establish its connection with the SKdV hierarchy, its Miura relations to similar generating PDEs for the modified and regular KdV hierarchies and its Lagrangian structure. Finally we demonstrate that its similarity reductions lead to the {\it full} Painlevé VI equation, i.e. with four arbitary parameters.

solv-int↗

arXiv1999

Nonexistence results for the Korteweg-deVries and Kadomtsev-Petviashvili equations

We study characteristic Cauchy problems for the Korteweg-deVries (KdV) equation $u_t=uu_x+u_{xxx}$, and the Kadomtsev-Petviashvili (KP) equation $u_{yy}=\bigl(u_{xxx}+uu_x+u_t\bigr)_x$ with holomorphic initial data possessing nonnegative Taylor coefficients around the origin. For the KdV equation with initial value $u(0,x)=u_0(x)$, we show that there is no solution holomorphic in any neighbourhood of $(t,x)=(0,0)$ in ${\mathbb C}^2$ unless $u_0(x)=a_0+a_1x$. This also furnishes a nonexistence result for a class of $y$-independent solutions of the KP equation. We extend this to $y$-dependent cases by considering initial values given at $y=0$, $u(t,x,0)=u_0(x,t)$, $u_y(t,x,0)=u_1(x,t)$, where the Taylor coefficients of $u_0$ and $u_1$ around $t=0$, $x=0$ are assumed nonnegative. We prove that there is no holomorphic solution around the origin in ${\mathbb C}^3$ unless $u_0$ and $u_1$ are polynomials of degree 2 or lower.

solv-int↗

arXiv1999

Mappings preserving locations of movable poles: a new extension of the truncation method to ordinary differential equations

The truncation method is a collective name for techniques that arise from truncating a Laurent series expansion (with leading term) of generic solutions of nonlinear partial differential equations (PDEs). Despite its utility in finding Backlund transformations and other remarkable properties of integrable PDEs, it has not been generally extended to ordinary differential equations (ODEs). Here we give a new general method that provides such an extension and show how to apply it to the classical nonlinear ODEs called the Painleve equations. Our main new idea is to consider mappings that preserve the locations of a natural subset of the movable poles admitted by the equation. In this way we are able to recover all known fundamental Backlund transformations for the equations considered. We are also able to derive Backlund transformations onto other ODEs in the Painleve classification.

solv-int↗

arXiv1998

Bäcklund transformations for the second Painlevé hierarchy: a modified truncation approach

The second Painlevé hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well-known second Painlevé equation, P2. In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlevé analysis to ordinary differential equations. We extend these techniques in order to derive auto-Bäcklund transformations for the second Painlevé hierarchy. We also derive a number of other Bäcklund transformations, including a Bäcklund transformation onto a hierarchy of P34 equations, and a little known Bäcklund transformation for P2 itself. We then use our results on Bäcklund transformations to obtain, for each member of the P2 hierarchy, a sequence of special integrals.

solv-int↗

arXiv1997

Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: a Review and Extensions of Tests for the Painlevé Property

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 equations have the Painlevé property, that is, all solutions are single-valued around all movable singularities. In this expository article, we review methods for analysing such singularity structure. In particular, we describe well known techniques of nonlinear regular-singular-type analysis, i.e. the Painlevé tests for ordinary and partial differential equations. Then we discuss methods of obtaining sufficiency conditions for the Painlevé property. Recently, extensions of \textit{irregular} singularity analysis to nonlinear equations have been achieved. Also, new asymptotic limits of differential equations preserving the Painlevé property have been found. We discuss these also.

solv-int↗

arXiv1997

The Discrete Painlevé I Hierarchy

The discrete Painlevé I equation (dP$\rm_I$) is an integrable difference equation which has the classical first Painlevé equation (P$\rm_I$) as a continuum limit. dP$\rm_I$ is believed to be integrable because it is the discrete isomonodromy condition for an associated (single-valued) linear problem. In this paper, we derive higher-order difference equations as isomonodromy conditions that are associated to the same linear deformation problem. These form a hierarchy that may be compared to hierarchies of integrable ordinary differential equations (ODEs). We strengthen this comparison by continuum limit calculations that lead to equations in the P$\rm_I$ hierarchy. We propose that our difference equations are discrete versions of higher-order Painlevé equations.

solv-int↗

arXiv1997

The Second Painlevé Equation in the Large-Parameter Limit I: Local Asymptotic Analysis

In this paper, we find all possible asymptotic behaviours of the solutions of the second Painlevé equation $y''=2y^3+xy +α$ as the parameter $α\to\infty$ in the local region $x\llα^{2/3}$. We prove that these are asymptotic behaviours by finding explicit error bounds. Moreover, we show that they are connected and complete in the sense that they correspond to all possible values of initial data given at a point in the local region.

solv-int↗

arXiv1996

The Coalescence Limit of the Second PainlevÉ Equation

In this paper, we study a well known asymptotic limit in which the second Painlevé equation (P_II) becomes the first Painlevé equation (P_I). The limit preserves the Painlevé property (i.e. that all movable singularities of all solutions are poles). Indeed it has been commonly accepted that the movable simple poles of opposite residue of the generic solution of P_{II} must coalesce in the limit to become movable double poles of the solutions of P_I, even though the limit naively carried out on the Laurent expansion of any solution of P_{II} makes no sense. Here we show rigorously that a coalescence of poles occurs. Moreover we show that locally all analytic solutions of P_I arise as limits of solutions of P_{II}.

solv-int↗

arXiv1996

Complex Blow-Up in Burgers' Equation: an Iterative Approach

We show that for a given holomorphic noncharacteristic surface S in two-dimensional complex space, and a given holomorphic function on S, there exists a unique meromorphic solution of Burgers' equation which blows up on S. This proves the convergence of the formal Laurent series expansion found by the Painlevé test. The method used is an adaptation of Nirenberg's iterative proof of the abstract Cauchy-Kowalevski theorem.

solv-int↗

arXiv1996

A Local Asymptotic Analysis of the First Discrete Painlevé Equation as the Discrete Independent Variable Approaches Infinity

The first discrete Painlevé equation (dPI), which appears in a model of quantum gravity, is an integrable nonlinear nonautonomous difference equation which yields the well known first Painlevé equation (PI) in a continuum limit. The asymptotic study of its solutions as the discrete time-step $n\to\infty$ is important both for physical application and for checking the accuracy of its role as a numerical discretization of PI. Here we show that the asymptotic analysis carried out by Boutroux (1913) for PI as its independent variable approaches infinity can also be achieved for dPI as its discrete independent variable approaches the same limit.

solv-int↗

arXiv1996

The Radius of Convergence and the Well-Posedness of the Painlevé Expansions of the Korteweg-deVries equation

In this paper we obtain explicit lower bounds for the radius of convergence of the Painlevé expansions of the Korteweg-de-Vries equation around a movable singularity manifold ${\Cal S}$ in terms of the sup norms of the arbitrary functions involved. We use this estimate to prove the well-posedness of the singular Cauchy problem on ${\Cal S}$ in the form of continuous dependence of the meromorphic solution on the arbitrary data.

solv-int↗