Sarbarish Chakravarty

Soliton solutions of the Kadomtsev-Petviashvili II equation

We study a general class of line-soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation by investigating the Wronskian form of its tau-function. We show that, in addition to the previously known line soliton solutions of KPII, this class also contains a large variety of multisoliton solutions, many of which exhibit nontrivial spatial interaction patterns. We also show that, in general, such solutions consist of unequal numbers of incoming and outgoing line solitons. From the asymptotic analysis of the tau function, we explicitly characterize the incoming and outgoing line solitons of this class of solutions. We illustrate these results by discussing several examples.

Integrable systems and reductions of the self-dual Yang–Mills equations

Many integrable equations are known to be reductions of the self-dual Yang–Mills equations. This article discusses some of the well known reductions including the standard soliton equations, the classical Painleve equations and integrable generalizations of the Darboux–Halphen system and Chazy equations. The Chazy equation, first derived in 1909, is shown to correspond to the equations studied independently by Ramanujan in 1916.

A self-dual Yang-Mills hierarchy and its reductions to integrable systems in

The self-dual Yang-Mills equations play a central role in the study of integrable systems. In this paper we develop a formalism for deriving a four dimensional integrable hierarchy of commuting nonlinear flows containing the self-dual Yang-Mills flow as the first member. We show that upon appropriate reduction and suitable choice of gauge group it produces virtually all well known hierarchies of soliton equations in 1+1 and 2+1 dimensions and can be considered as a “universal” integrable hierarchy. Prototypical examples of reductions to classical soliton equations are presented and related issues such as recursion operators, symmetries, and conservation laws are discussed.

1+1

In the previous papers (notably, Kodama Y 2004 J. Phys. A: Math. Gen. 37 11169–90, Biondini G and Chakravarty S 2006 J. Math. Phys. 47 033514), a large variety of line-soliton solutions of the Kadomtsev–Petviashvili II (KPII) equation was found. The line-soliton solutions are solitary waves which decay exponentially in the (x, y)-plane except along certain rays. In this paper, it is shown that those solutions are classified by asymptotic information of the solution as |y| → ∞. The present work then unravels some interesting relations between the line-soliton classification scheme and classical results in the theory of permutations.

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Four-wave mixing in wavelength-division-multiplexed soliton systems with damping and amplification is studied. An analytical model is introduced that explains the dramatic growth of the four-wave terms. The model yields a resonance condition relating the soliton frequency and the amplifier distance. It correctly predicts all essential features regarding the resonant growth of the four-wave contributions.

2+1

Multisoliton interactions are studied with an asymptotic expansion of the N-soliton solution in the limit of large frequency separation between the channels. In this limit the spectral distortion is small and the peak frequency shift of a channel is the sum of pairwise shifts as a result of interaction with other channels. These results, derived for collisions among an arbitrary number of channels, will be useful in estimating the limits on the minimum channel spacings and packet sizes for a wavelength-multiplexed optical communication system.

dimensions

A reduction of the self‐dual Yang–Mills (SDYM) equations is studied by imposing two space–time symmetries and by requiring that the connection one‐form belongs to a Lie algebra of formal matrix‐valued differential operators in an auxiliary variable. In this article, the scalar case and the canonical cases for 2×2 matrices are examined. In the scalar case, it is shown that the field equations can be reduced to the forced Burgers equation. In the matrix case, several well‐known 2+1 integrable equations are obtained. Also examined are certain transformation properties between the solutions of some of these 2+1 equations.

Classification of the line-soliton solutions of KPII

Abstract A new class of real, non-singular and rationally decaying potentials and eigenfunctions of the non-stationary Schrodinger equation and solutions of the KP I equation are constructed via binary Darboux transformations. These solutions are classified by the pole structure of the corresponding meromorphic eigenfunction and a set of integers including a quantity called the charge. The properties of the potential, eigenfunction and their relationship to the inverse scattering transform are discussed.

Four-wave mixing in wavelength-division-multiplexed soliton systems: damping and amplification

Collision-induced timing shifts in a wavelength-division multiplexed soliton system are computed when damping, amplification, filtering and positive dispersion management following the loss profile are included. A statistical analysis is presented which takes into account the resulting effect of the large number of collisions occurring in the fiber. Analytic expressions are derived for the root mean square timing jitter and the maximum length of error-free transmission with an arbitrary number of channels. An extensive analysis of system performance corresponding to situations with and without filters and/or dispersion management is carried out.

Multisoliton interactions and wavelength-division multiplexing

It is shown that classically known generalizations of the Chazy equation and Darboux–Halphen system are reductions of the self-dual Yang–Mills (SDYM) equations with an infinite-dimensional gauge algebra. The general ninth-order Darboux–Halphen system is reduced to a Schwarzian equation which governs conformal mappings of regions with piecewise circular sides. The generalized Chazy equation is shown to correspond to special mappings where either the triangles are equiangular or two of the angles are π/3.