arXiv subjects

Sarbarish Chakravarty

Publications and source records attributed to Sarbarish Chakravarty.

11 recordsLinked to original sources

arXiv2022

Classification of KPI Lumps

A large family of nonsingular rational solutions of the Kadomtsev-Petviashvili (KP) I equation are investigated. These solutions are constructed via the Gramian method and are identified as points in a complex Grassmannian. Each solution is a traveling wave moving with a uniform background velocity but have multiple peaks which evolve at a slower time scale in the co-moving frame. For large times, these peaks separate and form well-defined wave patterns in the $xy$-plane. The pattern formation are described by the roots of well-known polynomials arising in the study of rational solutions of Painlevé II and IV equations. This family of solutions are shown to be described by the classical Schur functions associated with partitions of integers and irreducible representations of the symmetric group of $N$ objects. It is then shown that there exists a one-to-one correspondence between the KPI rational solutions considered in this article and partitions of a positive integer $N$.

nlin.SI↗

arXiv2021

Dynamics of KPI lumps

A family of nonsingular rational solutions of the Kadomtsev-Petviashvili (KP) I equation are investigated. These solutions have multiple peaks whose heights are time-dependent and the peak trajectories in the $xy$-plane are altered after collision. Thus they differ from the standard multi-peaked KPI simple $n$-lump solutions whose peak heights as well as peak trajectories remain unchanged after interaction.The anomalous scattering occurs due to a non-trivial internal dynamics among the peaks in a slow time scale. This phenomena is explained by relating the peak locations to the roots of complex heat polynomials. It follows from the long time asymptotics of the solutions that the peak trajectories separate as $O(\sqrt{|t|})$ as $|t| \to \infty$, and all the peak heights approach the same constant value corresponding to that of the simple 1-lump solution. Consequently, a multi-peaked $n$-lump solution evolves to a superposition of $n$ 1-lump solutions asymptotically as $|t| \to \infty$.

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arXiv2018

On a reduction of the generalized Darboux-Halphen system

The equations for the general Darboux-Halphen system obtained as a reduction of the self-dual Yang-Mills can be transformed to a third-order system which resembles the classical Darboux-Halphen system with a common additive terms. It is shown that the transformed system can be further reduced to a constrained non-autonomous, non-homogeneous dynamical system. This dynamical system becomes homogeneous for the classical Darboux-Halphen case, and was studied in the context of self-dual Einstein's equations for Bianchi IX metrics. A Lax pair and Hamiltonian for this reduced system is derived and the solutions for the system are prescribed in terms of hypergeometric functions.

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arXiv2017

The Chazy XII Equation and Schwarz Triangle Functions

Dubrovin [Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348] showed that the Chazy XII equation $y'''- 2yy''+3y'^2 = K(6y'-y^2)^2$, $K \in \mathbb{C}$, is equivalent to a projective-invariant equation for an affine connection on a one-dimensional complex manifold with projective structure. By exploiting this geometric connection it is shown that the Chazy XII solution, for certain values of $K$, can be expressed as $y=a_1w_1+a_2w_2+a_3w_3$ where $w_i$ solve the generalized Darboux-Halphen system. This relationship holds only for certain values of the coefficients $(a_1,a_2,a_3)$ and the Darboux-Halphen parameters $(α, β, γ)$, which are enumerated in Table 2. Consequently, the Chazy XII solution $y(z)$ is parametrized by a particular class of Schwarz triangle functions $S(α, β, γ; z)$ which are used to represent the solutions $w_i$ of the Darboux-Halphen system. The paper only considers the case where $α+β+γ<1$. The associated triangle functions are related among themselves via rational maps that are derived from the classical algebraic transformations of hypergeometric functions. The Chazy XII equation is also shown to be equivalent to a Ramanujan-type differential system for a triple $(\hat{P}, \hat{Q},\hat{R})$.

math.CA↗

arXiv2013

Construction of KP solitons from wave patterns

We often observe that waves on the surface of shallow water form complex web-like patterns. They are examples of nonlinear waves, and these patterns are generated by nonlinear interactions among several obliquely propagating waves. In this note, we discuss how to construct an exact soliton solution of the KP equation from such web-pattern of shallow water wave. This can be regarded as an "inverse problem" in the sense that by measuring certain metric data of the solitary waves in the given pattern, it is possible to construct an exact KP soliton solution which can describe the non-stationary dynamics of the pattern.

nlin.SI↗

arXiv2009

On the construction of the KP line-solitons and their interactions

The line-soliton solutions of the Kadomtsev--Petviashvili (KP) equation are investigated in this article using the tau-function formalism. In particular, the Wronskian and the Grammian forms of the tau-function are discussed, and the equivalence of these two forms are established. Furthermore, the interaction properties of two special types of 2-soliton solutions of the KP equation are studied in details.

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arXiv2009

Parameterizations of the Chazy equation

The Chazy equation $y''' = 2yy'' - 3y'^2$ is derived from the automorphic properties of Schwarz triangle functions $S(α, β, γ; z)$. It is shown that solutions $y$ which are analytic in the fundamental domain of these triangle functions, only correspond to certain values of $α, β, γ$. The solutions are then systematically constructed. These analytic solutions provide all known and one new parametrization of the Eisenstein series $P, Q, R$ introduced by Ramanujan in his modular theories of signature 2, 3, 4 and 6.

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arXiv2008

A generating function for the N-soliton solutions of the Kadomtsev-Petviashvili II equation

This work describes a classification of the $N$-soliton solutions of the Kadomtsev-Petviashvili II equation in terms of chord diagrams of N chords joining pairs of 2N points. The different classes of N-solitons are enumerated by the distribution of crossings of the chords. The generating function of the chord diagrams is expressed as a continued fraction, special cases of which are moment generating functions for certain kinds of $q$-orthogonal polynomials.

nlin.SI↗

arXiv2007

Classification of the line-soliton solutions of KPII

In the previous papers (notably, Y. Kodama, J. Phys. A 37, 11169-11190 (2004), and G. Biondini and S. Chakravarty, J. Math. Phys. 47 033514 (2006)), we found a large variety of line-soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation. The line-soliton solutions are solitary waves which decay exponentially in $(x,y)$-plane except along certain rays. In this paper, we show that those solutions are classified by asymptotic information of the solution as $|y| \to \infty$. Our study then unravels some interesting relations between the line-soliton classification scheme and classical results in the theory of permutations.

nlin.SI↗

arXiv2006

Elastic and inelastic line-soliton solutions of the Kadomtsev-Petviashvili II equation

The Kadomtsev-Petviashvili II (KPII) equation admits a large variety of multi-soliton solutions which exhibit both elastic as well as inelastic types of interactions. This work investigates a general class of multi-solitons which were not previously studied, and which do not in general conserve the number of line solitons after interaction. The incoming and outgoing line solitons for these solutions are explicitly characterized by analyzing the $τ$-function generating such solutions. A special family of $N$-soliton solutions is also considered in this article. These solutions are characterized by elastic soliton interactions, in the sense that amplitude and directions of the individual line solitons as $y\to\infty$ are the same as those of the individual line solitons as $y\to-\infty$. It is shown that the solution space of these elastic $N$-soliton solutions can be classified into $(2N-1)!!$ disjoint sectors which are characterized in terms of the amplitudes and directions of the $N$ line solitons.

nlin.SI↗

arXiv2005

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 previously known line-soliton solutions, this class also contains a large variety of new multi-soliton 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.

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