NONLINEAR AND MODERN MATHEMATICAL PHYSICS: Proceedings of the 2nd International Workshop

Preface
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Remarks on combinatorial aspects of the KP equation
View Description Hide DescriptionWe survey several results connecting combinatorics and Wronskian solutions of the KP equation, contextualizing the successes of a recent approach introduced by Kodama, et. al. We include the necessary combinatorial and analytical background to present a formula for generalized KP solitons, compute several explicit examples, and indicate how such a perspective could be used to extend previous research relating linesoliton solutions of the KP equation with Grassmannians.

Conservation laws for a KuramotoSivashinsky equation with dispersive effects
View Description Hide DescriptionIn this paper, we found a class of nonlinearly selfadjoint of a generalized KuramotoSivashinsky equation with dispersive effects which are not selfadjoint. From a general theorem on conservation laws proved by Ibragimov we obtain conservation laws for this equation.

Introduction to the Painlevé property, test and analysis
View Description Hide DescriptionThis short survey presents the essential features of what is called Painlevé analysis, i.e. the set of methods based on the singularities of differential equations in order to perform their explicit integration. Full details can be found in The Painlevé handbook or in various lecture notes posted on arXiv.

Desargues maps and their reductions
View Description Hide DescriptionWe present recent developments on geometric theory of the Hirota system and of the noncommutative discrete Kadomtsev–Petviashvili (KP) hierarchy adding also some new results which make the picture more complete. We pay special attention to multidimensional consistency of the Desargues maps and of the resulting nonlinear noncommutative systems. In particular, we show threedimensional consistency of the noncommutative KP map in its edge formulation. We discuss also relation of Desargues maps and quadrilateral lattice maps. We study from that point of view reductions of the Hirota system to discrete BKP and CKP systems presenting also a novel constraint which leads to the Miwa equations. By imposing periodicity reduction of the discrete KP hierarchy we obtain nonisospectral versions of the modified lattice Gel’fand–Dikii equations. To close the picture from below, we apply additional selfsimilarity constraint on the nonisospectral nonautonomous modified lattice Korteweg–de Vries system to recover known qPainleve equation of type A _{2}+A _{1}.

The concept of quasiintegrability
View Description Hide DescriptionWe show that certain field theory models, although nonintegrable according to the usual definition of integrability, share some of the features of integrable theories for certain configurations. Here we discuss our attempt to define a “quasiintegrable theory”, through a concrete example: a deformation of the (integrable) sineGordon potential. The techniques used to describe and define this concept are both analytical and numerical. The zerocurvature representation and the abelianisation procedure commonly used in integrable field theories are adapted to this new case and we show that they produce asymptotically conserved charges that can then be observed in the simulations of scattering of solitons.

Nonlinear Schrödinger equations for BoseEinstein condensates
View Description Hide DescriptionThe GrossPitaevskii equation, or more generally the nonlinear Schrödinger equation, models the BoseEinstein condensates in a macroscopic gaseous superfluid wavematter state in ultracold temperature. We provide analytical study of the NLS with L ^{2} initial data in order to understand propagation of the defocusing and focusing waves for the BEC mechanism in the presence of electromagnetic fields. Numerical simulations are performed for the twodimensional GPE with anisotropic quadratic potentials.

Nonlinear selfadjointness and conservation laws for a porous medium equation with absorption
View Description Hide DescriptionWe give conditions for a general porous medium equation to be nonlinear selfadjoint. By using the property of nonlinear selfadjointness we construct some conservation laws associated with classical and nonclassical generators of a porous medium equation with absorption.

Discrete soliton equation hierarchy
View Description Hide DescriptionIt is known that the continuous soliton equations such as the KdV equation have hierarchy structures. A higher order KdV equation constituting the hierarchy includes higher order derivative terms in the equation. The Toda equation is characterized by discreteness in the space dimension. In the hierarchy for the discrete soliton equation such as the Toda equation, neither spatial derivatives nor higher order spatial derivatives appear. Despite of this difference, we show both continuous and discrete soliton equation hierarchies starting with the noncommutative zero curvature equation. Especially, we show the explicit form of the higher order Toda equation.

Mathematical caricature of large waves
View Description Hide DescriptionThe KadomtsevPetviiashvili equation is considered as a mathematical caricature of large and rogue waves.

A discrete spectral problem and associated two types of integrable coupling systems
View Description Hide DescriptionBased on semidirect sums of Lie subalgebra B and G, two higherdimensional 4×4 and 6×6 matrix Lie algebra sμ(4) and sμ(6) are constructed with the help of the known Lie algebra A _{1}. Two hierarchies of integrable coupling nonlinear equations with three potentials are proposed, which are derived from coupled discrete fourbyfour and sixbysix matrix spectral problems. Moreover, the corresponding 3Hamiltonian forms are deduced for each lattice equation in the resulting hierarchy by means of the discrete variational identity and two strong symmetry operators of the resulting hierarchy are given. finally, we prove that the hierarchies of the resulting Hamiltonian equations are all Liouville integrable discrete Hamiltonian systems.

Integrable couplings and matrix loop algebras
View Description Hide DescriptionWe will discuss how to generate integrable couplings from zero curvature equations associated with matrix spectral problems. The key elements are matrix loop algebras consisting of block matrices with blocks of the same size or different sizes. Hamiltonian structures are furnished by applying the variational identity defined over semidirect sums of Lie algebras. Illustrative examples include integrable couplings of the AKNS hierarchy by using the irreducible representations V _{2} and V _{3} of the special linear Lie algebra sl(2, ).

Evolution method and “differential hierarchy” of colored knot polynomials
View Description Hide DescriptionWe consider braids with repeating patterns inside arbitrary knots which provides a multiparametric family of knots, depending on the “evolution” parameter, which controls the number of repetitions. The dependence of knot (super)polynomials on such evolution parameters is very easy to find. We apply this evolution method to study of the families of knots and links which include the cases with just two parallel and antiparallel strands in the braid, like the ordinary twist and 2strand torus knots/links and counteroriented 2strand links. When the answers were available before, they are immediately reproduced, and an essentially new example is added of the “double braid”, which is a combination of parallel and antiparallel 2strand braids. This study helps us to reveal with the full clarity and partly investigate a mysterious hierarchical structure of the colored HOMFLY polynomials, at least, in (anti)symmetric representations, which extends the original observation for the figureeight knot to many (presumably all) knots. We demonstrate that this structure is typically respected by the tdeformation to the superpolynomials.

Soliton solutions of coupled systems by improved (G′/G)expansion method
View Description Hide DescriptionThe paper witnesses the extension of improved (G′/G)expansion method to generate traveling wave solutions of coupled systems. The proposed algorithm is extremely effective and is tested on two very important systems (namely coupled Higgs and Maccari equations) in mathematical physics. Numerical results reflect complete compatibility of suggested scheme.

Integrability in nonperturbative QFT
View Description Hide DescriptionExact nonperturbative partition functions of coupling constants and external fields exhibit huge hidden symmetry, reflecting the possibility to change integration variables in the functional integral. In many cases this implies also some nonlinear relations between correlation functions, typical for the taufunctions of integrable systems. To a variety of old examples, from matrix models to SeibergWitten theory and AdS/CFT correspondence, now adds the ChernSimons theory of knot invariants. Some knot polynomials are already shown to combine into taufunctions, the search for entire set of relations is still in progress. It is already known, that generic knot polynomials fit into the set of Hurwitz partition functions – and this provides one more stimulus for studying this increasingly important class of deformations of the ordinary KP/Toda τfunctions.

Rogue waves and enhanced downshifting in a winddriven sea
View Description Hide DescriptionIn this paper we investigate the effects of nonlinear damping with and without linear damping/forcing in a sea state modeled by higherorder NLS in a two unstable mode regime. In particular, we are interested in how the linear term affects downshifting, rogue wave formation, and the number of rogue waves. We find that irreversible downshifting occurs when the nonlinear damping is the dominant damping effect. In particular, when only nonlinear damping is present, permanent downshifting occurs for all values of the nonlinear damping parameter β, appearing abruptly for larger values of β. We find including linear damping weakens the nonlinear damping effect of downshifting while linear forcing enhances the downshifting.

Numerical solution of a twoway diffusion equation
View Description Hide DescriptionWe consider a boundaryvalue problem involving a twoway diffusion equation of the form , where D(x) is positive, and h(x) changes sign when −1 ≤x≤ 1. A problem of this type arises in three sources: 1. A countercurrent separator, 2. Kinetics of electron scattering, and 3. Dynamics of runaway electrons in tokamaks. A numerical solution method is proposed to solve the problem.

Three higherdimensional Virasoro integrable models: Multiple soliton solutions
View Description Hide DescriptionIn this work, we study three higherdimensional Virasoro integrable models, namely the (3+1)dimensional NizhnikNovikovVeselov equation, the (3+1)dimensional breaking soliton equation, and a (3+1)dimensional extended breaking soliton equation. The three equations are among the Virasoro integrable models. We use the simplified form of the Hirota’s method to establish multiple soliton solutions for each equation. We determine the constraint conditions between the coefficients of the spatial variables to guarantee the existence of the multiple soliton solutions for each model.

Symmetric solutions of the dispersionless Toda hierarchy and associated conformal dynamics
View Description Hide DescriptionUnder certain reality conditions, a general solution to the dispersionless Toda lattice hierarchy describes deformations of simplyconnected plane domains with a smooth boundary. The solution depends on an arbitrary (real positive) function of two variables which plays the role of a density or a conformal metric in the plane. We consider in detail the important class of symmetric solutions characterized by the density functions that depend only on the distance from the origin and that are positive and regular in an annulus . We construct the dispersionless taufunction which gives formal local solution to the inverse potential problem and to the Riemann mapping problem and discuss the associated conformal dynamics related to viscous flows in the HeleShaw cell.