TY - JOUR
AB - Back in 1967, Clifford  Gardner, John Greene,  Martin Kruskal andRobert Miura published  a seminal paper in Physical Review Letters whichwas to become a cornerstone  in the theory of integrablesystems. In 2006, the authors of this paper received the AMS Steele Prize. Inthis award the AMS pointed out that`In applications of mathematics, solitons and their descendants (kinks,anti-kinks,instantons, and breathers) have entered and changed such diverse fieldsas nonlinear optics, plasma physics, and ocean, atmospheric, andplanetary sciences. Nonlinearity has undergone a revolution: from anuisance to be eliminated, to a new tool to be exploited.'From this discovery the modern theory of integrability bloomed,leading scientists to a deep understanding of many nonlinearphenomena which is by no means reachable by perturbation methods or otherprevious tools from linear theories.Nonlinear phenomena appear everywhere in nature, their description andunderstanding is therefore of great interest both from the theoretical andapplicative point of view. If a nonlinear phenomenon can be represented by anintegrable system then we have at our disposal a variety of tools to achieve abetter mathematical description of the phenomenon. This special issue is largelydedicated to investigations of nonlinear phenomena which are related to theconcept of integrability, either involving integrable systems themselves orbecause they use techniques from the theory of integrability.The idea of this special issue originated during the 18th edition of theNonlinear Evolution Equations and Dynamical Systems (NEEDS) workshop, held atIsola Rossa, Sardinia, Italy, 16–23 May 2009(http://needs-conferences.net/2009/). The issue benefits from the occasionoffered by the meeting, in particular by its mini-workshops programme, andcontains invited review papers and contributed papers. It is worth pointing outthat there was an open call for papers and all contributions were peer reviewedaccording to the standards of the journal. The selection of papers in this issueaims to bring togetherrecent developments and findings, even though it consists of only a fraction oftheimpressive developments in recent years which have affected a broad range offields, including the theory of special functions, quantum integrable systems,numerical analysis, cellular automata, representations of quantum groups,symmetries of difference equations, discrete geometry, among others.The special issue begins with four review papers:Integrable models in nonlinear optics and soliton solutionsDegasperis [1] reviews integrable models in nonlinear optics. He presents anumber of approximate models which are integrable and illustrates the linksbetween the mathematical and applicative aspects of the theory of integrabledynamical systems. In particular he discusses the recent impact ofboomeronic-type wave equations on applications arising in the context of theresonant interaction of three waves.Hamiltonian PDEs: deformations, integrability, solutionsDubrovin [2] presents classification results for systems of nonlinearHamiltonianpartial differential equations (PDEs) in one spatial dimension. In particular heuses a perturbative approach to the theory of integrability of these systems anddiscusses their solutions. He conjectures universality of the critical behaviourfor the solutions, where the notion of universality refers to asymptoticindependence of the structure of solutions (at the point of gradientcatastrophe) from the choice of generic initial data as well as from the choiceof a generic PDE.KP solitons in shallow waterKodama [3] presents a survey of recent studies on soliton solutions of theKadomtsev–Petviashvili (KP) equation. A large variety of exact solitonsolutionsof the KP equation are presented and classified. The study includes numericalanalysis of the stability of the found solution as well as numerical simulationsof the initial value problems which indicate that a certain class of initialwaves  approach asymptotically these exact solutions of the KP equation. Theauthor discusses an application of the theory to the problem of theresonant interaction of solitary waves appearing in the reflection of anobliquely incident wave onto a vertical wall, known as the Mach reflectionproblem in shallow water. A beautiful explanation of the problem was presentedin a swimming pool experiment during NEEDS 2009.Smooth and peaked solitons of the CH equationHolm and Ivanov [4] discuss the relations between smooth and peaked solitonsolutions for the Camassa–Holm (CH) shallow water wave equation in one spatialdimension. They first present the derivation of the soliton solution for the CHequation by means of inverse scattering transform (IST); the solution isobtained in a form that admits the peakon limit. The canonical Hamiltonianformulation of the CH equation in action-angle variables is recovered using thescattering data. The authors review some of the geometric properties of the CHequation and conclude their review with the higher dimensional generalization ofthe dispersionless CH equation, known as EPDiff.  They also consider thepossible extensions of their approach in three open problems.Regular contributions to this issue cover a wide range of topicsrelated to integrable systems. Let us briefly illustrate some of the topicscovered by this issue.One of the main topics is the study of hierarchies of integrable equations. Themultifaceted idea of integrability of a particular PDE includes an approachwhose aim is to find an infinite set of independentconserved quantities, much in the spirit of Liouville integrability in classicalmechanics. The existence of these conserved quantities in involution, or of thecorresponding infinite set of commuting symmetries, leads to an infinite set ofcommuting flows; i.e., to the construction of a hierarchy of compatible PDEswith respect to an infinite set of times. Obviously onecan generalize or adapt this construction to different settings like theintegro-differential, discrete or super-symmetric ones. The emphasis is usuallyto find auxiliary linear systems defining an infinite set of linear commutingflows whose solutions, if some asymptotic conditions are imposed, are namedwave or Baker–Akhiezer functions. These linear flows determine the so calledLax equations, another infinite set of commuting equations whose compatibilityleads to the so called Zakharov–Shabat system. An alternative description ofthe hierarchies is achieved with the use of the bilinear equations directlylinked with the tau-function description of the hierarchy.There are two paradigmatic integrable hierarchies, namely the KP and2-dimensional Toda lattice (2DTL). These hierarchies are treated within thisvolume in three contributions. In particular, Takasaki [5] reconsiders theextendedToda hierarchy of Carlet, Dubrovin and Zhang in the light of  Ogawa's 2 + 1Dextension of the 1D Toda hierarchy. It turns out that the former may be thoughtof as some sort of dimensional reduction of the latter. This explains thestructure of the bilinear formalism proposed by Milanov.  Carlet and Manas[6]study the 2-component KP and 2D Toda hierarchies and solve explicitly severalimplicit constraints present in the usual Lax formulation of the hierarchy, thusidentifying a set of free dependent variables for such hierarchies. Finally, theKP hierarchy is considered in the paper by Lin et al [7],  whichexplores theextended flows of a q-deformed modified KP hierarchy leading to theintroduction of self-consistent sources. By a combination of the dressing methodand the method of variation of constants, the authors are able through adressing approach to find a scheme for the construction of solutions of thecorresponding integrable equations with self-consistent sources.The study of dispersionless integrable hierarchies is an active field ofresearch, and this special issue includes two papers devoted to the subject.Konopelchenko et al [8] describe critical and degenerate criticalpoints of ascalar function which obeys the Euler–Poisson–Darboux equation in terms of thehodograph solutions of the dispersionless coupled Korteweg–de Vrieshierarchies.Finally, Bogdanov [9] considers 2-component integrable generalizations ofthedispersionless 2D Toda lattice hierarchy connected with non-Hamiltonian vectorfields, similar to the Manakov–Santini hierarchy generalizing the dKPhierarchy.He presents the simplest 2-component generalization of the dispersionless 2DTLequation, being its differential reduction analogous to the Dunajskiinterpolating system.Some papers in the issue are concerned with methods to construct solutions ofintegrable systems, while others place more emphasis on studying properties ofspecific solutions of applicative interest. Among the first approach, the paperby Kaup and van Gorder [10] describes perturbation theory applied to theInverseScattering Transform in 3x eigenvalue problems of Zakharov–Shabat'stype. Schiebold [11] studies a projection method to construct solutions oftheAblowitz–Kaup–Newell–Segur (AKNS) system, which enables her to write explicitN-soliton solutions in closed form. An example of the second kind is the paperby Biondini and Wang [12], who study in detail the behaviour of linesolitonsolutions of the 2DTL, describing their directions and amplitudes and also therichness of their interactions, which include resonant soliton interactions andweb structure.An important field of study in integrable systems relates to the singularitystructure of the solutions to nonlinear equations. When all movablesingularities are poles, the system is said to have the Painleve property.Thesolutions may be multivalued but they can be analytically continued tomeromorphic functions on the universal cover of the punctured Riemann sphere(the punctures being the fixed singularities) and the spectral curve is anaffine algebraic curve. Benes and Previato [13] study the connectionbetween the Painleve property andalgebras of differential operators, extending an approach initiated by Flaschka.Solutions to some integrable systems can be constructed in terms of analyticobjects associated to a spectral algebraic curve. It is therefore of interest tostudy the Riemann surfaces of algebraic functions, a program illustrated in thepaper by Braden and Northover [14], who have implemented some algorithmsfor thispurpose in a popular symbolic computation software. In the paper by Zhilinski[15],the critical points of the energy momentum map in classical Hamiltonian problemswith nontrivial monodromy are shown to form regular lattices. The quantummechanical counterpart has similar lattices for the joint spectrum of thecommuting observables. Some examples are given in which these points formspecial geometric patterns. Claeys [16] uses analytic techniques andRiemann–Hilbertproblems to study the asymptotic behaviour when x and t tend to infinity ofa solution to the second member of the Painleve I hierarchy, which arises inmulticritical string model theory and random matrix theory. This solution isconjectured to describe the universal asymptotics for Hamiltonian perturbationsof hyperbolic equations near the point of gradient catastrophe for theunperturbed equation.Darboux and Backlund transformations were born more than a century ago in thecontext of the geometric theory of surfaces. In the past few decades they havebecome a useful element in the theory of integrability, with applications indifferent guises. Typically, they appear in dressing methods that showhow to construct new interesting solutions from known simple ones. A few of thecontributed papers to the issue make use of these transformations as one oftheir fundamental objects. Liu et al [17] use iterated Darbouxtransformations toconstruct compact representations of the multi-soliton solutions to thederivative nonlinear Schroedinger (DNLS) equation. Ragnisco and Zullo [18]constructBacklund transformations for the trigonometric classical Gaudin magnet in thepartially anisotropic (xxz) case, identifying the subcase of transformationsthat preserve the real character of the variables. The recently discoveredexceptional polynomials are complete polynomial systems that satisfySturm–Liouville problems but differ from the classical families of Hermite,Laguerre and Jacobi. Gomez-Ullate et al [19] prove that thefamilies ofexceptional orthogonal polynomials known to date can be obtained from theclassical ones via a  Darboux transformation, which becomes a useful tool toderive some of their properties.Integrability in the context of classical mechanics is associated to theexistence of a sufficient number of conserved quantities, which allows sometimesan explicit integration of the equations of motion. This is the case for themotion of the Chaplygin sleigh, a rigid body motion on a fluid with nonholonomicconstraints studied in the paper by Fedorov and Garcia-Naranjo [20], whoderiveexplicit solutions and study their asymptotic behaviour. In connection withclassical mechanics, some techniques of KAM theory have been used by Procesi[21] toderive normal forms for the NLS equation in its Hamiltonian formulation andprove existence and stability of quasi-periodic solutions in the case ofperiodic boundary conditions.Algebraic and group theoretic aspects of integrability are covered in a numberof papers in the issue. The quest for symmetries of a system of differentialequations usually allows us to reduce the order or the number of equations or tofind special solutions possesing that symmetry, but algebraic aspects ofintegrable systems encompass a wide and rich spectrum of techniques, asevidenced by the following contributions. Muriel and Romero [22] perform asystematicstudy of all second order nonlinear ODEs that are linearizable by generalizedSundman and point transformations, showing that the two classes are inequivalentand providing an explicit characterization thereof. Lie algebras are alsoprominent in the work of Gerdjikov et al [23], where a class ofintegrable PDEsassociated to symmetric spaces is studied in detail. In their approach, systemsof nonlinear integrable PDEs are obtained as reductions of generic integrablesystems corresponding to Lax operators with matrix coefficients. The reductionhere is carried out using a reduction group which reflects symmetries of the Laxoperator. These symmetries allow also a characterization of the correspondingRiemann–Hilbert data. Habibullin [24] employs algebraic techniques tostudy discretechains of differential–difference equations that are Darboux integrable, i.e.that admit a certain number of nontrivial first integrals. Musso [25]provides aunified algebraic framework for the rational, trigonometric and elliptic Gaudinmodels. The results are achieved using a generalization of the Gaudin algebrasand of the so-called coproduct method. Odesskii and Sokolov [26] present aclassification of all infinite (1+1)-dimensional hydrodynamic-type chains ofshift one. They establish a one-to-one correspondence between integrable chainsand infinite triangular Gibbons-Tsarev (GT) systems and thus reduce theclassification problem to a description of all GT-systems.In Korff's paper [27] we find a study of various algebraic andcombinatorialstructures that emerge in the statistical vertex model with infinite spin, anintegrable  model associated to a certain quantum affine algebra. In the crystallimit, this model is connected with the WZNW model in conformal field theory.The motivation for some of the submitted contributions arises also from fieldtheories in theoretical physics. Ferreira et al [28] construct soliton solutionswith non-zero topological charges to the Skyrme–Faddeev model in Yang–Millstheory. Using techniques of differential geometry and complex analysis, Mantonand Rink [29] explore vortex solutions on hyperbolic surfaces extending anapproachby Witten. These solutions can be interpreted as self-dual SU(2) Yang–Millsfields on R4.Shah and Woodhouse [30] use the Penrose-Ward correspondence from twistortheory torelate generalized anti self-duality equations to certain  isomonodromicproblems whose solutions are expressed in terms of generalized hypergeometricfunctions.Applications of integrable systems and nonlinear phenomena in other fields arealso present in some of the papers.Kanna et al [31] study the collision of soliton solutions tocoherently coupledNLS equations using a variant of the Hirota bilinearization method. Their results have applications in pulse shaping innonlinear optics. Calogero et al [32] present examples of systems ofODEs withquadratic nonlinearities that could describe rate equations in chemicaldynamics. They derive explicit conditions on the parameters of the problem forwhich the  solutions are periodic and isochronous. Ablowitz and Haut [33]study themotion of large amplitude water waves with surface tension using asymptoticexpansions and providing a comparison with experimental results.This issue is the result of the collaboration of many individuals.We would like to thank the editors and staff of the Journal of Physics A:Mathematical and Theoretical for their enthusiastic support and efficient helpduring the preparation of this issue.A key factor has been the work of many anonymous referees who performedcareful analysis and scrutiny of the research papers submitted to this issue,often making remarks which helped to improve their quality and readability. Theycarried out dedicated, altruistic work with a very high standard and thisissue would not exist without their contribution.Finally, we would like to thank the authors who responded to our open call,sending us their most recent results  and sharing with us the enthusiasm andinterest for this fascinating field of research.We hope that this collection of papers will provide a good overview for anyoneinterested in recent developments in the field of integrability and nonlinearphenomena.[1] Integrable models in nonlinear optics and solitonsolutionsDegasperis A[2] Hamiltonian PDEs: deformations, integrability,solutionsDubrovin B[3] Smooth and peaked solitons of the CHequationHolm D D and Ivanov R I[4] KP solitons in shallow waterKodama Y[5] Two extensions of 1D Toda hierarchyTakasaki K[6] On the Lax representation ofthe 2-component KP and 2D Toda hierarchiesGuido Carlet and Manuel Manas[7] The q-deformed mKP hierarchywith self-consistent sources, Wronskian solutions and solitonsLin R L, Peng H and Manas  M[8] Hodograph solutions of the dispersionless coupledKdV hierarchies, critical points and the Euler–Poisson–Darboux equationKonopelchenko B, Martinez Alonso L and E Medina[9] Non-Hamiltonian generalizations of thedispersionless 2DTL hierarchyBogdanov L V[10] Squared eigenfunctions and theperturbation theory for the nondegenerate N x N operator: a generaloutlineKaup D J and Van Gorder R A[11] The noncommutative AKNS system: projection tomatrix systems, countable superposition and soliton-like solutionsSchiebold C[12] On the soliton solutions of thetwo-dimensional Toda latticeBiondini G and Wang D[13] Differential algebra of thePainleve propertyBenes G N and Previato E[14] Klein's curveBraden H W and Northover T P[15] Quantum monodromy and pattern formationZhilinskii B[16] A symptotics for a special solution to the secondmember of the Painleve I hierarchyClaeys T[17] Darboux transformation for a two-componentderivative nonlinear Schroedinger equationLing L and Liu Q P[18] Backlund transformations as exactintegrable time discretizations for the trigonometric Gaudin modelRagnisco O and Zullo F[19] Exceptional orthogonalpolynomials and the DarbouxtransformationGomez-Ullate D, Kamran N and Milson R[20] The hydrodynamic ChaplyginsleighFedorov Y N and Garcia-Naranjo L C[21] A normal form for beam and non-local nonlinearSchroedinger equationsProcesi M[22] Nonlocal transformations andlinearization ofsecond-order ordinary differential equationsMuriel and Romero J L[23] Reductions ofintegrable equations on A.III-typesymmetric spacesGerdjikov V S, Mikhailov A V and Valchev T I[24] OnDarboux-integrable semi-discrete chainsHabibullin I, Zheltukhina N and Sakieva A[25] Loop coproducts, Gaudin models and PoissoncoalgebrasMusso F[26] Classification of integrablehydrodynamic chainsOdesskii A V and Sokolov V V[27] Noncommutative Schur polynomials and the crystal limitof the Uq sl(2)-vertex modelKorff C[28] Axially symmetric solitonsolutions in aSkyrme–Faddeev-type model with Gies's extensionFerreira L A, Sawado N and Toda K[29] Vortices on hyperbolic surfacesManton N S and Rink N A[30] Multivariate hypergeometriccascades, isomonodromy problems and Ward ansatzeShah M R and Woodhouse N J M[31] Coherently coupledbright optical solitons and their collisionsKanna T, Vijayajayanthi M and Lakshmanan M[32] Isochronous rateequations describing chemical reactionsCalogero F, Leyvraz F and Sommacal M[33] Asymptotic expansions for solitarygravity-capillarywaves in two and three dimensionsAblowitz M J and Haut T S
TI - Integrability and nonlinear phenomena
JF - Journal of Physics A: Mathematical and Theoretical
DO - 10.1088/1751-8121/43/43/430301
DA - 2010-10-29
UR - https://www.deepdyve.com/lp/iop-publishing/integrability-and-nonlinear-phenomena-4xmf6o87y3
SP - 430301
VL - 43
IS - 43
DP - DeepDyve
ER -