# Nonlinear Sciences - Exactly Solvable and Integrable Systems Publications (50)

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## Nonlinear Sciences - Exactly Solvable and Integrable Systems Publications

A systematic framework is presented for the construction of hierarchies of soliton equations. This is realised by considering scalar linear integral equations and their representations in terms of infinite matrices, which give rise to all (2+1)- and (1+1)-dimensional soliton hierarchies associated with scalar differential spectral problems. The integrability characteristics for the obtained soliton hierarchies, including Miura-type transforms, {\tau}-functions, Lax pairs as well as soliton solutions, are also derived within this framework. Read More

We discuss various universality aspects of numerical computations using standard algorithms. These aspects include empirical observations and rigorous results. We also make various speculations about computation in a broader sense. Read More

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable $(1+1)$-dimensional PDEs. According to the preprint arXiv:1212.2199, for any given $(1+1)$-dimensional evolution PDE one can define a sequence of Lie algebras $F^p$, $p=0,1,2,3,\dots$, such that representations of these algebras classify all ZCRs of the PDE up to local gauge equivalence. Read More

The complete group classification problem for the class of (1+1)-dimensional $r$th order general variable-coefficient Burgers-Korteweg-de Vries equations is solved for arbitrary values of $r$ greater than or equal to two. We find the equivalence groupoids of this class and its various subclasses obtained by gauging equation coefficients with equivalence transformations. Showing that this class and certain gauged subclasses are normalized in the usual sense, we reduce the complete group classification problem for the entire class to that for the selected maximally gauged subclass, and it is the latter problem that is solved efficiently using the algebraic method of group classification. Read More

A nonautonomous version of the ultradiscrete hungry Toda lattice with a finite lattice boundary condition is derived by applying reduction and ultradiscretization to a nonautonomous two-dimensional discrete Toda lattice. It is shown that the derived ultradiscrete system has a direct connection to the box-ball system with many kinds of balls and finite carrier capacity. Particular solutions to the ultradiscrete system are constructed by using the theory of some sort of discrete biorthogonal polynomials. Read More

We investigate $n$-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. The classification of such systems is reduced to the projective classification of linear congruences of lines in $\mathbb{P}^{n+2}$ satisfying additional geometric constraints. Algebraically, the problem can be reformulated as follows: for a vector space $W$ of dimension $n+2$, classify $n$-tuples of skew-symmetric 2-forms $A^{\alpha} \in \Lambda^2(W)$ such that \[ \phi_{\beta \gamma}A^{\beta}\wedge A^{\gamma}=0, \] for some non-degenerate symmetric $\phi$. Read More

**Authors:**Olivier Marchal

The purpose of this article is to analyze the connection between Eynard-Orantin topological recursion and formal WKB solutions of a $\hbar$-difference equation: $\Psi(x+\hbar)=\left(e^{\hbar\frac{d}{dx}}\right) \Psi(x)=L(x;\hbar)\Psi(x)$ with $L(x;\hbar)\in GL_2( (\mathbb{C}(x))[\hbar])$. In particular, we extend the notion of determinantal formulas and topological type property proposed for formal WKB solutions of $\hbar$-differential systems to this setting. We apply our results to a specific $\hbar$-difference system associated to the quantum curve of the Gromov-Witten invariants of $\mathbb{P}^1$ for which we are able to prove that the correlation functions are reconstructed from the Eynard-Orantin differentials computed from the topological recursion applied to the spectral curve $y=\cosh^{-1}\frac{x}{2}$. Read More

We analyze several types of soliton solutions to a family of Tzitzeica equations. To this end we use two methods for deriving the soliton solutions: the dressing method and Hirota method. The dressing method allows us to derive two types of soliton solutions. Read More

We start with the Lax representation for the Kaup-Kupersschmidt equation (KKE). We We outline the deep relation between the scalar Lax operator and the matrix Lax operators related to Kac-Moody algebras. Then we derive the MKdV equations gauge equivalent to the KKE. Read More

In the paper we first construct rational solutions for the Nijhoff-Quispel-Capel (NQC) equation by means of bilinear method. These solutions can be transferred to those of Q3$_\delta$ equation in the Adler-Bobenko-Suris (ABS) list. Then making use of degeneration relation we obtain rational solutions for Q2, Q1$_\delta$, H3$_\delta$, H2 and H1. Read More

We extend the global existence result for the derivative NLS equation to the case when the initial datum includes a finite number of solitons. This is achieved by an application of the B\"{a}cklund transformation that removes a finite number of zeros of the scattering coefficient. By means of this transformation, the Riemann--Hilbert problem for meromorphic functions can be formulated as the one for analytic functions, the solvability of which was obtained recently. Read More

**Authors:**Percy Deift

We describe a list of open problems in random matrix theory and the theory of integrable systems that was presented at the conference Asymptotics in Integrable Systems, Random Matrices and Random Processes and Universality, Centre de Recherches Mathematiques, Montreal, June 7-11, 2015. We also describe progress that has been made on problems in an earlier list presented by the author on the occasion of his 60th birthday in 2005 (see [Deift P., Contemp. Read More

It is possible to understand whether a given BPS spectrum is generated by a relevant deformation of a 4D N=2 SCFT or of an asymptotically free theory from the periodicity properties of the corresponding quantum monodromy. With the aim of giving a better understanding of the above conjecture, in this paper we revisit the description of framed BPS states of four-dimensional relativistic quantum field theories with eight conserved supercharges in terms of supersymmetric quantum mechanics. We unveil aspects of the deep interrelationship in between the Seiberg-dualities of the latter, the discrete symmetries of the theory in the bulk, and quantum discrete integrable systems. Read More

A chain of transformations is found which relates one new integrable case of the generalized short pulse equation of Hone, Novikov and Wang [arXiv:1612.02481] with the sine-Gordon equation. Read More

We consider the weakly asymmetric exclusion process with $N=L/2$ particles on a periodic lattice of $L$ sites, and hopping rates $1$ and $q=1-\mu/\sqrt{L}$ respectively in the forward and in the backward direction. Using Bethe ansatz, we obtain a systematic perturbative expansion of the spectral gap near $\mu=0$ by solving order by order a simple functional equation. A key point is that when $\mu\to0$, Bethe roots at a distance $1/\sqrt{L}$ from the edge of the Fermi sea should not be considered as a continuum, but converge instead at large $L$ to the complex zeroes of $1+\mathrm{erf}(x)$ after a rescaling by $\sqrt{L}$. Read More

We present auto and hetero B\"acklund transformations of the nonholonomic Veselova system using standard divisor arithmetic on the hyperelliptic curve of genus two. As a by-product one gets two natural integrable systems on the cotangent bundle to the unit two-dimensional sphere whose additional integrals of motion are polynomials in the momenta of fourth order. Read More

Resorting to the characteristic polynomial of Lax matrix for the Dym-type hierarchy, we define a trigonal curve, on which appropriate vector-valued Baker-Akhiezer function and meromorphic function are introduced. Based on the theory of trigonal curve and three kinds of Abelian differentials, we obtain the explicit Riemann theta function representations of the meromorphic function, from which we get the algebro-geometric constructions for the entire Dym-type hierarchy Read More

We introduce the seven families of stochastic systems of interacting particles in one-dimension corresponding to the seven families of irreducible reduced affine root systems. We prove that they are determinantal in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single function called the spatio-temporal correlation kernel. For the three families A, B and D, we identify the systems of stochastic differential equations solved by these determinantal processes, which will be regarded as the elliptic extensions of the Dyson model. Read More

A special class of multicomponent NLS equations, generalizing the vector NLS and related to the {\bf BD.I}-type symmetric are shown to be integrable through the inverse scattering method (ISM). The corresponding fundamental analytic solutions are constructing thus reducing the inverse scattering problem to a Riemann-Hilbert problem. Read More

The well known elliptic discrete Painlev\'e equation of Sakai is constructed by a standard translation on the $E_8^{(1)}$ lattice, given by nearest neighbor vectors. In this paper, we give a new elliptic discrete Painlev\'e equation obtained by translations along next-nearest-neighbor vectors. This equation is a generic (8-parameter) version of a 2-parameter elliptic difference equation found by reduction from Adler's partial difference equation, the so-called Q4 equation. Read More

The advances in geometric approaches to optical devices due to transformation optics has led to the development of cloaks, concentrators, and other devices. It has also been shown that transformation optics can be used to gravitational fields from general relativity. However, the technique is currently constrained to linear devices, as a consistent approach to nonlinearity (including both the case of a nonlinear background medium and a nonlinear transformation) remains an open question. Read More

This paper considers the planar figure of a combinatorial polytope or tessellation identified by the Coxeter symbol k_{i,j}, inscribed in a conic, satisfying the geometric constraint that each octahedral cell has a centre. It is movable on account of some constraints being satisfied as a consequence of the others, and a close connection to the birational group found originally by Coble in the different context of invariants for sets of points in projective space, allows to specify precisely the subset of vertices that may be freely chosen. This gives a unified geometric view of certain integrable discrete systems in one, two and three dimensions. Read More

Remarkable mathematical properties of the integrable nonlinear Schr\"odinger equation (NLSE) can offer advanced solutions for the mitigation of nonlinear signal distortions in optical fibre links. Fundamental optical soliton, continuous and discrete eigenvalues of the nonlinear spectrum have already been considered for transmission of information in fibre-optic channels. Here we propose to apply signal modulation to the kernel of the Gelfand-Levitan-Marchenko equations (GLME) that offers the advantage of a relatively simple decoder design. Read More

Heckman introduced $N$ operators on the space of polynomials in $N$ variables, such that these operators form a covariant set relative to permutations of the operators and variables, and such that Jack symmetric polynomials are eigenfunctions of the power sums of these operators. We introduce the analogues of these $N$ operators for Macdonald symmetric polynomials, by using Cherednik operators. The latter operators pairwise commute, and Macdonald polynomials are eigenfunctions of their power sums. Read More

The Heisenberg Spin Chain system, in the continuum limit, can be represented by the non-linear Schr\"odinger equation through the Hashimoto map. Inhomogeneity induced through localizing the nearest neighbor interaction strength can also be mapped similarly to an integro -differential generalization of the non-linear Schrodinger system [J. Phys. Read More

Novel hybrid Ermakov-Painlev\'{e} IV systems are introduced and an associated Ermakov invariant is used in establishing their integrability. B\"{a}cklund transformations are then employed to generate classes of exact solutions via the linked canonical Painlev\'{e} IV equation. Read More

The reverse space-time (RST) Sine-Gordon, Sinh-Gordon and nonlinear Schr\"odinger equations were recently introduced and shown to be integrable infinite-dimensional dynamical systems. The inverse scattering transform (IST) for rapidly decaying data was also constructed. In this paper, IST for these equations with nonzero boundary conditions (NZBCs) at infinity is presented. Read More

In this paper, the unified bilinear forms of the AKNS hierarchy and the KdV hierarchy are presented from their recursion operators. Via the compatibility between soliton equations and their auto-Backlund transformations, the bilinear integrable hierarchies are discretized and the discrete recursion operators are obtained. All the discrete recursion operators converge to the original continuous forms after a standard limit. Read More

We derive the solutions of the boundary Yang-Baxter equation associated with a supersymmetric nineteen vertex model constructed from the three-dimensional representation of the twisted quantum affine Lie superalgebra $U_{q}[\mathrm{osp}\left(2|2\right)^{\left(2\right)}]\simeq U_{q}[C\left(2\right)^{\left(2\right)}]$. We found three classes of solutions. The type I solution is characterized by three boundary free-parameters and all elements of the corresponding reflection $K$-matrix are different from zero. Read More

We show how to solve initial-boundary value problems for integrable nonlinear differential-difference equations on a finite set of integers. The method we employ is the discrete analogue of the unified transform (Fokas method). The implementation of this method to the Ablowitz-Ladik system yields the solution in terms of the unique solution of a matrix Riemann-Hilbert problem, which has a jump matrix with explicit $(n,t)$-dependence involving certain functions referred to as spectral functions. Read More

We employ the Ablowitz-Ladik system as an illustrative example in order to demonstrate how to analyze initial-boundary value problems for integrable nonlinear differential-difference equations via the unified transform (Fokas method). In particular, we express the solutions of the integrable discrete nonlinear Schr\"{o}dinger and integrable discrete modified Korteweg-de Vries equations in terms of the solutions of appropriate matrix Riemann-Hilbert problems. We also discuss in detail, for both the above discrete integrable equations, the associated global relations, linearizable boundary conditions, and Dirichlet to Neumann maps. Read More

We study MNLS related to the D.III-type symmetric spaces. Applying to them Mikhailov reduction groups of the type $\mathbb{Z}_r\times \mathbb{Z}_2$ we derive new types of 2-component NLS equations. Read More

We study the growth of degrees in many autonomous and non-autonomous lattice equations defined by quad rules with corner boundary values, some of which are known to be integrable by other characterisations. Subject to an enabling conjecture, we prove polynomial growth for a large class of equations which includes the Adler-Bobenko-Suris equations and Viallet's $Q_V$ and its non-autonomous generalization. Our technique is to determine the ambient degree growth of the projective version of the lattice equations and to conjecture the growth of their common factors at each lattice vertex, allowing the true degree growth to be found. Read More

We investigate dynamics of probe particles moving in the near-horizon limit of (2N+1)-dimensional extremal Myers-Perry black hole with arbitrary rotation parameters. We observe that in the most general case with nonequal nonvanishing rotational parameters the system admits separation of variables in N-dimensional ellipsoidal coordinates. This allows us to find solution of the corresponding Hamilton-Jacobi equation and write down the explicit expressions of Liouville constants of motion. Read More

We study quantum aspects of the recently constructed doubly lambda-deformed sigma-models representing the effective action of two WZW models interacting via current bilinears. We show that although the exact beta-functions and current anomalous dimensions are identical to those of the lambda-deformed models, this is not true for the anomalous dimensions of generic primary field operators in accordance with the fact that the two models differ drastically. Our proofs involve CFT arguments, as well as effective sigma-model action and gravity calculations. Read More

In this paper, we construct the additional symmetries of the fermionic $(2N,2M)$-Toda hierarchy basing on the generalization of the $N{=}(1|1)$ supersymmetric two dimensional Toda lattice hierarchy. These additional flows constitute a $w_{\infty}\times w_{\infty}$ Lie algebra. As a Bosonic reduction of the $N{=}(1|1)$ supersymmetric two dimensional Toda lattice hierarchy and the fermionic $(2N,2M)$-Toda hierarchy, we define a new extended fermionic $(2N,2M)$-Toda hierarchy which admits a Bosonic Block type superconformal structure. Read More

In this article, we will construct the additional perturbative quantum torus symmetry of the dispersionless BKP hierarchy basing on the $W_{\infty}$ infinite dimensional Lie symmetry. These results show that the complete quantum torus symmetry is broken from the BKP hierarchy to its dispersionless hierarchy. Further a series of additional flows of the multicomponent BKP hierarchy will be defined and these flows constitute an $N$-folds direct product of the positive half of the quantum torus symmetries. Read More

We study the asymptotic behaviour of the solutions of the fifth Painlev\'e equation as the independent variable approaches zero and infinity in the space of initial values. We show that the limit set of each solution is compact and connected and, moreover, that any solution with the essential singularity at zero has an infinite number of poles and zeroes, and any solution with the essential singularity at infinity has infinite number of poles and takes value $1$ infinitely many times. Read More

We study the excitation spectrum of two-component delta-function interacting bosons confined to a single spatial dimension, the Yang-Gaudin Bose gas. We show that there are pronounced finite-size effects in the dispersion relations of excitations, perhaps best illustrated by the spinon single particle dispersion which exhibits a gap at $2k_F$ and a finite-momentum roton minimum. Such features occur at energies far above the finite volume excitation gap, vanish slowly as $1/L$ for fixed spinon number, and can persist to the thermodynamic limit at fixed spinon density. Read More

A modern notion of integrability is that of multidimensional consistency (MDC), which classically implies the coexistence of (commuting) dynamical flows in several independent variables for one and the same dependent variable. This property holds for both continuous dynamical systems as well as for discrete ones defined in discrete space-time. Possibly the simplest example in the discrete case is that of a linear quadrilateral lattice equation, which can be viewed as a linearised version of the well-known lattice potential Korteweg-de Vries (KdV) equation. Read More

We conjecture recurrence relations satisfied by the degrees of some linearizable lattice equations. This helps to prove linear growth of these equations. We then use these recurrences to search for lattice equations that have linear growth and hence are linearizable. Read More

We study a quantum spin chain invariant by the superalgebra $osp(1|2)$. We derived non-linear integral equations for the row-to-row transfer matrix eigenvalue in order to analyze its finite size scaling behaviour and we determined its central charge. We have also studied the thermodynamical properties of the obtained spin chain via the non-linear integral equations for the quantum transfer matrix eigenvalue. Read More

We consider the factorization problem of matrix symbols depending analytically on parameters on a closed contour (i.e. a Riemann--Hilbert problem). Read More

In recent work it was shown how recursive factorisation of certain QRT maps leads to Somos-4 and Somos-5 recurrences with periodic coefficients, and to a fifth-order recurrence with the Laurent property. Here we recursively factorise the 12-parameter symmetric QRT map, given by a second-order recurrence, to obtain a system of three coupled recurrences which possesses the Laurent property. As degenerate special cases, we first derive systems of two coupled recurrences corresponding to the 5-parameter multiplicative and additive symmetric QRT maps. Read More

A novel third order nonlinear evolution equation is introduced. It is connected, via Baecklund transformations, with the Korteweg-deVries (KdV), modified Korteweg-deVries (mKdV) equation and other third order nonlinear evolution equations. Hence, it is termed KdV-type equation. Read More

We present a novel approximation method which can predict the number of solitons asymptotically appearing under arbitrary rapidly decreasing initial conditions. The number of solitons can be estimated without integration of original soliton equations. As an example, we take the one-dimensional nonlinear Schrodinger equation and estimate the behaviors of scattering amplitude in detail. Read More

Integrable deformations of the hyperbolic and trigonometric ${\mathrm{BC}}_n$ Sutherland models were recently derived via Hamiltonian reduction of certain free systems on the Heisenberg doubles of ${\mathrm{SU}}(n,n)$ and ${\mathrm{SU}}(2n)$, respectively. As a step towards constructing action-angle variables for these models, we here apply the same reduction to a different free system on the double of ${\mathrm{SU}}(2n)$ and thereby obtain a novel integrable many-body model of Ruijsenaars--Schneider--van Diejen type that is in action-angle duality with the respective deformed Sutherland model. Read More

We consider the population of critical points generated from the critical point of the master function with no variables, which is associated with the trivial representation of the twisted affine Lie algebra $A^{(2)}_{2n}$. The population is naturally partitioned into an infinite collection of complex cells $\mathbb{C}^m$, where $m$ are some positive integers. For each cell we define an injective rational map $\mathbb{C}^m \to M(A^{(2)}_{2n})$ of the cell to the space $M(A^{(2)}_{2n})$ of Miura opers of type $A^{(2)}_{2n}$. Read More

The classes of electrovacuum Einstein - Maxwell fields (with a cosmological constant), which metrics admit an Abelian two-dimensional isometry group $\mathcal{G}_2$ with non-null orbits and electromagnetic fields possess the same symmetry, are considered. For these fields we describe the structures of so called "nondynamical degrees of freedom" which presence as well as the presence of a cosmological constant change (in a strikingly similar ways) the dynamical equations and destroy their known integrable structures. Modifications of the known reduced forms of Einstein - Maxwell equations -- the Ernst equations and self-dual Kinnersley equations in the presence of non-dynamical degrees of freedom are found and subclasses of fields with non-dynamical degrees of freedom are considered for : (I) vacuum metrics with cosmological constant, (II) vacuum space-times with isometry groups $\mathcal{G}_2$ which orbits do not admit the orthogonal 2-surfaces (none-orthogonally-transitive isometry groups) and (III) electrovacuum fields with more general structures of electromagnetic fields than in the known integrable cases. Read More

We show how solutions to a large class of Riccati evolutionary nonlinear partial differential equations can be generated from the corresponding linearized equations. The key is an integral equation analogous to the Marchenko equation, or more generally dressing transformation, in integrable systems. We show explicitly how this can be achieved for scalar partial differential equations with nonlocal quadratic nonlinearities. Read More