Bäcklund transformations for a matrix second Painlevé equation

Higher-order Painlevé equations are a topic of much current interest. Here we are interested in deriving auto-Bäcklund transformations for one particular kind of higher-order Painlevé equation, namely, a matrix Painlevé equation. The extension of a recently derived approach to deal with the matrix second Painlevé equation considered here represents a further demonstration of that approach's efficacy.     

Nonlinear Sigma Models in (1+2) Dimensions and an Infinite Number of Conserved Currents

In this Letter, we treat nonlinear sigma models such as the C P¹-model, Q P¹-model, etc. in 1+2 dimensions. For submodels of such models, we definitely construct an infinite number of nontrivial conserved currents. Our result is a generalization of Alvarez, Ferreira and Guillen.     

Consistent Composition of Bäcklund Transformations Produces Confined Maps

We prove that difference equations arising from consistent compositions of Bäcklund transformations of the continuous Painlevé equations possess the singularity confinement property.     

Q-Deformed KP Hierarchy: Its Additional Symmetries and Infinitesimal Bäcklund Transformations

We study the additional symmetries associated with the q-deformed Kadomtsev–Petviashvili (q-KP) hierarchy. After identifying the resolvent operator as the generator of the additional symmetries, the q-KP hierarchy can be consistently reduced to the so-called q-deformed constrained KP (q-cKP) hierarchy. We then show that the additional symmetries acting on the wave function can be viewed as infinitesimal Bäcklund transformations by acting the vertex operator on the tau-function of the q-KP hierarchy. This establishes the Adler–Shiota–van Moerbeke formula for the q-KP hierarchy.     

On Darboux–Bäcklund Transformations for the Q-Deformed Korteweg–de Vries Hierarchy

We study Darboux–Bäcklund transformations (DBTs) for the q-deformed Korteweg–de Vries hierarchy by using the q-deformed pseudodifferential operators. The elementary DBTs are triggered by the gauge operators constructed from the (adjoint) wave functions of the associated linear systems. Iterating these elementary DBTs, we obtain not only q-deformed Wronskian-type but also binary-type representations of the tau-function of the hierarchy.     

Gap Probabilities for Double Intervals in Hermitian Random Matrix Ensembles as τ-Functions – Spectrum Singularity Case

The probability for the exclusion of eigenvalues from an interval (–x,x) symmetrical about the origin for a scaled ensemble of Hermitian random matrices, where the Fredholm kernel is a type of Bessel kernel with parameter a(a generalisation of the sine kernel in the bulk scaling case), is considered. It is shown that this probability is the square of a -function, in the sense of Okamoto, for the Painlevé system P_III. This then leads to a factorisation of the probability as the product of two -functions for the Painlevé system dash. A previous study has given a formula of this type but involving dash systems with different parameters consequently implying an identity between products of -functions or equivalently sums of Hamiltonians.     

Linear problems and Bäcklund transformations for the Hirota-Ohta system

The auxiliary linear problems are presented for all discretization levels of the Hirota-Ohta system. The structure of these linear problems coincides essentially with the structure of Nonlinear Schrödinger hierarchy. The squared eigenfunction constraints are found which relate Hirota-Ohta and Kulish-Sklyanin vectorial NLS hierarchies.     

Differentially Rotating Disks of Dust

We present a three-parameter family of solutions to the stationary axisymmetric Einstein equations that describe differentially rotating disks of dust. They have been constructed by generalizing the Neugebauer—Meinel solution of the problem of a rigidly rotating disk of dust. The solutions correspond to disks with angular velocities depending monotonically on the radial coordinate; both decreasing and increasing behaviour is exhibited. In general, the solutions are related mathematically to Jacobi's inversion problem and can be expressed in terms of Riemann theta functions. A particularly interesting two-parameter subfamily represents Bäcklund transformations to appropriate seed solutions of the Weyl class.     

On time-like surfaces of positive constant Gaussian curvature and imaginary principal curvatures

We establish the Bäcklund transformation for the construction of time-like surfaces with positive Gaussian curvature and imaginary principal curvatures. The construction can be realized by algebraic algorithm via Darboux transformations.     

Minimal Model Fusion Rules From 2-Groups

The fusion rules for the (p,q)-minimal model representations of the Virasoro algebra are shown to come from the group in the following manner. There is a partition into disjoint subsets and a bijection between and the sectors of the (p,q)-minimal model such that the fusion rules correspond to where .     

The integrability of nonisospectral and variable-coefficient KdV equation with binary Bell polynomials

Binary Bell polynomials are extended to systematically construct bilinear formalism, bilinear Bäcklund transformations, Lax pairs and infinite conservation laws of the nonisospectral and variable-coefficient KdV equation in a quick and natural way. Moreover, the infinite conservation laws are local and obtained through directly decoupling binary Bell polynomials.     

New exact solutions and Bäcklund transformation for Boiti-Leon-Manna-Pempinelli equation

Based on the binary Bell polynomials, the bilinear form for the Boiti-Leon-Manna-Pempinelli equation is obtained. The new exact solutions are presented with an arbitrary function in y, and soliton interaction properties are discussed by the graphical analysis. Further, the bilinear Bäcklund transformation is derived by the binary Bell polynomials, and the corresponding Lax pair is obtained by linearizing the bilinear equation.     

The exact solution and integrable properties to the variable-coefficient modified Korteweg-de Vries equation

In this paper, a variable-coefficient modified Korteweg-de Vries (vc-mKdV) equation is investigated. With the help of symbolic computation, the N-soliton solution is derived through the Hirota method. Then the bilinear Bäcklund transformations and Lax pairs are presented. At last, we show some interactions of solitary waves.     

Surfaces of Demoulin: differential geometry, Bäcklund transformation and integrability

The surfaces of Demoulin constitute an important subclass of surfaces in projective differential geometry which arise in many seemingly unrelated geometric constructions. Analytically, they are described by a two-component system which coincides with the D₃⁽¹⁾ Toda lattice. We review some of the most important geometric properties of the Demoulin surfaces and construct a Bäcklund transformation which may be specialized to the well-known Bäcklund transformation for the Tzitzeica equation governing affine spheres in affine geometry.     

New Positive and Negative Hierarchies of Integrable Differential-Difference Equations and Conservation Laws

Two hierarchies of nonlinear integrable positive and negative lattice equations are derived from a discrete spectrak problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct infinite conservation laws about the positive hierarchy.     

Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws

We introduce a general framework for kinetic BGK models. We assume to be given a system of hyperbolic conservation laws with a family of Lax entropies, and we characterize the BGK models that lead to this system in the hydrodynamic limit, and that are compatible with the whole family of entropies. This is obtained by a new characterization of Maxwellians as entropy minimizers that can take into account the simultaneous minimization problems corresponding to the family of entropies. We deduce a general procedure to construct such BGK models, and we show how classical examples enter the framework. We apply our theory to isentropic gas dynamics and full gas dynamics, and in both cases we obtain new BGK models satisfying all entropy inequalities.     

An improved method for Darboux transformation for the coupled inhomogeneous nonlinear Schrödinger system from plasma physics and nonlinear optics

The coupled inhomogeneous nonlinear Schrödinger-type system which can be used to control soliton propagation and interaction in certain plasmas and optical fibers is investigated. An improved method for Darboux transformation (DT) is presented in more general forms by constructing an improved Γ$\Gamma$-Riccati-type Bäcklund transformation (Γ$\Gamma$-R BT). With the N$N$th-iterated Γ$\Gamma$-R BT or the N$N$th-iterated DT, which is a compact representation for the N$N$-soliton-like solutions and can generate a series of analytic solutions from a pair of the seed solutions through algebraic manipulations, the analytic one-/two-soliton-like solutions are provided. With the choice of parameters for the soliton solutions, the dynamical characteristics of the influences of the inhomogeneous parameters on the propagation of the soliton pulses are discussed graphically.