We present discrete forms of the Painlev\'e transcendental equations ${\mathit{P}}_{\mathrm{III}}$,${\mathit{P}}_{\mathrm{IV}}$, and ${\mathit{P}}_{\mathrm{V}}$ that complement list already known ${\mathit{P}}_{\mathrm{I}}$ ${\mathit{P}}_{\mathrm{II}}$. These, most likely integrable, nonautonomous mappings go over to usual in continuous limit, while autonomous limit we recover system belong integrable family Quispel et al. Finally, show satisfy same reduction relations as transcendents,...

By constructing the general six-parameter bright two-soliton solution of integrable coupled nonlinear Schr\"odinger equation (Manakov model) using Hirota method, we find that solitons exhibit certain novel inelastic collision properties, which have not been observed in any other $(1+1)$-dimensional soliton system so far. In particular, identify exciting possibility switching between modes by changing phase. However, standard elastic property is regained with specific choices parameters.

In this paper the results of a search for bilinear equations type P(Dx, Dt)F ⋅ F=0, which have three-soliton solutions, are presented. Polynomials up to order 8 been studied. addition previously known cases KP, BKP, and DKP their reductions, new polynomial P=DxDt(Dx2 +√3DxDt+Dt2) +aDx2+bDxDt +cDt2 has found. Its complete integrability is not known, but it solutions. Infinite sequences models with linear dispersion manifolds also found, e.g., P=DxMDtNDyP, if some powers odd, P=DxMDtN(Dx2...

In this paper the results of a search for pairs bilinear equations type Ai(Dx,Dt)F⋅F +Bi(Dx,Dt)G⋅F +Ci(Dx,Dt)G⋅G=0, i=1,2, which have standard three-soliton solutions, are presented. The freedom to rotate in (F,G) space is fixed by one-soliton ansatz F=1, G=en, then Bi determine dispersion manifold while Ai and Ci auxiliary functions. it assumed that B1 B2 even proportional, quadratic. As new results, B1=aD3x Dt +DtDy+b, A2=−C2=DxDt, generalizations sine–Gordon model B1=DxDt+a with family...

We present a number of second order maps, which pass the singularity confinement test commonly used to identify integrable discrete systems, but nevertheless are non-integrable. As more sensitive integrability test, we propose analysis complexity (``algebraic entropy'') map using growth degree its iterates: is associated with polynomial while generic exponential for chaotic systems.

In this paper (second in a series) [for part I, see J. Math. Phys. 30, 1732 (1987)] the search for bilinear equations having three-soliton solutions continues. This time pairs of type P1(Dx,Dt)F⋅G=0, P2(Dx,Dt)F⋅G=0, where P1 is an odd polynomial and P2 quadratic, are considered. The main results following new systems: P1=aDx7+bDx5 +Dx2Dt+Dy, P2=Dx2; P1=aDx3+bDt3 +Dy, P2=DxDt; P1=DxDtDy +aDx+bDt, P2=DxDt. addition to these, several models with linear dispersion manifolds were obtained, as before.

Higher-order and multicomponent generalizations of the nonlinear Schrödinger equation are important in various applications, e.g., optics. One these equations, integrable Sasa-Satsuma equation, has particularly interesting soliton solutions. Unfortunately, construction multisoliton solutions to this presents difficulties due its complicated bilinearization. We discuss briefly some previous attempts then give correct bilinearization based on interpretation as a reduction three-component...

We introduce a noncanonical ("new-time") transformation which exchanges the roles of coupling constant and energy in Hamiltonian systems while preserving integrability. In this way we can construct new integrable and, for example, explain observed duality between H\'enon-Heiles Holt models. It is shown that sometimes connect weak- full-Painlev\'e Hamiltonians. also discuss quantum integrability find origin deformation $\ensuremath{-}\frac{5}{72}{\ensuremath{\hbar}}^{2}{x}^{\ensuremath{-}2}$.

Classical integrability and quantum are compared for two degrees of freedom Hamiltonian systems. We use c-number representatives operators the Moyal bracket commutator. Three different cases found: (i) representative mechanical second invariant is identical to classical invariant, (ii) O(ℏ2) corrections needed in obtain (iii) also potential must be deformed by an term. Several examples from Henon–Heiles Holt families integrable potentials included.

In recent years there have been new insights into the integrability of quadrilateral lattice equations, i.e. partial difference equations which are natural discrete analogues integrable differential in 1+1 dimensions. scalar (i.e. single-field) case, now exist classification results by Adler, Bobenko and Suris (ABS) leading to some examples addition 'of KdV type' that were known since late 1970s early 1980s. this paper, we review construction soliton solutions for KdV-type use those...

For two-dimensional lattice equations the standard definition of integrability is that it should be possible to extend map consistently three dimensions, i.e., "consistent around a cube" (CAC).Recently Adler, Bobenko and Suris conducted search based on this principle, together with additional assumptions symmetry "the tetrahedron property".We present here results for CAC assuming also same properties, but not property.

In Part I soliton solutions to the ABS list of multi-dimensionally consistent difference equations (except Q4) were derived using connection between Q3 equation and NQC equations, then by reductions. that work, central role was played a Cauchy matrix. this work we use different approach, derive N-soliton following Hirota's direct constructive method. This leads Casoratians bilinear equations. We give here details for H-series Q1; results have been given earlier.

New results are reported for the ground-state configurations of Faddeev-Skyrme model. We started minimization runs on a large set initial states and found topologically new ground Hopf charges $Q=4$ 5, symmetry-breaking deformation $Q=6.$ The corresponding energies improve fit to Vakulenko-Kapitanskii behavior $E\ensuremath{\propto}|Q{|}^{3/4}.$

For well over a decade now, there has been growing interest in the integrability of discrete systems, i.e., systems that can be described by ordinary or partial difference equations allowing exact methods solution. Such turn up wide variety areas and have an extensive range applications: mathematical physics, numerical analysis, computer science, biology, economics, combinatorics, statistical theory special functions, asymptotic geometry, design, quantum field theory, so on.

A detailed analysis of the constant quantum Yang–Baxter equation Rk1k2j1j2 Rl1k3k1j3Rl2l3k2k3= Rk2k3j2j3 Rk1l3j1k3Rl1l2k1k2 in two dimensions is presented, leading to an exhaustive list its solutions. The set 64 equations for 16 unknowns was first reduced by hand several subcases which were then solved computer using Gröbner-basis methods. Each solution transformed into a canonical form (based on various trace matrices R) final elimination duplicates and subcases. If we use homogeneous...

In this paper the results of a search for complex bilinear equations with two-soliton solutions are presented. The following basic types discussed: (a) nonlinear Schrödinger equation B(Dx, ...)G⋅F=0, A(Dx,Dt) F⋅F=GG*, and (b) Benjamin–Ono P(Dx, ...)F⋅F*=0. It is found that existence not automatic, but introduces conditions like usual three- four-soliton conditions. was limited by degree A=2, P≤4. main following: (1) (iaD3x+DxDt +iDy+b)G⋅F=0, D2xF⋅F=GG*; (2) (D2x+aD2y +iDt+b)G⋅F =0, DxDy...