Combinatorial R Matrices for a Family of Crystals: B(1)n,D(1)n,A(2)2n,and D(2)n + 1 Cases

For coherent families of crystals of affine Lie algebras of type B⁽¹⁾_n, D⁽¹⁾_n, A⁽²⁾_2n, and D⁽²⁾_n + 1 we describe the combinatorial R matrix using column insertion algorithms for B, C, D Young tableaux. This is a continuation of previous work by the authors (2000, in “Physical Combinatorics” (M. Kashiwara and T. Miwa, Eds.), Birkhäuser, Boston).     

Periodic,escape and chaotic orbits in the Copenhagen and the (n + 1)-body ring problems

This paper studies the use of recent computational techniques in the numerical study of qualitative properties of two degrees of freedom Hamiltonian systems. Two particular problems, the Copenhagen and the (n + 1)-body ring problems, are studied by using Chaos Indicators, the Crash Test and by computing the skeleton of symmetric periodic orbits.     

A (2 + 1)-dimensional integrable hierarchy and its extending integrable model

A loop algebra is constructed, whose subalgebra is first used to present a Lax pair. By making use of the Tu scheme by Tu Guizhang, a generalized (2 + 1)-dimensional KN hierarchy is worked out. Further, based on the associated relations between the subalgebras in the above loop algebra, an extending integrable model of the generalized (2 + 1)-dimensional KN hierarchy as above is produced.     

The geometry of SO(n)\SO0(n, 1)

Consider the Lie group SO₀(n, 1) with the left-invariant metric coming from the Killing-Cartan form. The maximal compact subgroup SO(n) of the isometry group acts from the left and right. This paper studies the geometry of the quotient space of the homogeneous submersion SO₀(n, 1) → SO(n)\SO₀(n, 1). It is a cohomogeneity one manifold, which can be expressed as a warped product. Its group of isometries, geodesics, and sectional curvatures are calculated.     

Generalizations of some conjectures of Sun on the relations between N(a, b, c, d; n) and T(a, b, c, d; n)

The Ramanujan Journal - We derive some new finite sums involving the sequence $$s_{2}\left( n\right) ,$$ the sum-of-digits of the expansion of n in base 2. These functions allow us to generalize...     

Explicit N-fold Darboux transformation and multi-soliton solutions for the (1 + 1)-dimensional higher-order Broer–Kaup system

In this paper, we present a new N-fold Darboux transformations of the (1 + 1)-dimensional higher-order Broer–Kaup (HBK) system with the help of a gauge transformation of the spectral problem. As an application, new explicit (2N − 1)-soliton solutions of the (1 + 1)-dimensional HBK system are obtained. Both the N-fold Darboux transformation and (2N − 1)-soliton solutions can be written explicitly in terms of Vandermonde-like determinants which are remarkable compactness and transparency.     

A new Jacobi elliptic function rational expansion method and its application to (1 + 1)-dimensional dispersive long wave equation

With the aid of computerized symbolic computation, a new elliptic function rational expansion method is presented by means of a new general ansätz and is very powerful to uniformly construct more new exact doubly-periodic solutions in terms of rational formal Jacobi elliptic function of nonlinear evolution equations (NLEEs). As an application of the method, we choose a (1 + 1)-dimensional dispersive long wave equation to illustrate the method. As a result, we can successfully obtain the solutions found by most existing Jacobi elliptic function methods and find other new and more general solutions at the same time. Of course, more shock wave solutions or solitary wave solutions can be gotten at their limit condition.     

Harmonic Analysis on SO (n, ℂ)/SO (n−1, ℂ), n≥3

In this article we compute the Plancherel measure for SO(n, ℂ)/SO(n − 1, ℂ) following the approach of Van den Ban. This result is required in order to calculate the explicit decomposition of the oscillator representation wn for the dual pair G = SL(2, ℂ) × SO(n, ℂ) and to prove that every wn(G)-invariant Hilbert subspace of the space of tempered distributions decomposes multiplicity free.     

On generating (2 + 1)-dimensional hierarchies of evolution equations

Two isospectral problems are constructed with the help of a 6-dimensional Lie algebra. By using the Tu scheme, a (1 + 1)-dimensional expanding integrable couplings of the KdV hierarchy is obtained and the corresponding Hamiltonian structure is established. In addition, the 2-order matrix operators proposed by Tuguizhang are extended to the case where some 4-order matrices are given. Based on the extension, a new hierarchy of 2 + 1 dimensions is obtained by the Hamiltonian operator of the above (1 + 1)-dimensional case and the TAH scheme. The new hierarchy of 2 + 1 dimensions can be reduced to a coupled (2 + 1)-dimensional nonlinear equation and furthermore it can be reduced to the (2 + 1)-dimensional KdV equation which has important physics applications. The Hamiltonian structure for the (2 + 1)-dimensional hierarchy is derived with the aid of an extended trace identity. To the best of our knowledge, generating the (2 + 1)-dimensional equation hierarchies by virtue of the TAH scheme has not been studied in detail except to previous little work by Tu et al.     

Optimal solutions of a (3 + 1)-dimensional B-Kadomtsev-Petviashvii equation

We explored and specialized new Lie infinitesimals for the (3 + 1)-dimensional B-Kadomtsev-Petviashvii (BKP) using the commutation product, which results a system of nonlinear ODEs manually solved. Through two stages of Lie symmetry reduction, (3 + 1)-dimensional BKP equation is reduced to nonsolvable nonlinear ODEs using various combinations of optimal Lie vectors. Using the integration and Riccati equation methods, we investigate new analytical solutions for these ODEs. Back substituting to the original variables generates new solutions for BKP. Some selected solutions illustrated through three-dimensional plots.     

An improved generalized F-expansion method and its application to the (2 + 1)-dimensional KdV equations

An improved generalized F-expansion method is proposed to seek exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the (2 + 1)-dimensional KdV equations to illustrate the validity and advantages of the proposed method. Many new and more general non-travelling wave solutions are obtained, including single and combined non-degenerate Jacobi elliptic function solutions, soliton-like solutions, trigonometric function solutions, each of which contains two arbitrary functions.     

A new generalized algebra method and its application in the (2 + 1) dimensional Boiti–Leon–Pempinelli equation

In the present paper, some types of general solutions of a first-order nonlinear ordinary differential equation with six degree are given and a new generalized algebra method is presented to find more exact solutions of nonlinear differential equations. As an application of the method and the solutions of this equation, we choose the (2 + 1) dimensional Boiti Leon Pempinelli equation to illustrate the validity and advantages of the method. As a consequence, more new types and general solutions are found which include rational solutions and irrational solutions and so on. The new method can also be applied to other nonlinear differential equations in mathematical physics.     

Integrable couplings of the (2 + 1)-dimensional dispersive long wave hierarchy and its Hamiltonian structure

A higher loop algebra is constructed, from which the integrable couplings associated with the (2 + 1)-dimensional dispersive long wave hierarchy is obtained with the help of the (2 + 1) zero curvature equation generated from one of reduced equations of the self-dual Yang–Mills equations. Furthermore, the Hamiltonian structure of the integrable couplings is worked out by taking use of the variational identity.     

Soliton-like and periodic form solutions to (2 + 1)-dimensional Toda equation

In this paper, we present a further extended tanh method for constructing exact solutions to nonlinear difference-differential equation(s) (NDDEs) and Lattice equations. By using this method via symbolic computation system MAPLE, we obtain abundant soliton-like and period-form solutions to the (2 + 1)-dimensional Toda equation. Solitary wave solutions are merely a special case in one family. This method can also be used to other nonlinear difference differential equations.     

A modified variable-coefficient projective Riccati equation method and its application to (2 + 1)-dimensional simplified generalized Broer–Kaup system

A modified variable-coefficient projective Riccati equation method is proposed and applied to a (2 + 1)-dimensional simplified and generalized Broer–Kaup system. It is shown that the method presented by Huang and Zhang [Huang DJ, Zhang HQ. Chaos, Solitons & Fractals 2005; 23:601] is a special case of our method. The results obtained in the paper include many new formal solutions besides the all solutions found by Huang and Zhang.