Explicit forms of the off-shell Jost solutions for general central potentials

We derive explicit forms of the regular solutions and the Jost solutions off the energy shell, which satisfy the inhomogeneous Schrödinger equation. The used forms of the Yukawa-like and Gauss-like potentials are related to the two known integral representations of the Hankel functions. The explicit form of the introduced fully off-shell Jost functions enables us to write it in the alternative integral forms, which contain the Jost solutions or the regular solution.     

Boltzmann Equations For Mixtures of Maxwell Gases: Exact Solutions and Power Like Tails

We consider the Boltzmann equations for mixtures of Maxwell gases. It is shown that in certain limiting case the equations admit self-similar solutions that can be constructed in explicit form. More precisely, the solutions have simple explicit integral representations. The most interesting solutions have finite energy and power like tails. This shows that power like tails can appear not just for granular particles (Maxwell models are far from reality in this case), but also in the system of particles interacting in accordance with laws of classical mechanics. In addition, non-existence of positive self-similar solutions with finite moments of any order is proven for a wide class of Maxwell models.     

The geometry of peaked solitons and billiard solutions of a class of integrable PDE's

The purpose of this Letter is to investigate the geometry of new classes of soliton-like solutions for integrable nonlinear equations. One example is the class of peakons introduced by Camassa and Holm [10] for a shallow water equation. We put this equation in the framework of complex integrable Hamiltonian systems on Riemann surfaces and draw some consequences from this setting. Amongst these consequences, one obtains new solutions such as quasiperiodic solutions,n-solitons, solitons with quasiperiodic background, billiard, andn-peakon solutions and complex angle representations for them. Also, explicit formulas for phase shifts of interacting soliton solutions are obtained using the method of asymptotic reduction of the corresponding angle representations. The method we use for the shallow water equation also leads to a link between one of the members of the Dym hierarchy and geodesic flow onN-dimensional quadrics. Other topics, planned for a forthcoming paper, are outlined.Research supported in part by DOE CHAMMP and HPCC programs.Research partially supported by the Department of Energy, the Office of Naval Research and the Fields Institute for Research in the Mathematical Sciences.     

Vandermonde-like determinants’ representations of Darboux transformations and explicit solutions for the modified Kadomtsev-Petviashvili equation

Recently, the (2+1)-dimensional modified Kadomtsev-Petviashvili (mKP) equation was decomposed into two known (1+1)-dimensional soliton equations by Dai and Geng [H.H. Dai, X.G. Geng, J. Math. Phys. 41 (2000) 7501]. In the present paper, a systematic and simple method is proposed for constructing three kinds of explicit N-fold Darboux transformations and their Vandermonde-like determinants’ representations of the two known (1+1)-dimensional soliton equations based on their Lax pairs. As an application of the Darboux transformations, three explicit multi-soliton solutions of the two (1+1)-dimensional soliton equations are obtained; in particular six new explicit soliton solutions of the (2+1)-dimensional mKP equation are presented by using the decomposition. The explicit formulas of all the soliton solutions are also expressed by Vandermonde-like determinants which are remarkably compact and transparent.     

Focusing of electromagnetic waves into a uniaxial crystal

We derive integral representations suitable for studying the focusing of electromagnetic waves through a plane interface into a uniaxial crystal. To that end we start from existing exact solutions for the transmitted fields due to an arbitrary three-dimensional (3D) wave that is incident upon a plane interface separating two uniaxial crystals with arbitrary orientation of the optical axis in each medium. Then we specialize to the case in which the medium of the incident wave is isotropic and derive explicit expressions for the dyadic Green's functions associated with the transmitted fields as well as integral representations suitable for asymptotic analysis and efficient numerical evaluation. Relevant integral representations for focused 3D electromagnetic waves are also given. Next we consider the special case in which (i) the incident field is a two-dimensional (2D) TM wave and (ii) the optical axis in the crystal lies in the plane of incidence, implying that we have a 2D vectorial problem, and derive dyadic Green's functions, integral representations suitable for asymptotic and numerical treatment, and integral representations for focused TM fields. Numerical results for focused 2D TM fields based on these integral representations as well as corresponding experimental results will be presented in forthcoming papers.     

Explicit algebraic Reynolds stress model for compressible flow turbulence

Algebraic Reynolds stress model (ARSM) is often employed in practical turbulent flow simulations. Most of previous works on ARSM have been carried out for incompressible flows. In the present paper, a new ARSM model is suggested for compressible flows. The model adopts a compressibility factor function involving the turbulent Mach number and the gradient Mach number. Compared to incompressible flow, explicit solution for ARSM for compressible flow can hardly be obtained due to dilatation terms. We propose approximate representations for these dilatation-related terms to obtain an explicit procedure for compressible flow turbulence. The model is applied to compressible mixing layer, supersonic flat-plate boundary and planar supersonic wake flow. It is found that the model works very well yielding results that are in good agreement with the DNS and the experimental data.     

A note on the quasi-periodic solutions of the modified Boussinesq hierarchy

Based on the theory of trigonal curve and the properties of three kinds of the Abel differentials on it, we deduce the explicit theta function representations of the Baker-Akhiezer function and the meromorphic function associated with the modified Boussinesq hierarchy. The modified Boussinesq flows are straightened using the Abel map and the Lagrange interpolation formula. The explicit theta function representations of solutions for the entire modified Boussinesq hierarchy are constructed with the aid of the asymptotic properties and the algebro-geometric characters of the meromorphic function.     

Algebro-geometric solutions for the two-component Hunter-Saxton hierarchy

This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Hunter-Saxton (HS2) hierarchy through an algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism, the hyperelliptic curve with finite number of genus, the Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the algebro-geometric solutions are obtained for the entire HS2 hierarchy.     

Explicit Solutions and Bifurcations for a Class of Generalized Boussinesq Wave Equation

In this paper,the generalized Boussinesq wave equation u tt-uxx+a(um) xx+buxxxx=0 is investigated by using the bifurcation theory and the method of phase portraits analysis.Under the different parameter conditions,the exact explicit parametric representations for solitary wave solutions and periodic wave solutions are obtained.     

Yang-Baxterization and New Solutions Associated with SU(2) Tensorial Space Vj⊗V1/2*

By applying extended diagrammatic technique to SU(2) tensorial space V^j⊗V^1/2 new solutions of the correspondent braid group representations and their Yang-Baxterization are established. The explicit forms of the new solutions of Yang-Baxter equation for j = 1 are given. Through this example the general feature of this new type of solutions is figured out.     

Darboux Transformations and New Explicit Solutions for a Blaszak-Marciniak Three-Field Lattice Equation

In this paper, the two different Darboux transformations for a Blaszak-Marciniak (BM) three-field lattice equation are constructed. As the applications of the obtained Darboux transformations, new explicit solutions for the BM lattice are given. We also discuss some properties for these new explicit solutions. Our analysis shows that the explicit solutions possess new characters.     

Bifurcations of Exact Traveling Wave Solutions for(2+1)-Dimensional HNLS Equation

For the (2+1)-Dimensional HNLS equation, what are the dynamical behavior of its traveling wave solutions and how do they depend on the parameters of the systems? This paper will answer these questions by using the methods of dynamical systems. Ten exact explicit parametric representations of the traveling wave solutions are given.     

Exact Eternal Solutions of the Boltzmann Equation

We construct two families of self-similar solutions of the Boltzmann equation in an explicit form. They turn out to be eternal and positive. They do not possess finite energy. Asymptotic properties of the solutions are also studied.     

Graded Lie algebras,representation theory,integrable mappings and systems. Non-abelian case

The exactly integrable systems connected with a semisimple series A for arbitrary grading are presented in an explicit form. Their general solutions are expressed in terms of the matrix elements of various fundamental representations of the An groups. The simplest example of such systems is the generalized Toda chain with matrices of arbitrary dimensions in each point of the lattice.     

Etiology of IRF models

We show that a class of 2D statistical mechanics models known as IRF models can be viewed as a subalgebra of the operator algebra of vertex models. Extending the Wigner calculus to quantum groups, we obtain an explicit intertwiner between two representations of this subalgebra.     

一维Tonks-Girardeau原子气区域中 Gross-Pitaevskii方程简化模型的精确行波解

利用动力系统方法研究一维Tonks-Girardeau原子气区域中Gross-Pitaevskii (GP)方程简化模型的一些精确行波解以及这些精确行波解的动力学行为, 研究系统的参数对行波解的动力学行为的影响. 在不同的参数条件下, 获得了一维Tonks-Girardeau原子气区域中GP方程简化模型的六个行波解的精确参数表达式. 关键词： 动力系统方法 孤立波解 周期波解 扭波解     

Bifurcation methods of dynamical systems for generalized Kadomtsov-Petviashvili-Benjamin-Bona-Mahony equation

By applying the bifurcation theory of dynamical system to the generalized KP-BBM equation, the phase portraits of the travelling wave system are obtained. It can be shown that singular straight line in the travelling wave system is the reason why smooth periodic waves converge to periodic cusp waves. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are obtained.      

Fock representations of quantum fields with generalized statistics

We develop a rigorous framework for constructing Fock representations of quantum fields obeying generalized statistics. The main features of these representations are investigated. Various aspects of the underlying mathematical structure are illustrated by means of explicit examples.     

Solutions to the Yang-Baxter Equation for the Spinor Representations of q-De

In this paper, both trigonometric and rational solutions to tlre Yang-Baxter equation associated with the spinor representations of the quantum D_l univcrsal enveloping algebras are obtained. The quantum Clebsch-Gordan matrices, the quantum projectors and the solutions are the block matrices with the dimensions of tire submatrices to Le 1 and 2^n-3 , 1l', where _l' = _l/2 + 1, if _l' is even, _l' = (_l + 1)/2, if _l is odd. Tlre csplicit forms of the submatrices with the same dimensions are indcpendcrlt of _l. As esamples, we discuss the solutions for the spinor representations of the quantum D₄ to D₇, and prcscnt tlre explicit forms of those submatrices with the dimensions 1, 2, 8 and 32. Tlre corresponding representations of the braid group and the link polynomials are also computed tlrrougll a standard method.     

Exponential representations of the canonical commutation relations

A class of representations of the canonical commutation relations is investigated. These representations, which are called exponential representations, are given by explicit formulas. Exponential representations are thus comparable to tensor product representations in that one may compute useful criteria concerning various properties. In particular, they are all locally Fock, and non-trivial exponential representations are globally disjoint from the Fock representation. Also, a sufficient condition is obtained for two exponential representations not to be disjoint. An example is furnished by Glimm's model for the :⁴: interaction for boson fields in three space-time dimensions.