Exactly Solvable And Integrable Systems

We consider travelling periodic and quasiperiodic wave solutions of a set of coupled nonlinear Schrödimger equations. In fibre optics these equations can be used to model single mode fibers with strong birefringence and two-mode optical fibres. Recently these equations appear as modes, which describe pulse-pulse interaction in wavelength-division-multiplexed channels of optical fiber transmission systems. Two phase quasi-periodic solutions for integrable Manakov system are given in tems of two-dimensional Kleinian functions. The reduction of quasi-periodic solutions to elliptic functions is dicussed. New solutions in terms of generalized Hermite polynomilas, which are associated with two-gap Treibich-Verdier potentials are found.

The Bethe Ansatz is a method that is used in quantum integrable models in order to solve them explicitly. This method is explained here in a general framework, which applies to 1D quantum spin chains, 2D statistical lattice models (vertex models) and relativistic field theories with 1 space dimension and 1 time dimension. The connection with quantum groups is expounded. Several applications are then presented. Finite size corrections are calculated via two methods: The Non-Linear Integral Equations, which are applied to the study of the states of the affine Toda model with imaginary coupling, and their interpolation between the high energy (ultra-violet) and low energy (infra-red) regions; and the Thermodynamic Bethe Ansatz Equations, along with the associated Fusion Equations, which are used to determine the thermodynamic properties of the generalized multi-channel Kondo model. The latter is then studied in more detail, still using the Bethe Ansatz and quantum groups, so as to characterize the spectrum of the low energy excitations.

Four results are given that address the existence, ambiguities and construction of a classical R-matrix given a Lax pair. They enable the uniform construction of R-matrices in terms of any generalized inverse of adL . For generic L a generalized inverse (and indeed the Moore-Penrose inverse) is explicitly constructed. The R-matrices are in general momentum dependent and dynamical. The construction applies equally to Lax matrices with spectral parameter.

Stimulated Raman scattering of a laser pump pulse seeded by a Stokes pulse generically leaves a two-level medium initially at rest in an excited state constituted of static solitons and radiation. The soliton birth manifests as sudden very large variations of the phase of the output pump pulse. This is proved by building the IST solution of SRS on the semi-line, which shows moreover that initial Stokes phase flips induce Raman spikes in the pump output also for short pulse experiments.

What is the connection of random matrices with integrable systems? Is this connection really useful? The answer to these questions leads to a new and unifying approach to the theory of random matrices. Introducing an appropriate time t-dependence in the probability distribution of the matrix ensemble, leads to vertex operator expressions for the n-point correlation functions (probabilities of n eigenvalues in infinitesimal intervals) and the corresponding Fredholm determinants (probabilities of no eigenvalue in a Borel subset E); the latter probability is a ratio of tau-functions for the KP-equation, whose numerator satisfy partial differential equations, which decouple into the sum of two parts: a Virasoro-like part depending on time only and a Vect(S^1)-part depending on the boundary points A_i of E. Upon setting t=0, and using the KP-hierarchy to eliminate t-derivatives, these PDE's lead to a hierarchy of non-linear PDE's, purely in terms of the A_i. These PDE's are nothing else but the KP hierarchy for which the t-partials, viewed as commuting operators, are replaced by non-commuting operators in the endpoints A_i of the E under consideration. When the boundary of E consists of one point and for the known kernels, one recovers the Painleve equations, found in prior work on the subject.

We study the rational solutions of the discrete version of Painleve's fourth equation d-PIV. The solutions are generated by applying Schlesinger transformations on the seed solutions -2z and -1/z. After studying the structure of these solutions we are able to write them in a determinantal form that includes an interesting parameter shift that vanishes in the continuous limit.

Three nonequivalent real forms of the complex twisted N=2 supersymmetric Toda chain hierarchy (solv-int/9907021) in real N=1 superspace are presented. It is demonstrated that they possess a global twisted N=2 supersymmetry. We discuss a new superfield basis in which the supersymmetry transformations are local. Furthermore, a representation of this hierarchy is given in terms of two twisted chiral N=2 superfields. The relations to the s-Toda hierarchy by H. Aratyn, E. Nissimov and S. Pacheva (solv-int/9801021) as well as to the modified and derivative NLS hierarchies are established.

We present here the iteration procedure for the determination of free energy ϵ 2 -expansion using the theory of KdV - type equations. In our approach we use the conservation laws for KdV - type equations depending explicitly on times t 1 , t 2 ,... to find the ϵ 2 -expansion of u(x, t 1 , t 2 ,...) after the infinite number of shifts of u(x,0,0,...)≡x along t 1 , t 2 ,... in recurrent form. The formulas for the free energy expansion are just obtained then as a result of quite simple integration procedure applied to u n (x) .

The vectorial fundamental transformation for the Darboux equations is reduced to the symmetric case. This is combined with the orthogonal reduction of Lame type to obtain those vectorial Ribaucour transformations which preserve the Egoroff reduction. We also show that a permutability property holds for all these transformations. Finally, as an example, we apply these transformations to the Cartesian background.

We discuss the Boussinesq system with t 5 stationary, within a general framework for the analysis of stationary flows of n-Gel'fand-Dickey hierarchies. We show how a careful use of its bihamiltonian structure can be used to provide a set of separation coordinates for the corresponding Hamilton--Jacobi equations.