Exactly Solvable And Integrable Systems

We show the complete integrability and the existence of optical solitons of higher order nonlinear Schrodinger equation by inverse scattering method for a wide range of values of coefficients. This is achieved first by invoking a novel connection between the integrability of a nonlinear evolution equation and the dimensions of a family of matrix Lax pairs. It is shown that Lax pairs of different dimensions lead to the same evolution equation only with the coefficients of the terms in different integer ratios. Optical solitons, thus obtained by inverse scattering method, have been found by solving an n dimensional eigenvalue problem.

Orthogonal and symplectic matrix integrals are investigated. It is shown that the matrix integrals can be considered as a τ -function of the coupled KP hierarchy, whose solution can be expressed in terms of pfaffians.

We consider the polynomials ϕ n (z)= κ n ( z n + b n−1 z n−1 +>...) orthonormal with respect to the weight exp( λ − − √ (z+1/z))dz/2πiz on the unit circle in the complex plane. The leading coefficient κ n is found to satisfy a difference-differential (spatially discrete) equation which is further proved to approach a third order differential equation by double scaling. The third order differential equation is equivalent to the Painlevé II equation. The leading coefficient and second leading coefficient of ϕ n (z) can be expressed asymptotically in terms of the Painlevé II function.

The semiclassical limit of the algebraic quantum inverse scattering method is used to solve the theory of the Gaudin model. Via Off-shell Bethe ansatz equations of an integrable representation of the graded osp(1|2) vertex model we find the spectrum of N-1 independent Hamiltonians of Gaudin. Integral representations of the N-point correlators are presented as solutions of the Knizhnik-Zamolodchikov equation. These results are extended for highest representations of the osp(1|2) Gaudin algebra.

The Painlevé analysis introduced by Weiss, Tabor and Carnevale (WTC) in 1983 for nonlinear partial differential equations (PDE's) is an extension of the method initiated by Painlevé and Gambier at the beginning of this century for the classification of algebraic nonlinear differential equations (ODE's) without movable critical points. In these lectures we explain the WTC method in its invariant version introduced by Conte in 1989 and its application to solitonic equations in order to find algorithmically their associated Bäcklund transformation. A lot of remarkable properties are shared by these so-called ``integrable'' equations but they are generically no more valid for equations modelising physical phenomema. Belonging to this second class, some equations called ``partially integrable'' sometimes keep remnants of integrability. In that case, the singularity analysis may also be useful for building closed form analytic solutions, which necessarily % Conte agree with the singularity structure of the equations. We display the privileged role played by the Riccati equation and systems of Riccati equations which are linearisable, as well as the importance of the Weierstrass elliptic function, for building solitary waves or more elaborate solutions.

Using the Painlevé analysis, we investigate the integrability properties of a system of two coupled nonlinear Schrödinger equations that describe the propagation of orthogonally polarized optical waves in an isotropic medium. Besides the well-known integrable vector nonlinear Schrödinger equation, we show that there exist a new set of equations passing the Painlevé test where the self and cross phase modulational terms are of different magnitude. We introduce the Hirota bilinearization and the Bäcklund transformation to obtain soliton solutions and prove integrability by making a change of variables. The conditions on the third-order susceptibility tensor χ (3) imposed by these new integrable equations are explained.

We consider some natural connections which arise between right-flat (p, q) paraconformal structures and integrable systems. We find that such systems may be formulated in Lax form, with a "Lax p-tuple" of linear differential operators, depending a spectral parameter which lives in (q-1)-dimensional complex projective space. Generally, the differential operators contain partial derivatives with respect to the spectral parameter.

We give a review of some recent work on generalization of the Bethe ansatz in the case of 2+1 -dimensional models of quantum field theory. As such a model, we consider one associated with the tetrahedron equation, i.e. the 2+1 -dimensional generalization of the famous Yang--Baxter equation. We construct some eigenstates of the transfer matrix of that model. There arise, together with states composed of point-like particles analogous to those in the usual 1+1 -dimensional Bethe ansatz, new string-like states and string-particle hybrids.

The perturbation theory based on the Riemann-Hilbert problem is developed for the modified nonlinear Schr{ö}dinger equation which describes the propagation of femtosecond optical pulses in nonlinear single-mode optical fibers. A detailed analysis of the adiabatic approximation to perturbation-induced evolution of the soliton parameters is given. The linear perturbation and the Raman gain are considered as examples.

There exist many situations where an ordinary differential equation admits a movable critical singularity which the test of Kowalevski and Gambier fails to detect. Some possible reasons are: existence of negative Fuchs indices, insufficient number of Fuchs indices, multiple family, absence of an algebraic leading order. Mainly giving examples, we present the methods which answer all these questions. They are all based on the theorem of perturbations of Poincaré and computerizable.