Exactly Solvable And Integrable Systems

We study a small piece of two dimensional Toda lattice as a complex dynamical system. In particular the Julia set, which appears when the piece is deformed, is shown analytically how it disappears as the system approaches to the integrable limit.

It is shown that the McCall-Hahn theory of self-induced transparency in coherent optical pulse propagation can be identified with the complex sine-Gordon theory in the sharp line limit. We reformulate the theory in terms of the deformed gauged Wess-Zumino-Witten sigma model and address various new aspects of self-induced transparency.

A straightforward algorithm for the symbolic computation of higher-order symmetries of nonlinear evolution equations and lattice equations is presented. The scaling properties of the evolution or lattice equations are used to determine the polynomial form of the higher-order symmetries. The coefficients of the symmetry can be found by solving a linear system. The method applies to polynomial systems of PDEs of first-order in time and arbitrary order in one space variable. Likewise, lattices must be of first order in time but may involve arbitrary shifts in the discretized space variable. The algorithm is implemented in Mathematica and can be used to test the integrability of both nonlinear evolution equations and semi-discrete lattice equations. With our Integrability Package, higher-order symmetries are obtained for several well-known systems of evolution and lattice equations. For PDEs and lattices with parameters, the code allows one to determine the conditions on these parameters so that a sequence of higher-order symmetries exist. The existence of a sequence of such symmetries is a predictor for integrability.

An algorithm to compute polynomial conserved densities of polynomial nonlinear lattices is presented. The algorithm is implemented in Mathematica and can be used as an automated integrability test. With the code diffdens.m, conserved densities are obtained for several well-known lattice equations. For systems with parameters, the code allows one to determine the conditions on these parameters so that a sequence of conservation laws exist.

A new method for the computation of conserved densities of nonlinear differential-difference equations is applied to Toda lattices and discretizations of the Korteweg-de Vries and nonlinear Schrodinger equations. The algorithm, which can be implemented in computer algebra languages such as Mathematica, can be used as an indicator of integrability.

All possible 1-parametric classical and transcendent degenerated solutions of the fourth Painleve equation with the corresponding connection formulae of the asymptotic parameters are described.

Conservation laws of the nonlinear Schrödinger equation are studied in the presence of higher-order nonlinear optical effects including the third-order dispersion and the self-steepening. In a context of group theory, we derive a general expression for infinitely many conserved currents and charges of the coupled higher-order nonlinear Schrödinger equation. The first few currents and charges are also presented explicitly. Due to the higher-order effects, conservation laws of the nonlinear Schrödinger equation are violated in general. The differences between the types of the conserved currents for the Hirota and the Sasa-Satsuma equations imply that the higher-order terms determine the inherent types of conserved quantities for each integrable cases of the higher-order nonlinear Schrödinger equation.

We present a systematic construction of integrable third order systems based on the coupling of an integrable second order equation and a Riccati equation. This approach is the extension of the Gambier method that led to the equation that bears his name. Our study is carried through for both continuous and discrete systems. In both cases the investigation is based on the study of the singularities of the system (the Painlevé method for ODE's and the singularity confinement method for mappings).

The inhomogeneity of the media or the external forces usually destroy the integrability of a system. We propose a systematic construction of a class of quantum models, which retains their exact integrability inspite of their explicit inhomogeneity. Such models include variable mass sine-Gordon model, cylindrical NLS, spin chains with impurity, inhomogeneous Toda chain, the Ablowitz-Ladik model etc.

One dimensional nonlinear lattices with harmonic long range interaction potentials (LRIP) having an inverse power kernel type, are studied. For the nearest neighbour nonlinear interaction we consider the anharmonic potential of the Fermi-Pasta-Ulam problem and the \phi^3+\phi^4 potential as well. The continuum limit is obtained following the method used by Ishimori and several Boussinesq and KdV type equations with supplementary Hilbert transform terms are found. These nonlocal terms are introduced by the LRIP. For the \phi^3+\phi^4 nearest neighbour interactions the continuum approximation turns out to admit exact bilinearization in Hirota formalism. Exact rational nonsingular solutions are found. The integrability of these nonlocal equations and the connection with perturbed KdV are also discussed.