Exactly Solvable And Integrable Systems

We give a Lax description for the system of polytropic gas equations. The special structure of the Lax function naturally leads to the two infinite sets of conserved charges associated with this system. We obtain closed form expressions for the conserved charges as well as the generating functions for them. We show how the study of these generating functions can naturally lead to the recursion relation between the conserved quantities as well as the higher order Hamiltonian structures.

Existence of a new class of soliton solutions is shown for higher order nonlinear Schrodinger equation, describing thrid order dispersion, Kerr effect and stimulated Raman scattering. These new solutions have been obtaiened by invoking a group of nonlinear transformations acting on localised stable solutions. Stability of these solutions has been studied for different values of the arbitrary coefficients, involved in the recursion relation and consequently, different values of coefficient lead to different transmission rates for almost same input power. Another series solution containing even powers of localised stable solution is shown to exist for higher order nonlinear Schrodinger equation.

A realization of discrete conjugate net is presented by using correlation functions of strings in a gauge covariant form.

A spectral mapping theorem is proved that resolves a key problem in applying invariant manifold theorems to nonlinear Schr\" odinger type equations. The theorem is applied to the operator that arises as the linearization of the equation around a standing wave solution. We cast the problem in the context of space-dependent nonlinearities that arise in optical waveguide problems. The result is, however, more generally applicable including to equations in higher dimensions and even systems. The consequence is that stable, unstable, and center manifolds exist in the neighborhood of a (stable or unstable) standing wave, such as a waveguide mode, under simple and commonly verifiable spectral conditions.

We construct a recursion operator for the system ( u t , v t )=( u 4 + v 2 ,1/5 v 4 ) , for which only one local symmetry is known and we show that the action of the recursion operator on ( u t , v t ) is a local function.

A fairly complete list of Toda-like integrable lattice systems, both in the continuous and discrete time, is given. For each system the Newtonian, Lagrangian and Hamiltonian formulations are presented, as well as the 2x2 Lax representation and r-matrix structure. The material is given in the "no comment" style, in particular, all proofs are omitted.

We derive a zero-curvature formalism for a combined sine-Gordon (sG) and modified Korteweg-de Vries (mKdV) equation which yields a local sGmKdV hierarchy. In complete analogy to other completely integrable hierarchies of soliton equations, such as the KdV, AKNS, and Toda hierarchies, the sGmKdV hierarchy is recursively constructed by means of a fundamental polynomial formalism involving a spectral parameter. We further illustrate our approach by developing the basic algebro-geometric setting for the sGmKdV hierarchy, including Baker-Akhiezer functions, trace formulas, Dubrovin-type equations, and theta function representations for its algebro-geometric solutions. Although we mainly focus on sG-type equations, our formalism also yields the sinh-Gordon, elliptic sine-Gordon, elliptic sinh-Gordon, and Liouville-type equations combined with the mKdV hierarchy.

We consider here two discrete versions of the modified KdV equation. In one case, some solitary wave solutions, Bäcklund transformations and integrals of motion are known. In the other one, only solitary wave solutions were given, and we supply the corresponding results for this equation. We also derive the integrability of the second equation and give a transformation which links the two models.

A new construction is given for obtaining R-matrices which solve the McGuire-Yang-Baxter equation in such a way that the spectral parameters do not possess the difference property. A discussion of the derivation of the supersymmetric U model is given in this context such that applied chemical potential and magnetic field terms can be coupled arbitrarily. As a limiting case the Bariev model is obtained.

A zero-field Ising model with ferromagnetic coupling constants on the so-called Labyrinth tiling is investigated. Alternatively, this can be regarded as an Ising model on a square lattice with a quasi-periodic distribution of up to eight different coupling constants. The duality transformation on this tiling is considered and the self-dual couplings are determined. Furthermore, we analyze the subclass of exactly solvable models in detail parametrizing the coupling constants in terms of four rapidity parameters. For those, the self-dual couplings correspond to the critical points which, as expected, belong to the Onsager universality class.