Exactly Solvable And Integrable Systems

It has recently been argued that there might exist a four-parameter analytic solution to the Bianchi IX cosmological model, which would extend the three-parameter solution of Belinskii et al. to one more arbitrary constant. We perform the perturbative Painlevé test in the proper time variable, and confirm the possible existence of such an extension.

Let q 1 , q 2 ,..., q N be the coordinates of N particles on the circle, interacting with the integrable potential ∑ N j<k ℘( q j − q k ) , where ℘ is the Weierstrass elliptic function. We show that every symmetric elliptic function in q 1 , q 2 ,..., q N is a meromorphic function in time. We give explicit formulae for these functions in terms of genus N−1 theta functions.

We investigate the integrability of Nonlinear Partial Differential Equations (NPDEs). The concepts are developed by firstly discussing the integrability of the KdV equation. We proceed by generalizing the ideas introduced for the KdV equation to other NPDEs. The method is based upon a linearization principle which can be applied on nonlinearities which have a polynomial form. We illustrate the potential of the method by finding solutions of the (coupled) nonlinear Schrödinger equation and the Manakov equation which play an important role in optical fiber communication. Finally, it is shown that the method can also be generalized to higher-dimensions.

For the equations y ′′ =P(x,y)+3Q(x,y) y ′ +3R(x,y) y ′ 2 +S(x,y) y ′ 3 the problem of equivalence is considered. Some classical results are resumed in order to prepare the background for the study of special subclass of such equations, which arises in the theory of dynamical systems admitting the normal shift.

Point transformations for the ordinary differential equations of the form y ′′ =P(x,y)+3Q(x,y) y ′ +3R(x,y)( y ′ ) 2 +S(x,y)( y ′ ) 3 are considered. Some classical results are resumed. Solution for the equivalence problem for the equations of general position is described.

For the unitary ensembles of N×N Hermitian matrices associated with a weight function w there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For the orthogonal and symplectic ensembles of Hermitian matrices there are 2×2 matrix kernels, usually constructed using skew-orthogonal polynomials, which play an analogous role. These matrix kernels are determined by their upper left-hand entries. We derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles. We also show that whenever w ′ /w is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w ′ /w . General formulas are obtained for these extra terms. We do not use skew-orthogonal polynomials in the derivations.

Using a moving space curve formalism, geometrical as well as gauge equivalence between a (2+1) dimensional spin equation (M-I equation) and the (2+1) dimensional nonlinear Schrödinger equation (NLSE) originally discovered by Calogero, discussed then by Zakharov and recently rederived by Strachan, have been estabilished. A compatible set of three linear equations are obtained and integrals of motion are discussed. Through stereographic projection, the M-I equation has been bilinearized and different types of solutions such as line and curved solitons, breaking solitons, induced dromions, and domain wall type solutions are presented. Breaking soliton solutions of (2+1) dimensional NLSE have also been reported. Generalizations of the above spin equation are discussed.

We present the explicit formulae, describing the structure of symmetries and formal symmetries of any scalar (1+1)-dimensional evolution equation. Using these results, the formulae for the leading terms of commutators of two symmetries and two formal symmetries are found. The generalization of these results to the case of system of evolution equations is also discussed.

We use the Calogero equation to illustrate the following two aspects of the Painleve analysis of nonlinear PDEs. First, if a nonlinear equation passes the Painleve test for integrability, the singular expansions of its solutions around characteristic hypersurfaces can be neither single-valued functions of independent variables nor single-valued functionals of data. Second, if the truncation of singular expansions of solutions is consistent, the truncation not necessarily leads to the simplest, or elementary, auto-Backlund transformation related to the Lax pair.

A new method for finding solutions of the nonlinear Shrödinger equation is proposed. Comutative multiplicative group of the nonlinear transformations, which operate on stationary localized solutions, enables a consideration of fractal subspaces in the solution space, stability and deterministic chaos. An increase of the transmission rate at the optical fiber communications can be based on new forms of localized stationary solutions, without significant change of input power. The estimated transmission rate is 50 Gbit/s, for certain available soliton transmission systems.