Analytic-bilinear approach for construction and study of integrable hierarchies, in particular, the KP hierarchy is discussed. It is based on the generalized Hirota identity. This approach allows to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations. Resolution of these functional equations leads to the τ-function and addition formulae t...

is that of suitably defining a trigonometric integral with the property that, if the series (1.1) converges everywhere to a function ƒ(x), then f(x) is necessarily integrable and the coefficients, an and bn, given in the usual Fourier form. It is well known that a series may converge everywhere to a function which is not Lebesgue summable nor even Denjoy integrable (completely totalisable, [3])...

Under certain conditions on an integrable function P having a real-valued Fourier transform P̂ and such that P(0)=0, we obtain an estimate which describes the oscillation of P̂ in [−C‖P ′‖∞/‖P ‖∞, C‖P ′‖∞/‖P ‖∞], whereC is an absolute constant, independent of P. Given > 0 and an integrable function with a non-negative Fourier transform, this estimate allows us to construct a finite linear combina...

For the constant frequency case, we introduce a special type of Ermakov-Pinney equations with nonlinear dissipation based on the corresponding Chiellini integrable Abel equation. General solutions of these equations are obtained following the Abel equation route. Based on particular solutions, we also provide general solutions containing a factor with the phase of the Milne type. In addition, t...

We study the geometry of completely integrable bi-Hamiltonian systems, and in particular, the existence of a bi-Hamiltonian structure for a completely integrable Hamiltonian system. We show that under some natural hypothesis, such a structure exists in a neighborhood of an invariant torus if, and only if, the graph of the Hamiltonian function is a hypersurface of translation, relative to the af...

In this note the following generalization of Fatou's lemma is proved: Lemma. Let {fn)n_l be a sequence of integrable functions on a measure space S with values in R+, the nonnegative orthant of a d-dimensional Euclidean space, for which ffn—*aGiR+. Then there exists an integrable function f, from S to R+, such that a.e. f(s) is a limit point of VnisV^and ff^a.

Abstract. Let W be an integrable positive Hermitian q × q-matrix valued function on the dual group of a discrete abelian group G such that W is integrable. Generalizing results of T. Nakazi and ofA. G. Miamee [N] andM. Pourahmadi [MiP] for q = 1 we establish a correspondencebetween trigonometric approximation problems in L(W ) and certain approximation problems in L(W). The result is applied to...

Analytic-bilinear approach for construction and study of integrable hierarchies is discussed. Generalized multicomponent KP and 2D Toda lattice hierarchies are considered. This approach allows to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations. Generalized hierarchy incorporates basic hierarchy, modified hierarchy, singularity manifol...

Duality in the integrable systems arising in the context of Seiberg-Witten theory shows that their tau-functions indeed can be seen as generating functions for the mutually Poisson-commuting hamiltonians of the dual systems. We demonstrate that the Θ-function coefficients of their expansion can be expressed entirely in terms of the co-ordinates of the Seiberg-Witten integrable system, being, th...

An expansion method was used to write a MATHEMATICA program to compute the energy levels and eigenfunctions of a 2-D quantum billiard system with arbitrary shape and dirichlet boundary conditions. One integrable system, the full circle, and one non-integrable system, the stadium, were examined. Chaotic properties were sought in nearest-neighbor energy level spacing distributions (NND). It was o...