Di Piazza and Preiss asked whether every Pettis integrable function defined on [0, 1] and taking values in a weakly compactly generated Ba-nach space is McShane integrable. In this paper we answer this question in the negative.

We denote by Lloc(IR) the space of locally integrable functions f : IR 7→ IR. These are the Lebesgue measurable functions which are integrable over every bounded interval. The support of a function φ, denoted by Supp(φ), is the closure of the set {x ; φ(x) 6= 0} where φ does not vanish. By C∞ c (IR) we denote the space of continuous functions with compact support, having continuous derivatives ...

the first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. this method can be applied to non integrable equations as well as to integrable ones. in this paper, the first integral method is used to construct exact solutions of the 2d ginzburg-landau equation.

The repartition of the separation between energy levels of various isotropic S=1/2 antiferromagnetic chains is studied numerically with the aim of investigating the transition from integrable to non-integrable systems. We begin by displaying the level separation distribution of the integrable Bethe chain. Then two non-integrable systems, two coupled chains and a next-nearest-neighbor coupled ch...

For some integrable systems, such as the open Toda molecule, the spectral curve of the Lax representation becomes the graph C = {(λ, z) | z = A(λ)} of a function A(λ). Those integrable systems provide an interesting “toy model” of separation of variables. Examples of this type of integrable systems are presented along with generalizations for which A(λ) lives on a cylinder, a torus or a Riemann...

I review the appearence of integrable structures in the formulation of exact nonperturbative solutions to 4d supersymmetric quantum gauge theories. Various examples of N ≥ 2 SUSY Yang-Mills nonperturbative solutions are adequately described in terms of the (deformations of the) finite-gap solutions to integrable models: through the generating differential and the τ -function. One of the basic d...

Integrable σ-models, such as the principal chiral model, ZT -coset models for T ∈ Z≥2 and their various integrable deformations, are examples of non-ultralocal integrable field theories described by r/s-systems with twist function. In this general setting, and when the Lie algebra g underlying the r/s-system is of classical type, we construct an infinite algebra of local conserved charges in in...

For an arbitrary in nite-dimensional Banach space X, we construct examples of strongly-measurable X-valued Pettis integrable functions whose indefinite Pettis integrals are nowhere weakly di erentiable; thus, for these functions the Lebesgue Di erentiation Theorem fails rather spectacularly. We also relate the degree of nondi erentiability of the inde nite Pettis integral to the cotype of X, fr...