Considering the coupled envelope equations in nonlinear couplers, the question of integrability is attempted. It is explicitly shown that Hirota’s bilinear method is one of the simple and alternative techniques to Painlevé analysis to obtain the integrability conditions of the coupled nonlinear Schrödinger (CNLS) type equations. We also show that the coupled Hirota equation introduced by Tasgal...

Several integrability tests for discrete equations will be reviewed. All tests considered can be applied directly to a given discrete equation and do not rely on the a priori knowledge of the existence of related structures such as Lax pairs. Specifically, singularity confinement, algebraic entropy, Nevanlinna theory, Diophantine integrability and discrete systems over finite fields will be des...

A new formulation of Calogero-Moser models based on root systems and their Weyl group is presented. The general construction of the Lax-pairs and the proof of the integrability applicable to all models based on the simply-laced algebras (ADE) are given for two types which we will call ‘root’ and ‘minimal’. The root type Lax pair is new; the matrices used in its construction bear a resemblance t...

Using a recently developed method, based on a generalization of the zero curvature representation of Zakharov and Shabat, we study the integrability structure in the Abelian Higgs model. It is shown that the model contains integrable sectors, where integrability is understood as the existence of infinitely many conserved currents. In particular, a gauge invariant description of the weak and str...

We introduce weighted amalgam-type spaces and analyze their relations with some known spaces. Integrability results for the Fourier transform of a function derivative from one those are proved. The obtained applied to integrability trigonometric series sequence coefficients bounded variation.

Normal forms were introduced by Poincaré as a tool to integrate nonlinear systems; by now we know this is in general impossible, but it turned out that the usefulness of normal forms goes well beyond integrability. An introduction to normal forms is provided e.g. in [3, 5, 26, 41, 43, 47, 63]; see also [10, 33]. An introduction to normal forms for Hamiltonian systems is given in appendix 7 of [...

Equivalence of the spectral gap, exponential integrability of hitting times and Lyapunov conditions are well known. We give here the correspondance (with quantitative results) for reversible diffusion processes. As a consequence, we generalize results of Bobkov in the one dimensional case on the value of the Poincaré constant for logconcave measures to superlinear potentials. Finally, we study ...

Abstract. This paper studies the global regularity theory for degenerate nonlinear parabolic partial differential equations. Our objective is to show that weak solutions belong to a higher Sobolev space than assumed a priori if the complement of the domain satisfies a capacity density condition and if the boundary values are sufficiently smooth. Moreover, we derive integrability estimates for t...

We investigate the local integrability and linearizability of three dimensional Lotka-Volterra equations at the origin. Necessary and sufficient conditions for both integrability and linearizability are obtained for (1,−1, 1), (2,−1, 1) and (1,−2, 1)-resonance. To prove sufficiency, we mainly use the method of Darboux with extensions for inverse Jacobi multipliers, and the linearizability of a ...

We suggest that trialgebraic symmetries might be a sensible starting point for a notion of integrability for two dimensional spin systems. For a simple trialgebraic symmetry we give an explicit condition in terms of matrices which a Hamiltonian realizing such a symmetry has to satisfy and give an example of such a Hamiltonian which realizes a trialgebra recently given by the authors in another ...