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On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

A. Sakhnovich (2010)

Mathematical Modelling of Natural Phenomena

A general theorem on the GBDT version of the Bäcklund-Darboux transformation for systems depending rationally on the spectral parameter is treated and its applications to nonlinear equations are given. Explicit solutions of direct and inverse problems for Dirac-type systems, including systems with singularities, and for the system auxiliary to the N-wave equation are reviewed. New results on explicit construction of the wave functions for radial...

On the growth of Sobolev norms for the cubic Szegő equation

Patrick Gérard, Sandrine Grellier (2014/2015)

Séminaire Laurent Schwartz — EDP et applications

We report on a recent result establishing that trajectories of the cubic Szegő equation in Sobolev spaces with high regularity are generically unbounded, and moreover that, on solutions generated by suitable bounded subsets of initial data, every polynomial bound in time fails for high Sobolev norms. The proof relies on an instability phenomenon for a new nonlinear Fourier transform describing explicitly the solutions to the initial value problem, which is inherited from the Lax pair structure enjoyed...

On the heat kernel and the Korteweg--de Vries hierarchy

Plamen Iliev (2005)

Annales de l’institut Fourier

We give explicit formulas for Hadamard's coefficients in terms of the tau-function of the Korteweg-de Vries hierarchy. We show that some of the basic properties of these coefficients can be easily derived from these formulas.

On the hierarchies of higher order mKdV and KdV equations

Axel Grünrock (2010)

Open Mathematics

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces H ^ s r ℝ defined by the norm v 0 H ^ s r ℝ : = ξ s v 0 ^ L ξ r ' , ξ = 1 + ξ 2 1 2 , 1 r + 1 r ' = 1 . Local well-posedness for the jth equation is shown in the parameter range 2 ≥ 1, r > 1, s ≥ 2 j - 1 2 r ' . The proof uses an appropriate variant of the Fourier restriction norm method. A counterexample is discussed to show that the Cauchy problem for equations of this type is in general ill-posed in the C 0-uniform sense, if s < 2 j - 1 2 r ' . The results for r = 2 - so far in...

On the integrability of the generalized Yang-Mills system

A. Lesfari, A. Elachab (2004)

Applicationes Mathematicae

We consider a hamiltonian system which, in a special case and under the gauge group SU(2), can be considered as a reduction of the Yang-Mills field equations. We prove explicitly, using the Lax spectral curve technique and the van Moerbeke-Mumford method, that the flows generated by the constants of motion are straight lines on the Jacobi variety of a genus two Riemann surface.

On the long time behavior of KdV type equations

Nikolay Tzvetkov (2003/2004)

Séminaire Bourbaki

In a series of recent papers, Martel and Merle solved the long-standing open problem on the existence of blow up solutions in the energy space for the critical generalized Korteweg- de Vries equation. Martel and Merle introduced new tools to study the nonlinear dynamics close to a solitary wave solution. The aim of the talk is to discuss the main ideas developed by Martel-Merle, together with a presentation of previously known closely related results.

On the mobility and efficiency of mechanical systems

Gershon Wolansky (2007)

ESAIM: Control, Optimisation and Calculus of Variations

It is shown that self-locomotion is possible for a body in Euclidian space, provided its dynamics corresponds to a non-quadratic Hamiltonian, and that the body contains at least 3 particles. The efficiency of the driver of such a system is defined. The existence of an optimal (most efficient) driver is proved.


Opening gaps in the spectrum of strictly ergodic Schrödinger operators

Artur Avila, Jairo Bochi, David Damanik (2012)

Journal of the European Mathematical Society

We consider Schrödinger operators with dynamically defined potentials arising from continuous sampling along orbits of strictly ergodic transformations. The Gap Labeling Theorem states that the possible gaps in the spectrum can be canonically labelled by an at most countable set defined purely in terms of the dynamics. Which labels actually appear depends on the choice of the sampling function; the missing labels are said to correspond to collapsed gaps. Here we show that for any collapsed gap,...

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