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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.

Perron-Frobenius operators and the Klein-Gordon equation

Francisco Canto-Martín, Håkan Hedenmalm, Alfonso Montes-Rodríguez (2014)

Journal of the European Mathematical Society

For a smooth curve Γ and a set Λ in the plane 2 , let A C ( Γ ; Λ ) be the space of finite Borel measures in the plane supported on Γ , absolutely continuous with respect to the arc length and whose Fourier transform vanishes on Λ . Following [12], we say that ( Γ , Λ ) is a Heisenberg uniqueness pair if A C ( Γ ; Λ ) = { 0 } . In the context of a hyperbola Γ , the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets Λ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly address the...

Stability and Instability of Solitary Wave Solutions of a Nonlinear Dispersive System of Benjamin-Bona-Mahony Type

Hakkaev, Sevdzhan (2003)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 35B35, 35B40, 35Q35, 76B25, 76E30.This paper concerns the orbital stability and instability of solitary waves of the system of coupling equations of Benjamin-Bona-Mahony type. By applying the abstract results of Grillakis, Shatah and Strauss and detailed spectral analysis, we obtain the existence and stability of the solitary waves.Partially Supported by Grant MM-810/98 of MESC and by Scientefic Research Grant 19/12.03.2003 of Shumen University.

Stability of periodic waves in Hamiltonian PDEs

Sylvie Benzoni-Gavage, Pascal Noble, L. Miguel Rodrigues (2013)

Journées Équations aux dérivées partielles

Partial differential equations endowed with a Hamiltonian structure, like the Korteweg–de Vries equation and many other more or less classical models, are known to admit rich families of periodic travelling waves. The stability theory for these waves is still in its infancy though. The issue has been tackled by various means. Of course, it is always possible to address stability from the spectral point of view. However, the link with nonlinear stability  - in fact, orbital stability, since we are...

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