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Initial boundary value problem for the mKdV equation on a finite interval

Anne Boutet de Monvel, Dmitry Shepelsky (2004)

Annales de l’institut Fourier

We analyse an initial-boundary value problem for the mKdV equation on a finite interval ( 0 , L ) by expressing the solution in terms of the solution of an associated matrix Riemann-Hilbert problem in the complex k -plane. This RH problem is determined by certain spectral functions which are defined in terms of the initial-boundary values at t = 0 and x = 0 , L . We show that the spectral functions satisfy an algebraic “global relation” which express the implicit relation between all boundary values in terms of spectral...

Long time asymptotics of the Camassa–Holm equation on the half-line

Anne Boutet de Monvel, Dmitry Shepelsky (2009)

Annales de l’institut Fourier

We study the long-time behavior of solutions of the initial-boundary value (IBV) problem for the Camassa–Holm (CH) equation u t - u t x x + 2 u x + 3 u u x = 2 u x u x x + u u x x x on the half-line x 0 . The paper continues our study of IBV problems for the CH equation, the key tool of which is the formulation and analysis of associated Riemann–Hilbert factorization problems. We specify the regions in the quarter space-time plane x > 0 , t > 0 having qualitatively different asymptotic pictures, and give the main terms of the asymptotics in terms of spectral data...

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...

Solitons and large time behavior of solutions of a multidimensional integrable equation

Anna Kazeykina (2013)

Journées Équations aux dérivées partielles

Novikov-Veselov equation is a (2+1)-dimensional analog of the classic Korteweg-de Vries equation integrable via the inverse scattering translform for the 2-dimensional stationary Schrödinger equation. In this talk we present some recent results on existence and absence of algebraically localized solitons for the Novikov-Veselov equation as well as some results on the large time behavior of the “inverse scattering solutions” for this equation.

Some constructive applications of Λ 2 -representations to integration of PDEs

A. Popov, S. Zadadaev (2000)

Annales Polonici Mathematici

Two new applications of Λ 2 -representations of PDEs are presented: 1. Geometric algorithms for numerical integration of PDEs by constructing planimetric discrete nets on the Lobachevsky plane Λ 2 . 2. Employing Λ 2 -representations for the spectral-evolutionary problem for nonlinear PDEs within the inverse scattering problem method.

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