EUDML

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Sur les fonctions harmoniques et holomorphes de classe $H^{1}$ et le problème de Ginzburg-Landau en deux dimensions

Séminaire Équations aux dérivées partielles (Polytechnique)

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 $x \geq 0$ . 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...

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...