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

Anne Boutet de Monvel[1]; Dmitry Shepelsky

  • [1] Université Paris 7, Institut de Mathématiques de Jussieu, case 7012, 2 place Jussieu, 75251 Paris (France), Institute for Low Temperature Physics, Mathematical Division, 47 Lenin Avenue, 61103 Kharkov (Ukraine)

Annales de l’institut Fourier (2004)

  • Volume: 54, Issue: 5, page 1477-1495
  • ISSN: 0373-0956

Abstract

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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 data.

How to cite

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Boutet de Monvel, Anne, and Shepelsky, Dmitry. "Initial boundary value problem for the mKdV equation on a finite interval." Annales de l’institut Fourier 54.5 (2004): 1477-1495. <http://eudml.org/doc/116149>.

@article{BoutetdeMonvel2004,
abstract = {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 data.},
affiliation = {Université Paris 7, Institut de Mathématiques de Jussieu, case 7012, 2 place Jussieu, 75251 Paris (France), Institute for Low Temperature Physics, Mathematical Division, 47 Lenin Avenue, 61103 Kharkov (Ukraine)},
author = {Boutet de Monvel, Anne, Shepelsky, Dmitry},
journal = {Annales de l’institut Fourier},
keywords = {modified Korteweg-de Vries equation; initial-boundary value problem; global relation; finite interval; Riemann-Hilbert problem; Riemann-Hilbert problem.},
language = {eng},
number = {5},
pages = {1477-1495},
publisher = {Association des Annales de l'Institut Fourier},
title = {Initial boundary value problem for the mKdV equation on a finite interval},
url = {http://eudml.org/doc/116149},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Boutet de Monvel, Anne
AU - Shepelsky, Dmitry
TI - Initial boundary value problem for the mKdV equation on a finite interval
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 5
SP - 1477
EP - 1495
AB - 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 data.
LA - eng
KW - modified Korteweg-de Vries equation; initial-boundary value problem; global relation; finite interval; Riemann-Hilbert problem; Riemann-Hilbert problem.
UR - http://eudml.org/doc/116149
ER -

References

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  2. A. Boutet de Monvel, A.S. Fokas, D. Shepelsky, Analysis of the global relation for the nonlinear Schrödinger equation on the half-line, Lett. Math. Phys 65 (2003), 199-212 Zbl1055.35107MR2033706
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  10. A.S. Fokas, A.R. Its, L.-Y. Sung, The nonlinear Schrödinger equation on the half-line Zbl1181.37095MR2150354
  11. X. Zhou, The Riemann-Hilbert problem and inverse scattering, SIAM J. Math. Anal 20 (1989), 966-986 Zbl0685.34021MR1000732
  12. X. Zhou, Inverse scattering transform for systems with rational spectral dependence, J. Differential Equations 115 (1995), 277-303 Zbl0816.35104MR1310933
  13. V.E. Zakharov, A.B. Shabat, A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I, Funct. Anal. Appl. 8 (1974), 226-235 Zbl0303.35024
  14. V.E. Zakharov, A.B. Shabat, A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering. II, Funct. Anal. Appl. 13 (1979), 166-174 Zbl0448.35090

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