On considère les équations de Korteweg–de Vries généralisées dans les cas sous-critique et critique. Soit solutions de type solitons de l'équation, correspondant à des vitesses 0<c1<c2<⋯<cN. Dans cette Note, on donne les idées principales de la démonstration du résultat suivant. Etant donnés , il existe une et une seule solution ϕ de l'équation de KdV généralisée telle que ‖ϕ(t)−∑Rj(t)‖H1→0 quand t→+∞. Les preuves complètes seront publiées plus tard.
We consider the generalized Korteweg–de Vries equations in the subcritical and critical cases. Let Rj(t,x)=Qcj(x−cjt−xj) be N soliton solutions of this equation, with corresponding speeds 0<c1<c2<⋯<cN. In this Note, we give a sketch of the proof of the following result. Given , there exists one and only one solution ϕ of the generalized KdV equation such that ‖ϕ(t)−∑Rj(t)‖H1→0 as t→+∞. Complete proofs will appear later.
@article{CRMATH_2004__338_6_457_0,
author = {Martel, Yvan},
title = {Solutions de type multi-soliton des \'equations de {KdV} g\'en\'eralis\'ees},
journal = {Comptes Rendus. Math\'ematique},
pages = {457--460},
publisher = {Elsevier},
volume = {338},
number = {6},
year = {2004},
doi = {10.1016/j.crma.2003.12.029},
language = {fr},
url = {http://www.numdam.org/articles/10.1016/j.crma.2003.12.029/}
}
TY - JOUR AU - Martel, Yvan TI - Solutions de type multi-soliton des équations de KdV généralisées JO - Comptes Rendus. Mathématique PY - 2004 SP - 457 EP - 460 VL - 338 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2003.12.029/ DO - 10.1016/j.crma.2003.12.029 LA - fr ID - CRMATH_2004__338_6_457_0 ER -
%0 Journal Article %A Martel, Yvan %T Solutions de type multi-soliton des équations de KdV généralisées %J Comptes Rendus. Mathématique %D 2004 %P 457-460 %V 338 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2003.12.029/ %R 10.1016/j.crma.2003.12.029 %G fr %F CRMATH_2004__338_6_457_0
Martel, Yvan. Solutions de type multi-soliton des équations de KdV généralisées. Comptes Rendus. Mathématique, Tome 338 (2004) no. 6, pp. 457-460. doi : 10.1016/j.crma.2003.12.029. http://www.numdam.org/articles/10.1016/j.crma.2003.12.029/
[1] Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Comm. Pure Appl. Math., Volume 46 (1993), pp. 527-620
[2] Element of Soliton Theory, Wiley, New York, 1980
[3] Y. Martel, Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg–de Vries equations, Préprint
[4] Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Rational Mech. Anal., Volume 157 (2001), pp. 219-254
[5] Blow up in finite time and dynamics of blow up solutions for the L2-critical generalized KdV equations, J. Amer. Math. Soc., Volume 15 (2002), pp. 617-664
[6] Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations, Commun. Math. Phys., Volume 231 (2002), pp. 347-373
[7] Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity, Commun. Math. Phys., Volume 129 (1990), pp. 223-240
[8] Existence of blow-up solutions in the energy space for the critical generalized Korteweg–de Vries equation, J. Amer. Math. Soc., Volume 14 (2001), pp. 555-578
[9] The Korteweg–de Vries equation: a survey of results, SIAM Rev., Volume 18 (1976), pp. 412-459
[10] Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., Volume 39 (1986), pp. 51-68
Cité par Sources :
