Two soliton collision for nonlinear Schrödinger equations in dimension 1
Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 3, pp. 357-384.

We study the collision of two solitons for the nonlinear Schrödinger equation iψ t =-ψ xx +F(|ψ| 2 )ψ, F(ξ)=-2ξ+O(ξ 2 ) as ξ0, in the case where one soliton is small with respect to the other. We show that in general, the two soliton structure is not preserved after the collision: while the large soliton survives, the small one splits into two outgoing waves that for sufficiently long times can be controlled by the cubic NLS: iψ t =-ψ xx -2|ψ| 2 ψ.

@article{AIHPC_2011__28_3_357_0,
     author = {Perelman, Galina},
     title = {Two soliton collision for nonlinear {Schr\"odinger} equations in dimension 1},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {357--384},
     publisher = {Elsevier},
     volume = {28},
     number = {3},
     year = {2011},
     doi = {10.1016/j.anihpc.2011.02.002},
     mrnumber = {2795711},
     zbl = {1217.35176},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2011.02.002/}
}
TY  - JOUR
AU  - Perelman, Galina
TI  - Two soliton collision for nonlinear Schrödinger equations in dimension 1
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2011
SP  - 357
EP  - 384
VL  - 28
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2011.02.002/
DO  - 10.1016/j.anihpc.2011.02.002
LA  - en
ID  - AIHPC_2011__28_3_357_0
ER  - 
%0 Journal Article
%A Perelman, Galina
%T Two soliton collision for nonlinear Schrödinger equations in dimension 1
%J Annales de l'I.H.P. Analyse non linéaire
%D 2011
%P 357-384
%V 28
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2011.02.002/
%R 10.1016/j.anihpc.2011.02.002
%G en
%F AIHPC_2011__28_3_357_0
Perelman, Galina. Two soliton collision for nonlinear Schrödinger equations in dimension 1. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 3, pp. 357-384. doi : 10.1016/j.anihpc.2011.02.002. http://www.numdam.org/articles/10.1016/j.anihpc.2011.02.002/

[1] T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math. vol. 10 (2003) | MR | Zbl

[2] V.S. Buslaev, G.S. Perelman, Scattering for the nonlinear Schrödinger equation: states close to a soliton, St. Petersburg Math. J. 4 no. 6 (1993), 1111-1143 | MR | Zbl

[3] J. Holmer, J. Marzuola, M. Zworski, Fast soliton scattering by delta impurities, Comm. Math. Phys. 274 no. 1 (2007), 187-216 | MR | Zbl

[4] J. Holmer, J. Marzuola, M. Zworski, Soliton splitting by external delta potentials, J. Nonlinear Sci. 17 no. 4 (2007), 349-367 | MR | Zbl

[5] S. Kamvissis, Long time behavior for the focusing nonlinear Schrödinger equation with real spectral singularities, Comm. Math. Phys. 180 no. 2 (1996), 325-341 | MR | Zbl

[6] J. Krieger, W. Schlag, Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension, J. Amer. Math. Soc. 19 no. 4 (2006), 815-920 | MR | Zbl

[7] Y. Martel, F. Merle, T.-P. Tsai, Stability in H 1 of the sum of K solitary waves for some nonlinear Schrödinger equations, Duke Math. J. 133 no. 3 (2006), 405-466 | Zbl

[8] Y. Martel, F. Merle, Stability of two soliton collision for nonintegrable gKdV, Comm. Math. Phys. 286 no. 1 (2009), 39-79 | MR | Zbl

[9] Y. Martel, F. Merle, Stability of two soliton collision for quartic gKdV, Annals of Math., in press. | MR

[10] G. Perelman, A remark on soliton–potential interactions for nonlinear Schrödinger equations, Math. Res. Lett. 16 no. 3 (2009), 477-486 | MR | Zbl

[11] A. Tovbis, S. Venakides, The eigenvalue problem for the focusing nonlinear Schrödinger equation: new solvable cases, Phys. D 146 no. 1–4 (2000), 150-164 | MR | Zbl

[12] M.I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39 no. 1 (1986), 51-68 | MR | Zbl

Cité par Sources :