Asymptotic stability of small solitons for one-dimensional nonlinear Schrödinger equations
Yvan Martel1
1 Laboratoire de mathématiques de Versailles, UVSQ, Université Paris-Saclay, CNRS, 45 avenue des États-Unis, 78035 Versailles Cedex, France
Séminaire Laurent Schwartz — EDP et applications (2023-2024), Exposé no. 10, 22 p.

We review results from two recent articles [34, 35] on the asymptotic completeness of small standing solitary waves for a class of one-dimensional nonlinear Schrödinger equations. The models considered are perturbations of the integrable cubic 1D Schrödinger equation. The notion of internal modes plays an important role and part of the discussion concerns their existence. In case an internal mode exists, the proof of asymptotic stability is more delicate and relies on a nonlinear variant of the Fermi golden rule.

Publié le :
DOI : 10.5802/slsedp.170
Yvan Martel 1

1 Laboratoire de mathématiques de Versailles, UVSQ, Université Paris-Saclay, CNRS, 45 avenue des États-Unis, 78035 Versailles Cedex, France 
@article{SLSEDP_2023-2024____A7_0,
     author = {Yvan Martel},
     title = {Asymptotic stability of small solitons for one-dimensional~nonlinear {Schr\"odinger} equations},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:10},
     pages = {1--22},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2023-2024},
     doi = {10.5802/slsedp.170},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.170/}
}
TY  - JOUR
AU  - Yvan Martel
TI  - Asymptotic stability of small solitons for one-dimensional nonlinear Schrödinger equations
JO  - Séminaire Laurent Schwartz — EDP et applications
N1  - talk:10
PY  - 2023-2024
SP  - 1
EP  - 22
PB  - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.170/
DO  - 10.5802/slsedp.170
LA  - en
ID  - SLSEDP_2023-2024____A7_0
ER  - 
%0 Journal Article
%A Yvan Martel
%T Asymptotic stability of small solitons for one-dimensional nonlinear Schrödinger equations
%J Séminaire Laurent Schwartz — EDP et applications
%Z talk:10
%D 2023-2024
%P 1-22
%I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
%U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.170/
%R 10.5802/slsedp.170
%G en
%F SLSEDP_2023-2024____A7_0
Yvan Martel. Asymptotic stability of small solitons for one-dimensional nonlinear Schrödinger equations. Séminaire Laurent Schwartz — EDP et applications (2023-2024), Exposé no. 10, 22 p. doi : 10.5802/slsedp.170. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.170/

[1] V.S. Buslaev and G. Perelman, On nonlinear scattering of states which are close to a soliton. Astérisque 210 (1992), 49–63. | Zbl

[2] V.S. Buslaev and G.S. Perelman, Nonlinear scattering: the states which are close to a soliton. Journal of Mathematical Sciences 77 (1995), 3161–3169. | DOI

[3] T. Cazenave, Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics 10. American Mathematical Society (2003). | DOI | Zbl

[4] T. Cazenave and P.L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations. Comm. Math. Phys. 85 (1982), 549–561. | DOI | Zbl

[5] S.-M. Chang, S. Gustafson, K. Nakanishi and T.-P. Tsai, Spectra of linearized operators for NLS solitary waves. SIAM J. Math. Anal. 39 (2007/08), 1070–1111. | DOI | Zbl

[6] G. Chen and F. Pusateri, The 1d nonlinear Schrödinger equation with a weighted L 1 potential. Analysis& PDE 15 (2022), 937–982. | DOI | Zbl

[7] G. Chen, Long-time dynamics of small solutions to 1d cubic nonlinear Schrödinger equations with a trapping potential. | arXiv

[8] M. Coles and S. Gustafson, A degenerate edge bifurcation in the 1D linearized nonlinear Schrödinger equation. Discrete Contin. Dyn. Syst. 36 (2016), 2991–3009. | DOI | Zbl

[9] C. Collot and P. Germain, Asymptotic stability of solitary waves for one dimensional nonlinear Schrödinger equations. | arXiv

[10] S. Cuccagna and M. Maeda, Coordinates at small energy and refined profiles for the Nonlinear Schrödinger Equation. Ann. PDE 7 (2021). | DOI | Zbl

[11] S. Cuccagna and M. Maeda, A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete Contin. Dyn. Syst., Series S 14 (2021) 1693–1716. | Zbl

[12] S. Cuccagna and M. Maeda, On selection of standing wave at small energy in the 1D Cubic Schrödinger Equation with a trapping potential. Commun. Math. Phys. 396 (2022), 1135-1186. | DOI | Zbl

[13] S. Cuccagna and M. Maeda, Asymptotic stability of kink with internal modes under odd perturbation. NoDEA, Nonlinear Differ. Equ. Appl. 30 (2023). | DOI | Zbl

[14] S. Cuccagna and M. Maeda, The asymptotic stability on the line of ground states of the pure power NLS with 0<|p-3|≪1 (2024). | arXiv

[15] S. Cuccagna and M. Maeda, On the asymptotic stability of ground states of the pure power NLS on the line at 3rd and 4th order Fermi Golden Rule (2024). | arXiv

[16] S. Cuccagna and D.E. Pelinovsky, The asymptotic stability of solitons in the cubic NLS equation on the line. Applicable Analysis 93 (2014), 791-822. | DOI | Zbl

[17] P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979), 121–251. | DOI | Zbl

[18] P. Germain and F. Pusateri, Quadratic Klein-Gordon equations with a potential in one dimension. Forum Math. Pi 10 (2022) 172 p. | DOI | Zbl

[19] M. Grillakis, J. Shatah and W.A. Strauss, Stability Theory of solitary waves in the presence of symmetry, I, J. Funct. Anal. 74 (1987), 160–197. | DOI | Zbl

[20] S. Gustafson, K. Nakanishi, and T.-P. Tsai, Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves. Int. Math. Res. Not. (2004), 3559–3584. | Zbl

[21] S. Gustafson, K. Nakanishi and T.-P. Tsai, Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schrödinger maps on ℝ 2 . Commun. Math. Phys. 300 (2010), 205–242. | DOI | Zbl

[22] Y.S. Kivshar and B.A. Malomed, Dynamics of solitons in nearly integrable systems. Rev. Mod. Phys. 61 (1989), 763. | DOI

[23] M. Kowalczyk and Y. Martel, Kink dynamics under odd perturbations for (1+1)-scalar field models with one internal mode. To appear in Math. Res. Lett.

[24] M. Kowalczyk, Y. Martel and C. Muñoz, Kink dynamics in the ϕ 4 model: asymptotic stability for odd perturbations in the energy space. J. Amer. Math. Soc. 30 (2017), 769–798. | DOI | Zbl

[25] M. Kowalczyk, Y. Martel and C. Muñoz, On asymptotic stability of nonlinear waves. Sémin. Laurent Schwartz, EDP Appl. 2016-2017, Exp. No. 18, 27 p. (2017). | DOI | Zbl

[26] M. Kowalczyk, Y. Martel and C. Muñoz, Soliton dynamics for the 1D NLKG equation with symmetry and in the absence of internal modes. J. Eur. Math. Soc. 24 (2022), 2133–2167. | DOI | Zbl

[27] M. Kowalczyk, Y. Martel, C. Muñoz and H. Van Den Bosch, A sufficient condition for asymptotic stability of kinks in general (1+1)-scalar field models. Ann. PDE 7 (2021). | DOI | Zbl

[28] J. Krieger and W. Schlag, Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension. J. Amer. Math. Soc. 19 (2006), 815–920. | DOI | Zbl

[29] Y. Li and J. Lührmann, Soliton dynamics for the 1D quadratic Klein-Gordon equation with symmetry. J. Differ. Equations 344 (2023), 172–202. | DOI | Zbl

[30] Asymptotic stability of solitary waves for the 1D focusing cubic Schrödinger equation under even perturbations. | arXiv

[31] H. Lindblad, J. Lührmann, W. Schlag and A. Soffer, On modified scattering for 1D quadratic Klein-Gordon equations with non-generic potentials. Int. Math. Res. Not. (2023), 5118–5208. | DOI | Zbl

[32] H. Lindblad, J. Lührmann, A. Soffer, Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities. Arch. Ration. Mech. Anal. 241 (2021), 1459–1527. | DOI | Zbl

[33] Y. Martel, Linear problems related to asymptotic stability of solitons of the generalized KdV equations. SIAM J. Math. Anal. 38 (2006), 759–781. | DOI | Zbl

[34] Y. Martel, Asymptotic stability of solitary waves for the 1D cubic-quintic Schrödinger equation with no internal mode. Prob. Math. Phys. 3 (2022), 839–867. | DOI | Zbl

[35] Y. Martel, Asymptotic stability of small solitary waves for the one-dimensional cubic-quintic Schrödinger equation. Invent. Math. 237 (2024), 1253-1328. | DOI | Zbl

[36] Y. Martel and F. Merle, A Liouville theorem for the critical generalized Korteweg-de Vries equation. J. Math. Pures Appl. 79 (2000), 339–425. | DOI | Zbl

[37] Y. Martel and F. Merle, Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157 (2001), 219–254. | DOI | Zbl

[38] Y. Martel and F. Merle, Multi solitary waves for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23 (2006), 849–864. | DOI | Zbl

[39] C. Maulén and C. Muñoz. Asymptotic stability of the fourth order ϕ 4 kink for general perturbations in the energy space. | arXiv

[40] M. Melgaard, On bound states for systems of weakly coupled Schrödinger equations in one space dimension. J. Math. Phys. 43 (2002), 5365–5385. | DOI | Zbl

[41] F. Merle and P. Raphaël, Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation. Geom. Funct. Anal. 13 (2003), 591–642. | DOI | Zbl

[42] T. Mizumachi, Asymptotic stability of small solitary waves to 1D nonlinear Schrödinger equations with potential. J. Math. Kyoto Univ. 48 (2008), 471–497. | DOI | Zbl

[43] I.P. Naumkin, Sharp asymptotic behavior of solution for cubic nonlinear Schrödinger equations with a potential. J. Math. Phys. 57 (2016), 05501. | DOI | Zbl

[44] M. Ohta, Stability and instability of standing waves for one-dimensional nonlinear Schrödinger equations with double power nonlinearity. Kodai Math. J. 18 (1995), 68–74. | DOI | Zbl

[45] E. Olmedilla, Multiple pole solutions of the nonlinear Schrödinger equation. Phys. D 25 (1987), 330–346. | DOI | Zbl

[46] D.E. Pelinovsky, Y.S. Kivshar and V.V. Afanasjev, Internal modes of envelope solitons. Phys. D 116 (1998), 121–142. | DOI | Zbl

[47] M. Reed and B. Simon, Analysis of Operators IV. Methods of Modern Mathematical Physics. Academic Press, 1978. | Zbl

[48] G. Rialland, Asymptotic stability of solitary waves for the 1D near-cubic non-linear Schrödinger equation in the absence of internal modes. Nonlinear Anal., TMA, Ser. A 241 Article ID 113474, 30 p. (2024). | DOI | Zbl

[49] G. Rialland, Asymptotic stability of solitons for near-cubic NLS equation with an internal mode. | arXiv

[50] W. Schlag, Dispersive estimates for Schrödinger operators: A survey. Mathematical aspects of nonlinear dispersive equations. Ann. of Math. Stud. 163, Princeton Univ. Press, Princeton, NJ, 2007. | Zbl

[51] I.M. Sigal, Non-linear Wave and Schrödinger equations. I. Instability of Periodic and Quasiperiodic Solutions. Commun. Math. Phys. 153 (1993), 297–320. | DOI | Zbl

[52] B. Simon, The bound state of weakly coupled Schrödinger operators in one and two dimensions. Ann. Phys. 97 (1976), 279–288. | DOI | Zbl

[53] A. Soffer and M.I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations. Commun. Math. Phys. 133 (1990), 119–146. | DOI | Zbl

[54] A. Soffer and M.I. Weinstein, Time dependent resonance theory. Geom. Funct. Anal. 8 (1998), 1086–1128. | DOI | Zbl

[55] A. Soffer and M.I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 136 (1999), 9–74. | DOI | Zbl

[56] M.I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math. 29 (1986), 51–68. | DOI | Zbl

[57] T. Zakharov and A.B. Shabat. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34 (1972), 62–69.

Cité par Sources :