A review on asymptotic stability of solitary waves in nonlinear dispersive problems in dimension one
[Étude de la stabilité asymptotique des ondes solitaires dans les problèmes dispersifs non linéaires en dimension un]
Pierre Germain1
1 Department of Mathematics, Huxley Building, South Kensington Campus, Imperial College London, London SW7 2AZ, United Kingdom
Journées équations aux dérivées partielles (2024), Exposé no. 6, 22 p.

Cet article de survol s’intéresse à la stabilité asymptotique des ondes solitaires d’équations dispersives non-linéaires. Nous nous attacherons plus particulièrement à l’équation de Schrödinger non-linéaire, à la notion de stabilité asymptotique complète (qui demande que la solution se décompose asymptotiquement en une onde solitaire et une radiation décroissante) et aux méthodes spectrales. Nous tenterons aussi de présenter l’état de l’art dans un contexte plus général, incluant l’équation de Klein–Gordon non-linéaire, la notion de stabilité asymptotique locale et les méthodes de viriel.

We review asymptotic stability of solitary waves for nonlinear dispersive equations set on the line. Our focus is threefold: first, the nonlinear Schrödinger equation; second, the notion of full asymptotic stability (which states that perturbations of a solitary wave decompose globally into a solitary wave and a decaying solution); and third, spectral methods. Besides this focus, we summarize the state of the art in a broader context, including nonlinear Klein–Gordon equations, the notion of local asymptotic stability, and virial methods.

Publié le :
DOI : 10.5802/jedp.687
Keywords: nonlinear dispersive equations, solitary waves, asymptotic stability
Mot clés : équations dispersives non linéaires, ondes solitaires, stabilité asymptotique
Pierre Germain 1

1 Department of Mathematics, Huxley Building, South Kensington Campus, Imperial College London, London SW7 2AZ, United Kingdom 
@incollection{JEDP_2024____A6_0,
     author = {Pierre Germain},
     title = {A review on asymptotic stability of solitary waves in nonlinear dispersive problems in dimension one},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     note = {talk:6},
     pages = {1--22},
     publisher = {R\'eseau th\'ematique AEDP du CNRS},
     year = {2024},
     doi = {10.5802/jedp.687},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.687/}
}
TY  - JOUR
AU  - Pierre Germain
TI  - A review on asymptotic stability of solitary waves in nonlinear dispersive problems in dimension one
JO  - Journées équations aux dérivées partielles
N1  - talk:6
PY  - 2024
SP  - 1
EP  - 22
PB  - Réseau thématique AEDP du CNRS
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.687/
DO  - 10.5802/jedp.687
LA  - en
ID  - JEDP_2024____A6_0
ER  - 
%0 Journal Article
%A Pierre Germain
%T A review on asymptotic stability of solitary waves in nonlinear dispersive problems in dimension one
%J Journées équations aux dérivées partielles
%Z talk:6
%D 2024
%P 1-22
%I Réseau thématique AEDP du CNRS
%U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.687/
%R 10.5802/jedp.687
%G en
%F JEDP_2024____A6_0
Pierre Germain. A review on asymptotic stability of solitary waves in nonlinear dispersive problems in dimension one. Journées équations aux dérivées partielles (2024), Exposé no. 6, 22 p. doi : 10.5802/jedp.687. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.687/

[1] Miguel A. Alejo; Luca Fanelli; Claudio Muñoz Stability and instability of breathers in the U (1) Sasa–Satsuma and nonlinear Schrödinger models, Nonlinearity, Volume 34 (2021) no. 5, pp. 3429-3484 | DOI | MR

[2] Adrian Ankiewicz; Nail Akhmediev Higher-order integrable evolution equation and its soliton solutions, Phys. Lett. A, Volume 378 (2014) no. 4, pp. 358-361 | DOI | MR

[3] Dario Bambusi; Scipio Cuccagna On dispersion of small energy solutions of the nonlinear Klein Gordon equation with a potential, Am. J. Math., Volume 133 (2011) no. 5, pp. 1421-1468 | DOI | MR

[4] Henri Berestycki; Pierre-Louis Lions Nonlinear scalar field equations, I existence of a ground state, Arch. Ration. Mech. Anal., Volume 82 (1983), pp. 313-345 | DOI | MR

[5] Michael Borghese; Robert Jenkins; Kenneth D. T.-R. McLaughlin Long time asymptotic behavior of the focusing nonlinear Schrödinger equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 35 (2018) no. 4, pp. 887-920 | DOI | Numdam | MR

[6] Vladimir Buslaev; Galina Perelman Scattering for the nonlinear Schrödinger equation: states close to a soliton, St. Petersbg. Math. J., Volume 4 (1993), pp. 1111-1142

[7] Vladimir Buslaev; Catherine Sulem On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 20 (2003) no. 3, pp. 419-475 | DOI | Numdam | MR

[8] Thierry Cazenave Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, 2003 | MR

[9] Thierry Cazenave; Pierre-Louis Lions Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., Volume 85 (1982), pp. 549-561 | DOI | MR

[10] Shu-Ming Chang; Stephen Gustafson; Kenji Nakanishi; Tai-Peng Tsai Spectra of linearized operators for NLS solitary waves, SIAM J. Math. Anal., Volume 39 (2008) no. 4, pp. 1070-1111 | DOI | MR

[11] Gong Chen Long-time dynamics of small solutions to 1d cubic nonlinear Schrödinger equations with a trapping potential (2021) | arXiv

[12] Gong Chen; Jiaqi Liu; Bingying Lu Long-time asymptotics and stability for the sine-Gordon equation (2020) | arXiv

[13] Gong Chen; Fabio Pusateri The 1-dimensional nonlinear Schrödinger equation with a weighted L1 potential, Anal. PDE, Volume 15 (2022) no. 4, pp. 937-982 | DOI | MR

[14] Gong Chen; Fabio Pusateri On the 1 d cubic NLS with a non-generic potential (2022) | arXiv

[15] Matta Coles; Stephen Gustafson A degenerate edge bifurcation in the 1D linearized nonlinear Schrödinger equation, Discrete Contin. Dyn. Syst., Volume 36 (2016) no. 6, pp. 2991-3009 | DOI | MR

[16] Charles Collot; Pierre Germain Asymptotic Stability of Solitary Waves for One Dimensional Nonlinear Schr∖" odinger Equations (2023) | arXiv

[17] Scipio Cuccagna; Masaya Maeda A survey on asymptotic stability of ground states of nonlinear Schrodinger equations II, Discrete Contin. Dyn. Syst., Ser. S, Volume 14 (2021) no. 5, pp. 1693-1716 | MR

[18] Scipio Cuccagna; Masaya Maeda Asymptotic stability of kink with internal modes under odd perturbation, NoDEA, Nonlinear Differ. Equ. Appl., Volume 30 (2023) no. 1, 1, 47 pages | MR

[19] Scipio Cuccagna; Masaya Maeda The asymptotic stability on the line of ground states of the pure power NLS with 0<|p-3|≪1 (2024) | arXiv

[20] Scipio Cuccagna; Masaya 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

[21] Scipio Cuccagna; Masaya Maeda; Federico Murgante; Stefano Scrobogna On asymptotic stability on a center hypersurface at the solition for even solutions of the NLKG when 2>p>5 3 (2023) | arXiv

[22] Scipio Cuccagna; Masaya Maeda; Stefano Scrobogna Small energy stabilization for 1D nonlinear Klein Gordon equations, J. Differ. Equations, Volume 350 (2023), pp. 52-88 | DOI | MR

[23] Scipio Cuccagna; Dmitry E. Pelinovsky The asymptotic stability of solitons in the cubic NLS equation on the line, Appl. Anal., Volume 93 (2014) no. 4, pp. 791-822 | DOI | MR

[24] Percy Deift; Eugene Trubowitz Inverse scattering on the line, Commun. Pure Appl. Math., Volume 32 (1979), pp. 121-251 | DOI | MR

[25] Jean-Marc Delort Modified scattering for odd solutions of cubic nonlinear Schrödinger equations with potential in dimension one, 2016 (https://hal.science/hal-01396705)

[26] Jean-Marc Delort; Nader Masmoudi Long-time dispersive estimates for perturbations of a kink solution of one-dimensional cubic wave equations, Memoirs of the European Mathematical Society, 1, European Mathematical Society, 2022 | DOI | MR

[27] B. Erdoĝan; Wilhelm Schlag Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II, J. Anal. Math., Volume 99 (2006) no. 1, pp. 199-248 | DOI | MR

[28] Pierre Germain Space-time resonances, Journ. Équ. Dériv. Partielles, Volume 2010 (2010), 8, 10 pages | Numdam

[29] Pierre Germain; Nader Masmoudi; Jalal Shatah Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not., Volume 2009 (2009) no. 3, pp. 414-432 | MR

[30] Pierre Germain; Nader Masmoudi; Jalal Shatah Global solutions for the gravity water waves equation in dimension 3, Ann. Math., Volume 175 (2012) no. 2, pp. 691-754 | DOI | Zbl

[31] Pierre Germain; Fabio Pusateri Quadratic Klein-Gordon equations with a potential in one dimension, Forum Math. Pi, Volume 10 (2022), e17, 172 pages | MR

[32] Pierre Germain; Fabio Pusateri; Frédéric Rousset The nonlinear Schrödinger equation with a potential, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 35 (2018) no. 6, pp. 1477-1530 | DOI | Numdam | MR

[33] Pierre Germain; Fabio Pusateri; Katherine Zhiyuan Zhang On 1d quadratic Klein–Gordon equations with a potential and symmetries, Arch. Ration. Mech. Anal., Volume 247 (2023) no. 2, 17, 39 pages | MR

[34] Michael Goldberg Transport in the one-dimensional Schrödinger equation, Proc. Am. Math. Soc., Volume 135 (2007) no. 10, pp. 3171-3179 | DOI | MR

[35] Manoussos Grillakis; Jalal Shatah; Walter Strauss Stability theory of solitary waves in the presence of symmetry, II, J. Funct. Anal., Volume 94 (1990) no. 2, pp. 308-348 | DOI

[36] Nakao Hayashi; Pavel I. Naumkin Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations, Am. J. Math., Volume 120 (1998) no. 2, pp. 369-389 | DOI | MR

[37] Mihaela Ifrim; Daniel Tataru Global bounds for the cubic nonlinear Schrödinger equation (NLS) in one space dimension, Nonlinearity, Volume 28 (2015) no. 8, p. 2661-265 | DOI | MR

[38] Adilbek Kairzhan; Fabio Pusateri Asymptotic stability near the soliton for quartic Klein–Gordon equation in 1D, Pure Appl. Anal., Volume 5 (2023) no. 4, pp. 795-832 | DOI | MR

[39] Jun Kato; Fabio Pusateri A new proof of long-range scattering for critical nonlinear Schrödinger equations, Differ. Integral Equ., Volume 24 (2011), pp. 923-940 | MR

[40] Herbert Koch; Dongxiao Yu Asymptotic stability of the sine-Gordon kinks under perturbations in weighted Sobolev norms (2023) | arXiv

[41] E. A. Kopylova; A. I. Komech On asymptotic stability of kink for relativistic Ginzburg–Landau equations, Arch. Ration. Mech. Anal., Volume 202 (2011), pp. 213-245 | DOI | MR

[42] E. A. Kopylova; A. I. Komech On asymptotic stability of moving kink for relativistic Ginzburg–Landau equation, Commun. Math. Phys., Volume 302 (2011), pp. 225-252 | DOI | MR

[43] Michał Kowalczyk; Yvan Martel Kink dynamics under odd perturbations for (1+ 1)-scalar field models with one internal mode (2022) | arXiv

[44] Michał Kowalczyk; Yvan Martel; Claudio Muñoz Kink dynamics in the Φ 4 model: asymptotic stability for odd perturbations in the energy space, J. Am. Math. Soc., Volume 30 (2017) no. 3, pp. 769-798 | DOI | MR

[45] Michał Kowalczyk; Yvan Martel; Claudio Muñoz Nonexistence of small, odd breathers for a class of nonlinear wave equations, Lett. Math. Phys., Volume 107 (2017), pp. 921-931 | DOI | MR

[46] Michał Kowalczyk; Yvan Martel; Claudio Muñoz On asymptotic stability of nonlinear waves, Sémin. Laurent Schwartz, EDP Appl., Volume 2016-2017 (2017), 18, 27 pages | Numdam

[47] Michał Kowalczyk; Yvan Martel; Claudio Muñoz Soliton dynamics for the 1D NLKG equation with symmetry and in the absence of internal modes, J. Eur. Math. Soc., Volume 24 (2021) no. 6, pp. 2133-2167 | DOI | MR

[48] Michał Kowalczyk; Yvan Martel; Claudio Muñoz; Hanne Van Den Bosch A sufficient condition for asymptotic stability of kinks in general (1+ 1)-scalar field models, Ann. PDE, Volume 7 (2021), pp. 1-98 | MR

[49] Joachim Krieger; Kenji Nakanishi; Wilhelm Schlag Global dynamics above the ground state energy for the one-dimensional NLKG equation., Math. Z., Volume 272 (2012), pp. 297-316 | DOI | MR

[50] Joachim Krieger; Wilhelm Schlag Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension, J. Am. Math. Soc., Volume 19 (2006) no. 4, pp. 815-920 | DOI | MR

[51] Tristan Léger; Fabio Pusateri Internal modes and radiation damping for quadratic Klein-Gordon in 3D (2021) | arXiv

[52] Tristan Léger; Fabio Pusateri Internal mode-induced growth in 3d nonlinear Klein–Gordon equations, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., Volume 33 (2022) no. 3, pp. 695-727 | DOI | MR | Zbl

[53] Yongming Li Dispersive estimates for 1D matrix Schrödinger operators with threshold resonance, Calc. Var. Partial Differ. Equ., Volume 63 (2024) no. 8, pp. 1-54 | MR

[54] Yongming Li; Jonas Lührmann Soliton dynamics for the 1D quadratic Klein-Gordon equation with symmetry, J. Differ. Equations, Volume 344 (2023), pp. 172-202 | MR

[55] Yongming Li; Jonas Luhrmann Asymptotic stability of solitary waves for the 1D focusing cubic Schr∖" odinger equation under even perturbations (2024) | arXiv

[56] Hans Lindblad; Jonas Lührmann; Wilhelm Schlag; Avy Soffer On modified scattering for 1D quadratic Klein–Gordon equations with non-generic potentials, Int. Math. Res. Not., Volume 2023 (2023) no. 6, pp. 5118-5208 | DOI | MR

[57] Hans Lindblad; Jonas Lührmann; Avy Soffer Decay and Asymptotics for the One-Dimensional Klein–Gordon Equation with Variable Coefficient Cubic Nonlinearities, SIAM J. Math. Anal., Volume 52 (2020) no. 6, pp. 6379-6411 | DOI | MR

[58] Hans Lindblad; Jonas Lührmann; Avy Soffer Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities, Arch. Ration. Mech. Anal., Volume 241 (2021) no. 3, pp. 1459-1527 | DOI | MR

[59] Hans Lindblad; Avy Soffer Scattering and small data completeness for the critical nonlinear Schrödinger equation, Nonlinearity, Volume 19 (2006), pp. 345-353 | DOI | MR

[60] Jonas Luhrmann; Wilhelm Schlag Asymptotic stability of the sine-Gordon kink under odd perturbations (2021) | arXiv

[61] Jonas Luhrmann; Wilhelm Schlag On codimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation (2023) | arXiv

[62] Yvan Martel Interaction of solitons from the PDE point of view, Proceedings of the International Congress of Mathematicians (ICM 2018) (In 4 Volumes) Proceedings of the International Congress of Mathematicians 2018, World Scientific (2018), pp. 2439-2466

[63] Yvan Martel Asymptotic stability of solitary waves for the 1D cubic-quintic Schrödinger equation with no internal mode, Probab. Math. Phys., Volume 3 (2023) no. 4, pp. 839-867 | DOI | MR

[64] Yvan Martel Asymptotic stability of small solitons for one-dimensional nonlinear Schrödinger equations, Sémin. Laurent Schwartz, EDP Appl., Volume 2023-2024 (2024), 10, 22 pages | Zbl

[65] Yvan Martel Asymptotic stability of small standing solitary waves of the one-dimensional cubic-quintic Schrödinger equation, Invent. Math., Volume 237 (2024) no. 3, pp. 1-76 | MR

[66] Yvan Martel; Frank Merle Multi solitary waves for nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 23 (2006) no. 6, pp. 849-864 | DOI | Numdam | MR

[67] Satoshi Masaki; Jason Murphy; Jun-ichi Segata Modified scattering for the 1d cubic NLS with a repulsive delta potential (2017) | arXiv

[68] Satoshi Masaki; Jason Murphy; Jun-ichi Segata Asymptotic stability of solitary waves for the 1d NLS with an attractive delta potential (2020) | arXiv

[69] Satoshi Masaki; Jason Murphy; Jun-ichi Segata Stability of small solitary waves for the one-dimensional NLS with an attractive delta potential, Anal. PDE, Volume 13 (2020) no. 4, pp. 1099-1128 | DOI | MR

[70] Tetsu Mizumachi Asymptotic stability of small solitary waves to 1D nonlinear Schrödinger equations with potential, J. Math. Kyoto Univ., Volume 48 (2008) no. 3, pp. 471-497 | MR

[71] Jason Murphy A review of modified scattering for the 1d cubic NLS, RIMS Kôkyûroku Bessatsu, Volume B88 (2021), pp. 119-146 | MR

[72] Ivan P. Naumkin Sharp asymptotic behavior of solutions for cubic nonlinear Schrödinger equations with a potential, J. Math. Phys., Volume 57 (2016) no. 5, 051501, 31 pages | MR | Zbl

[73] Ivan P. Naumkin Nonlinear Schrödinger equations with exceptional potentials, J. Differ. Equations, Volume 265 (2018) no. 9, pp. 4575-4631 | DOI | MR

[74] E. Olmedilla Multiple pole solutions of the non-linear Schrödinger equation, Phys. D: Nonlinear Phenom., Volume 25 (1987) no. 1-3, pp. 330-346 | DOI | MR

[75] José M. Palacios; Fabio Pusateri Local energy control in the presence of a zero-energy resonance (2024) | arXiv

[76] Jaime Angulo Pava Nonlinear dispersive equations: existence and stability of solitary and periodic travelling wave solutions, American Mathematical Society, 2009 no. 156 | DOI | MR

[77] Dmitry E. Pelinovsky; Yuri S. Kivshar; Vsevolod V. Afanasjev Internal modes of envelope solitons, Phys. D: Nonlinear Phenom., Volume 116 (1998) no. 1-2, pp. 121-142 | DOI | MR

[78] Galina Perelman On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré, Volume 2 (2001) no. 4, pp. 605-673 | DOI | MR

[79] Fabio Pusateri; Avraham Soffer Bilinear estimates in the presence of a large potential and a critical NLS in 3d, Memoirs of the American Mathematical Society, 1498, American Mathematical Society, 2024

[80] Guillaume Rialland Asymptotic stability of solitary waves for the 1D near-cubic non-linear Schrödinger equation in the absence of internal modes, Nonlinear Anal., Theory Methods Appl., Volume 241 (2024), 113474 | MR

[81] Guillaume Rialland Asymptotic stability of solitons for near-cubic NLS equation with an internal mode (2024) | arXiv

[82] Aaron Saalmann Asymptotic stability of N-solitons in the cubic NLS equation, J. Hyperbolic Differ. Equ., Volume 14 (2017) no. 03, pp. 455-485 | DOI | MR

[83] Junkichi Satsuma; Nobuo Yajima B. Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media, Prog. Theor. Phys. Suppl., Volume 55 (1974), pp. 284-306 | DOI | MR

[84] Wilhelm Schlag Dispersive estimates for Schrödinger operators: a survey, Mathematical aspects of nonlinear dispersive equations (Annals of Mathematics Studies), Volume 163, Princeton University Press, 2005, pp. 255-285

[85] Aleksei Shabat; Vladimir Zakharov Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP, Volume 34 (1972) no. 1, pp. 62-69

[86] Avy Soffer Soliton dynamics and scattering, Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume III: Invited lectures, European Mathematical Society (2006), pp. 459-471 | MR

[87] Avy Soffer; Michael I. Weinstein Resonances, radiation damping and instabilitym in Hamiltonian nonlinear wave equations, Invent. Math., Volume 136 (1999) no. 1, pp. 9-74 | DOI | MR

[88] Gavin Stewart Asymptotics for the cubic 1D NLS with a slowly decaying potential (2024) | arXiv

[89] Catherine Sulem; Pierre-Louis Sulem The nonlinear Schrödinger equation: self-focusing and wave collapse, Applied Mathematical Sciences, 139, Springer, 2007 | MR

[90] Terence Tao Nonlinear dispersive equations: local and global analysis, American Mathematical Society, 2006 no. 106 | MR

[91] Terence Tao Why are solitons stable?, Bull. Am. Math. Soc., Volume 46 (2009) no. 1, pp. 1-33 | MR

[92] Tai-Peng Tsai; Horng-Tzer Yau Asymptotic dynamics of nonlinear Schrödinger equations: Resonance-dominated and dispersion-dominated solutions, Commun. Pure Appl. Math., Volume 55 (2002) no. 2, pp. 153-216 | DOI | MR

[93] Michael I. Weinstein Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., Volume 16 (1985) no. 3, pp. 472-491 | DOI | MR

[94] D. Yafaev Mathematical scattering theory. Analytic theory., Mathematical Surveys and Monographs, 158, American Mathematical Society, 2010 | DOI

Cité par Sources :