Comptes Rendus
Équations aux dérivées partielles
Analysis of stability and instability for standing waves of the double power one dimensional nonlinear Schrödinger equation
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 867-892.

For the double power one dimensional nonlinear Schrödinger equation, we establish a complete classification of the stability or instability of standing waves with positive frequencies. In particular, we fill out the gaps left open by previous studies. Stability or instability follows from the analysis of the slope criterion of Grillakis, Shatah and Strauss. The main new ingredients in our approach are a reformulation of the slope and the explicit calculation of the slope value in the zero-frequency case. Our theoretical results are complemented with numerical experiments.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.351
Classification : 35Q55, 35B35
Mots clés : nonlinear Schrödinger equation, double power nonlinearity, standing waves, stability, orbital stability

Perla Kfoury 1 ; Stefan Le Coz 1 ; Tai-Peng Tsai 2

1 Institut de Mathématiques de Toulouse ; UMR5219, Université de Toulouse ; CNRS, UPS IMT, F-31062 Toulouse Cedex 9,France
2 Department of Mathematics, University of British Columbia, Vancouver BC, Canada V6T 1Z2
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{CRMATH_2022__360_G8_867_0,
     author = {Perla Kfoury and Stefan Le Coz and Tai-Peng Tsai},
     title = {Analysis of stability and instability for standing waves of the double power one dimensional nonlinear {Schr\"odinger} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {867--892},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.351},
     language = {en},
}
TY  - JOUR
AU  - Perla Kfoury
AU  - Stefan Le Coz
AU  - Tai-Peng Tsai
TI  - Analysis of stability and instability for standing waves of the double power one dimensional nonlinear Schrödinger equation
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 867
EP  - 892
VL  - 360
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.351
LA  - en
ID  - CRMATH_2022__360_G8_867_0
ER  - 
%0 Journal Article
%A Perla Kfoury
%A Stefan Le Coz
%A Tai-Peng Tsai
%T Analysis of stability and instability for standing waves of the double power one dimensional nonlinear Schrödinger equation
%J Comptes Rendus. Mathématique
%D 2022
%P 867-892
%V 360
%I Académie des sciences, Paris
%R 10.5802/crmath.351
%G en
%F CRMATH_2022__360_G8_867_0
Perla Kfoury; Stefan Le Coz; Tai-Peng Tsai. Analysis of stability and instability for standing waves of the double power one dimensional nonlinear Schrödinger equation. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 867-892. doi : 10.5802/crmath.351. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.351/

[1] Milton Abramowitz; Irene A. Stegun Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, 55, U.S. Government Printing Office, Washington, D.C., 1964, xiv+1046 pages

[2] Govin P. Agrawal Nonlinear fiber optics, Optics and Photonics, Academic Press Inc., 2007

[3] Nail Akhmediev; Adrian Ankiewicz; Roger Grimshaw Hamiltonian-versus-energy diagrams in soliton theory, Phys. Rev. E, Volume 59 (1999) no. 5, p. 6088

[4] Jaime Angulo Pava; César A. Hernández Melo On stability properties of the cubic-quintic Schrödinger equation with δ-point interaction, Commun. Pure Appl. Anal., Volume 18 (2019) no. 4, pp. 2093-2116 | DOI

[5] Jaime Angulo Pava; César A. Hernández Melo; Ramón G. Plaza Orbital stability of standing waves for the nonlinear Schrödinger equation with attractive delta potential and double power repulsive nonlinearity, J. Math. Phys., Volume 60 (2019) no. 7, 071501, 23 pages | DOI

[6] Xavier Antoine; Weizhu Bao; Christophe Besse Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Commun., Volume 184 (2013) no. 12, pp. 2621-2633 | DOI

[7] Jacopo Bellazzini; Luigi Forcella; Vladimir Georgiev Ground state energy threshold and blow-up for NLS with competing nonlinearities (2020) | arXiv

[8] Henri Berestycki; Thierry Cazenave Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris, Volume 293 (1981) no. 9, pp. 489-492

[9] Henri Berestycki; Pierre-Louis Lions Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal., Volume 82 (1983) no. 4, pp. 313-345 | DOI

[10] Christophe Besse A relaxation scheme for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., Volume 42 (2004) no. 3, pp. 934-952

[11] Rémi Carles; Christian Klein; Christof Sparber On soliton (in-)stability in multi-dimensional cubic-quintic nonlinear Schrödinger equations (2020) (21 pages) | HAL

[12] Thierry Cazenave Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society; Courant Institute of Mathematical Sciences, 2003

[13] Thierry Cazenave; Pierre-Louis Lions Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., Volume 85 (1982) no. 4, pp. 549-561

[14] Andrew Comech; Dmitry Pelinovsky Purely nonlinear instability of standing waves with minimal energy, Commun. Pure Appl. Math., Volume 56 (2003) no. 11, pp. 1565-1607

[15] Stephan De Bièvre; François Genoud; Simona Rota-Nodari Orbital stability: analysis meets geometry, Nonlinear optical and atomic systems (Lecture Notes in Mathematics), Volume 2146, Springer, 2015, pp. 147-273 | DOI

[16] Stephan De Bièvre; Simona Rota-Nodari Orbital Stability via the Energy–Momentum Method: The Case of Higher Dimensional Symmetry Groups, Arch. Ration. Mech. Anal., Volume 231 (2019) no. 1, pp. 233-284 | DOI

[17] Noriyoshi Fukaya; Masayuki Hayashi Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities, Trans. Am. Math. Soc., Volume 374 (2021) no. 2, pp. 1421-1447 | DOI

[18] Reika Fukuizumi Stability and instability of standing waves for nonlinear Schrödinger equations, Ph. D. Thesis, Tohoku Mathematical Publications 25 (2003)

[19] François Genoud; Boris A. Malomed; Rada M. Weishäupl Stable NLS solitons in a cubic-quintic medium with a delta-function potential, Nonlinear Anal., Theory Methods Appl., Volume 133 (2016), pp. 28-50 | DOI

[20] Manoussos Grillakis; Jalal Shatah; Walter Strauss Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., Volume 74 (1987) no. 1, pp. 160-197 | DOI

[21] 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

[22] Zihua Guo; Cui Ning; Yifei Wu Instability of the solitary wave solutions for the generalized derivative nonlinear Schrödinger equation in the critical frequency case, Math. Res. Lett., Volume 27 (2020) no. 2, pp. 339-375 | DOI | MR

[23] Masayuki Hayashi Sharp thresholds for stability and instability of standing waves in a double power nonlinear Schrödinger equation (2021) | arXiv

[24] Nakao Hayashi; Tohru Ozawa On the derivative nonlinear Schrödinger equation, Physica D, Volume 55 (1992) no. 1-2, pp. 14-36 | DOI

[25] Ilya D. Iliev; Kiril P. Kirchev Stability and instability of solitary waves for one-dimensional singular Schrödinger equations, Differ. Integral Equ., Volume 6 (1993), pp. 685-703

[26] Perla Kfoury; Stefan Le Coz; Tai-Peng Tsai Stability-of-standing-waves-of-the-double-power-1D-NLS, 2021 https://github.com/perlakfoury/stability-of-standing-waves-of-the-double-power-1d-nls

[27] Stefan Le Coz; Yvan Martel; Pierre Raphaël Minimal mass blow up solutions for a double power nonlinear Schrödinger equation, Rev. Mat. Iberoam., Volume 32 (2016) no. 3, pp. 795-833

[28] Stefan Le Coz; Yifei Wu Stability of Multisolitons for the Derivative Nonlinear Schrödinger Equation, Int. Math. Res. Not., Volume 2018 (2018) no. 13, pp. 4120-4170 | DOI

[29] Mathieu Lewin; Simona Rota Nodari The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications, Calc. Var. Partial Differ. Equ., Volume 59 (2020) no. 6, pp. 1-49

[30] Fei Justina Liu; Tai-Peng Tsai; Ian Zwiers Existence and stability of standing waves for one dimensional NLS with triple power nonlinearities, Nonlinear Anal., Theory Methods Appl., Volume 211 (2021), 112409, 34 pages | DOI

[31] Masaya Maeda Stability and instability of standing waves for 1-dimensional nonlinear Schrödinger equation with multiple-power nonlinearity, Kodai Math. J., Volume 31 (2008) no. 2, pp. 263-271 | DOI

[32] Masaya Maeda Stability of bound states of Hamiltonian PDEs in the degenerate cases, J. Funct. Anal., Volume 263 (2012) no. 2, pp. 511-528 | DOI | MR

[33] Masahito Ohta Instability of standing waves for the generalized Davey-Stewartson system, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 62 (1995) no. 1, pp. 69-80

[34] Masahito Ohta Instability of bound states for abstract nonlinear Schrödinger equations, J. Funct. Anal., Volume 261 (2011) no. 1, pp. 90-110 | DOI | MR

[35] Masahito Ohta; Takahiro Yamaguchi Strong instability of standing waves for nonlinear Schrödinger equations with double power nonlinearity, SUT J. Math., Volume 51 (2015) no. 1, pp. 49-58

[36] Phan Van Tin On the derivative nonlinear Schrödinger equation on the half line with Robin boundary condition, J. Math. Phys., Volume 62 (2021) no. 8, 081502, 24 pages | DOI

[37] Michael I. Weinstein Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., Volume 87 (1982/83) no. 4, pp. 567-576

[38] Michael I. Weinstein Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., Volume 16 (1985), pp. 472-491 | Zbl

[39] Yifei Wu Instability of the standing waves for the nonlinear Klein-Gordon equations in one dimension (2018) | arXiv

Cité par Sources :

Commentaires - Politique