Abstract
We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the initial data u0 ∈ X, where \(X \in \{M_{2,q}^s(\mathbb{R}),\,{H^\sigma}(\mathbb{T}),\,{H^{{s_1}}}(\mathbb{R}) + {H^{{s_2}}}(\mathbb{T})\}\) and q ∈ [1, 2], s ⩾ 0, or σ ⩾ 0, or s2 ⩾ s1 ⩾ 0. Moreover, if M s2,q (ℝ) ↪ L3(ℝ), or if \(\sigma \geqslant {1 \over 6}\), or if \({s_1} \geqslant {1 \over 6}\) and \({s_2} > {1 \over 2}\) we how that the Cauchy problem is unconditionally wellposed in X. Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ the normal form reduction via the differentiation by parts technique and build upon our previous work.
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References
A. V. Babin, A. A. Ilyin, E. S. Titi: On the regularization mechanism for the periodic Korteweg-de Vries equation. Commun. Pure Appl. Math. 64 (2011), 591–648.
M. Ben-Artzi, H. Koch, J.-C. Saut: Dispersion estimates for fourth order Schrödinger equations. C. R. Acad. Sci., Paris, Sér. I, Math. 330 (2000), 87–92.
Á. Bényi, K. Gröchenig, K. Okoudjou, L. G. Rogers: Unimodular Fourier multipliers for modulation spaces. J. Funct. Anal. 246 (2007), 366–384.
T. Boulenger, E. Lenzmann: Blowup for biharmonic NLS. Ann. Scient. Éc. Norm. Supér. (4) 50 (2017), 503–544.
L. Chaichenets, D. Hundertmark, P. Kunstmann, N. Pattakos: Knocking out teeth in one-dimensional periodic nonlinear Schrödinger equation. SIAM J. Math. Anal. 51 (2019), 3714–3749.
L. Chaichenets, D. Hundertmark, P. Kunstmann, N. Pattakos: Nonlinear Schrödinger equation, differentiation by parts and modulation spaces. J. Evol. Equ. 19 (2019), 803–843.
L. Chaichenets, N. Pattakos: The global Cauchy problem for the NLS with higher order anisotropic dispersion. Glasg. Math. J. 63 (2021), 45–53.
A. Choffrut, O. Pocovnicu: Ill-posedness for the cubic nonlinear half-wave equation and other fractional NLS on the real line. Int. Math. Res. Not. 2018 (2018), 699–738.
A. Chowdury, D. J. Kedziora, A. Ankiewicz, N. Akhmediev: Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms. Phys. Rev. E 90 (2014), Article ID 032922.
M. Christ: Nonuniqueness of weak solutions of the nonlinear Schrödinger equation. Available at https://arxiv.org/abs/math/0503366 (2005), 13 pages.
M. Christ: Power series solution of a nonlinear Schrödinger equation. Mathematical Aspects of Nonlinear Dispersive Equations. Annals of Mathematics Studies 163. Princeton University Press, Princeton, 2007, pp. 131–155.
J. Colliander, T. Oh: Almost sure well-posedness of the cubic nonlinear Schrödinger equation below \({L^2}(\mathbb{T})\). Duke Math. J. 161 (2012), 367–414.
H. G. Feichtinger: Modulation spaces on locally compact abelian groups. Proceedings of International Conference on Wavelets and Applications 2002. New Delhi Allied Publishers, Delhi, 2003, pp. 99–140.
G. Fibich, B. Ilan, G. Papanicolaou: Self-focusing with fourth-order dispersion. SIAM J. Appl. Math. 62 (2002), 1437–1462.
M. Gubinelli: Rough solutions for the periodic Korteweg-deVries equation. Commun. Pure Appl. Anal. 11 (2012), 709–733.
S. Guo: On the 1D cubic nonlinear Schrödinger equation in an almost critical space. J. Fourier Anal. Appl. 23 (2017), 91–124.
Z. Guo, S. Kwon, T. Oh: Poincaré-Dulac normal form reduction for unconditional well-posedness of the periodic cubic NLS. Commun. Math. Phys. 322 (2013), 19–48.
Z. Guo, T. Oh: Non-existence of solutions for the periodic cubic NLS below L 2. Int. Math. Res. Not. 2018 (2018), 1656–1729.
G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers. Clarendon Press, New York, 1979.
S. Herr, V. Sohinger: Unconditional uniqueness results for the nonlinear Schrödinger equation. Commun. Contemp. Math. 21 (2019), Article ID 1850058, 33 pages.
Z.-Z. Kang, T.-C. Xia, W.-X. Ma: Riemann-Hilbert approach and N-soliton solution for an eighth-order nonlinear Schrödinger equation in an optical fiber. Adv. Difference Equ. 2019 (2019), Article ID 188, 14 pages.
V. I. Karpman: Stabilization of soliton instabilities by higher order dispersion: Fourth-order nonlinear Schrödinger type equations. Phys. Rev. E 53 (1996), 1336–1339.
V. I. Karpman, A. G. Shagalov: Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion. Physica D 144 (2000), 194–210.
T. Kato: On nonlinear Schrödinger equations. II. H S-solutions and unconditional well-posedness. J. Anal. Math. 67 (1995), 281–306.
N. Kishimoto: Unconditional uniqueness of solutions for nonlinear dispersive equations. Available at https://arxiv.org/abs/1911.04349 (2019), 48 pages.
A. Miyachi, F. Nicola, S. Rivetti, A. Tabacco, N. Tomita: Estimates for unimodular Fourier multipliers on modulation spaces. Proc. Am. Math. Soc. 137 (2009), 3869–3883.
T. Oh, N. Tzvetkov: Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation. Probab. Theory Relat. Fields 169 (2017), 1121–1168.
T. Oh, Y. Wang: Global well-posedness of the periodic cubic fourth order NLS in negative Sobolev spaces. Forum Math. Sigma 6 (2018), Article ID e5, 80 pages.
T. Oh, Y. Wang: On the ill-posedness of the cubic nonlinear Schrödinger equation on the circle. An. Ştiinţ. Univ. Al. I. Cuza Iaşi., Ser. Nouă, Mat. 64 (2018), 53–84.
N. Pattakos: NLS in the modulation space M 2,q(ℝ). J. Fourier Anal. Appl. 25 (2019), 1447–1486.
B. Pausader: The cubic fourth-order Schrödinger equation. J. Func. Anal. 256 (2009), 2473–2517.
B. Pausader, S. Shao: The mass-critical fourth-order Schrödinger equation in high dimensions. J. Hyperbolic Differ. Equ. 7 (2010), 651–705.
Y. V. Sedletsky, I. S. Gandzha: A sixth-order nonlinear Schrödinger equation as a reduction of the nonlinear Klein-Gordon equation for slowly modulated wave trains. Nonlinear Dyn. 94 (2018), 1921–1932.
A. Vargas, L. Vega: Global wellposedness for 1D non-linear Schrödinger equation for data with an infinite L 2 norm. J. Math. Pures Appl., IX. Sér. 80 (2001), 1029–1044.
B. Wang, H. Hudzik: The global Cauchy problem for the NLS and NLKG with small rough data. J. Differ. Equations 232 (2007), 36–73.
Y. Yue, L. Huang, Y. Chen: Modulation instability, rogue waves and spectral analysis for the sixth-order nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 89 (2020), Article ID 105284, 14 pages.
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Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Project-ID 258734477, SFB 1173.
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Klaus, F., Kunstmann, P. & Pattakos, N. Unconditional uniqueness of higher order nonlinear Schrödinger equations. Czech Math J 71, 709–742 (2021). https://doi.org/10.21136/CMJ.2021.0078-20
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DOI: https://doi.org/10.21136/CMJ.2021.0078-20