Ill-posedness in $ H^s $ for a defocusing power-type derivative Schrödinger equation with lower-order linear perturbations

Senyue Luo; Meilan Qiu; Fangfang Deng; Senyue Luo; Meilan Qiu; Fangfang Deng

doi:10.3934/math.2026350

AIMS Mathematics

2026, Volume 11, Issue 3: 8507-8520. doi: 10.3934/math.2026350

Previous Article Next Article

Research article Topical Sections

Ill-posedness in $ H^s $ for a defocusing power-type derivative Schrödinger equation with lower-order linear perturbations

1.
Department of Basic Education, The Open University of Guangdong, Guangdong Polytechnic Institute, 510091, Guangdong, China
2.
School of Mathematics and Statistics, Huizhou University, Guangdong, Huizhou, 516007, China

Received: 09 January 2026 Revised: 15 March 2026 Accepted: 19 March 2026 Published: 30 March 2026
MSC : 35B30, 35C07, 35Q55

This work studied the one-dimensional defocusing power-type derivative Schrödinger equation with lower-order linear perturbations
$ \mathrm{i} u_t+u_{xx}- \mathrm{i} |u|^k u_x+\alpha u_x+\beta u = 0, \qquad (t, x)\in \mathbb{R}\times \mathbb{R}, $
where $ k\ge 2 $ and $ \alpha, \beta\in \mathbb{R} $ are constants. An explicit family of solitary traveling-wave solutions is first constructed within an exactly integrable traveling-wave reduction, and their $ H^s $ regularity and parameter dependence are characterized. A traveling-wave-based ill-posedness mechanism is then implemented: two solutions associated with nearby parameter sets are produced so that their initial data are arbitrarily close in $ H^s $, while their profiles remain separated by a uniform positive lower bound in $ H^s $ at some positive time. As a result, the solution flow map fails to be uniformly continuous below a certain regularity threshold. These results indicate that the presence of lower-order linear perturbations does not improve the low-regularity stability threshold for this DNLS-type equation.
- DNLS-type equation,
- solitary waves,
- ill-posedness
Citation: Senyue Luo, Meilan Qiu, Fangfang Deng. Ill-posedness in $ H^s $ for a defocusing power-type derivative Schrödinger equation with lower-order linear perturbations[J]. AIMS Mathematics, 2026, 11(3): 8507-8520. doi: 10.3934/math.2026350

Related Papers:

Abstract

This work studied the one-dimensional defocusing power-type derivative Schrödinger equation with lower-order linear perturbations

$ \mathrm{i} u_t+u_{xx}- \mathrm{i} |u|^k u_x+\alpha u_x+\beta u = 0, \qquad (t, x)\in \mathbb{R}\times \mathbb{R}, $

where $ k\ge 2 $ and $ \alpha, \beta\in \mathbb{R} $ are constants. An explicit family of solitary traveling-wave solutions is first constructed within an exactly integrable traveling-wave reduction, and their $ H^s $ regularity and parameter dependence are characterized. A traveling-wave-based ill-posedness mechanism is then implemented: two solutions associated with nearby parameter sets are produced so that their initial data are arbitrarily close in $ H^s $, while their profiles remain separated by a uniform positive lower bound in $ H^s $ at some positive time. As a result, the solution flow map fails to be uniformly continuous below a certain regularity threshold. These results indicate that the presence of lower-order linear perturbations does not improve the low-regularity stability threshold for this DNLS-type equation.

References

[1]	H. Bahouri, G. Perelman, Global well-posedness for the derivative nonlinear Schrödinger equation, Invent. Math., 229 (2022), 639–688. https://doi.org/10.1007/s00222-022-01113-0 doi: 10.1007/s00222-022-01113-0
[2]	H. A. Biagioni, F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc., 353 (2001), 3649–3659. https://doi.org/10.1090/S0002-9947-01-02754-4 doi: 10.1090/S0002-9947-01-02754-4
[3]	J. E. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness result for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649–669. https://doi.org/10.1137/S0036141001384387 doi: 10.1137/S0036141001384387
[4]	J. E. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, A refined global well-posedness result for Schrödinger equations with derivatives, SIAM J. Math. Anal., 34 (2002), 64–86. https://doi.org/10.1137/S0036141001394541 doi: 10.1137/S0036141001394541
[5]	A. Grünrock, Bi- and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS, Int. Math. Res. Notices, 2005 (2005), 2525–2558. https://doi.org/10.1155/IMRN.2005.2525 doi: 10.1155/IMRN.2005.2525
[6]	A. Grünrock, S. Herr, Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal., 39 (2008), https://doi.org/10.1137/070689139 doi: 10.1137/070689139
[7]	Z. H. Guo, Y. F. Wu, Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{1/2}(\mathbb{R})$, Discrete Cont. Dyn., 37 (2017), 257–264. https://doi.org/10.3934/dcds.2017010 doi: 10.3934/dcds.2017010
[8]	N. Hayashi, T. Ozawa, On the derivative nonlinear Schrödinger equation, Physica D, 55 (1992), 14–36. https://doi.org/10.1016/0167-2789(92)90185-P doi: 10.1016/0167-2789(92)90185-P
[9]	N. Hayashi, T. Ozawa, Finite energy solution of nonlinear Schrödinger equations of derivative type, SIAM J. Math. Anal., 25 (1994), https://doi.org/10.1137/S0036141093246129 doi: 10.1137/S0036141093246129
[10]	S. Herr, On the Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition, Int. Math. Res. Notices, 2006 (2006), 96763. https://doi.org/10.1155/IMRN/2006/96763 doi: 10.1155/IMRN/2006/96763
[11]	C. Hao, Well-posedness for one-dimensional derivative nonlinear Schrödinger equations, Commun. Pur. Appl. Anal., 6 (2007), 997–1021. https://doi.org/10.3934/cpaa.2007.6.997 doi: 10.3934/cpaa.2007.6.997
[12]	M. Hayashi, T. Ozawa, Well-posedness for a generalized derivative nonlinear Schrödinger equation, J. Differ. Equations, 261 (2016), 5424–5445. https://doi.org/10.1016/j.jde.2016.08.018 doi: 10.1016/j.jde.2016.08.018
[13]	D. J. Kaup, A. C. Newell, An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys., 19 (1978), 798–801. https://doi.org/10.1063/1.523737 doi: 10.1063/1.523737
[14]	X. Liu, G. Simpson, C. Sulem, Stability of solitary waves for a generalized derivative nonlinear Schrödinger equation, J. Nonlinear Sci., 23 (2013), 557–583. https://doi.org/10.1007/s00332-012-9161-2 doi: 10.1007/s00332-012-9161-2
[15]	J. Moses, B. Malomed, F. Wise, Self-steepening of ultrashort optical pulses without self-phase-modulation, Phys. Rev. A, 76 (2007), 021802. https://doi.org/10.1103/PhysRevA.76.021802 doi: 10.1103/PhysRevA.76.021802
[16]	K. Mio, T. Ogino, K. Minami, S. Takeda, Modified nonlinear Schrödinger equation for Alfvén waves propagating along the magnetic field in cold plasmas, J. Phys. Soc. Japan, 41 (1976), 265–271. https://doi.org/10.1143/JPSJ.41.265 doi: 10.1143/JPSJ.41.265
[17]	G. Santos, Existence and uniqueness of solution for a generalized nonlinear derivative Schrödinger equation, J. Differ. Equations, 259 (2015), 2030–2060. https://doi.org/10.1016/j.jde.2015.03.023 doi: 10.1016/j.jde.2015.03.023
[18]	H. Takaoka, Well-posedness for the one-dimensional Schrödinger equation with the derivative nonlinearity, Adv. Differential Equations, 4 (1999), 561–580. https://doi.org/10.57262/ade/1366031032 doi: 10.57262/ade/1366031032
[19]	H. Takaoka, Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces, Electron. J. Differ. Equ., 2001 (2001), 1–23.
[20]	M. Tsutsumi, I. Fukuda, On solutions of the derivative nonlinear Schrödinger equation. Existence and uniqueness theorem, Funkcial. Ekvac., 23 (1980), 259–277. https://doi.org/10.24546/0100499343 doi: 10.24546/0100499343
[21]	B. X. Wang, Globally well and ill posedness for non-elliptic derivative Schrödinger equations with small rough data, J. Funct. Anal., 265 (2013), 3009–3052. https://doi.org/10.1016/j.jfa.2013.08.009 doi: 10.1016/j.jfa.2013.08.009
[22]	Y. Z. Wang, Y. Zine, Norm inflation for the derivative nonlinear Schrödinger equation, CR Math., 362 (2024), 1857–1871. https://doi.org/10.5802/crmath.566 doi: 10.5802/crmath.566
[23]	Y. F. Wu, Global well-posedness on the derivative nonlinear Schrödinger equation, Anal. PDE, 8 (2015), 1101–1112. https://doi.org/10.2140/apde.2015.8.1101 doi: 10.2140/apde.2015.8.1101

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)