Small and large data scattering for the dispersion-managed NLS

Jumpei Kawakami; Jason Murphy

doi:10.3934/dcds.2025118

Article Contents

2026, Volume 47: 256-285. Doi: 10.3934/dcds.2025118

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Small and large data scattering for the dispersion-managed NLS

Jumpei Kawakami^1, and
Jason Murphy^2, ,

1.
Division of Mathematics and Mathematical Sciences, Kyoto University, Japan
2.
Department of Mathematics, University of Oregon, USA

^*Corresponding author: Jason Murphy
^*Corresponding author: Jason Murphy

Received: December 2024

Revised: April 2025

Early access: August 4, 2025

Published online: August 4, 2025

J. K. was supported by JST SPRING, Grant Number JPMJSP2110. J. M. was supported by NSF grant DMS-2350225. We thank the anonymous referees for their careful reading of this manuscript.

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Abstract

We prove several scattering results for dispersion-managed nonlinear Schrödinger equations. In particular, we establish small-data scattering for both 'intercritical' and 'mass-subcritical' powers by suitable modifications of the standard approach via Strichartz estimates. In addition, we prove scattering for arbitrary data in a weighted Sobolev space for intercritical powers by establishing a pseudoconformal energy estimate. We also rule out (unmodified) scattering for sufficiently low powers. Finally, we give some remarks concerning blowup for the focusing equation.

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Mathematics Subject Classification: Primary: 35Q55.

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References

[1]	G. P. Agrawal, Nonlinear Fiber Optics, Second Edition (Optics and Photonics), Academic Press, San Diego, 1995.
[2]	J. E. Barab, Nonexistence of asymptotically free solutions for a nonlinear schrödinger equation, J. Math. Phys., 25 (1984), 3270-3273. doi: 10.1063/1.526074.
[3]	T. Cazenave, Semilinear SchrÖdinger Equations, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 10 (2003), 323. doi: 10.1090/cln/010.
[4]	T. Cazenave and F. Weissler, Rapidly decaying solutions of the nonlinear schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100. doi: 10.1007/BF02099529.
[5]	M.-R. Choi, Y. Hong and Y.-R. Lee, Global existence versus finite time blowup dichotomy for the dispersion managed NLS, Nonlinear Anal., 251 (2025), 16. doi: 10.1016/j.na.2024.113696.
[6]	M.-R. Choi, D. Hundertmark and Y.-R. Lee, Well-posedness of dispersion managed nonlinear schrödinger equations, J. Math. Anal. Appl., 522 (2023), 37. doi: 10.1016/j.jmaa.2022.126938.
[7]	M.-R. Choi and Y.-R. Lee, Averaging of dispersion managed nonlinear schrödinger equations, Nonlinearity, 35 (2022), 2121-2133. doi: 10.1088/1361-6544/ac5464.
[8]	M.-R. Choi, K. Lee and Y.-R. Lee, Scattering for the dispersion managed nonlinear schrödinger equation, Preprint, 38 (2025), 555–576, arXiv: 2401.16717.
[9]	M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized korteweg-de vries equation, J. Funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C.
[10]	I. Gabitov and S. K. Turitsyn, Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation, Opt. Lett., 21 (1996), 327-329. doi: 10.1364/OL.21.000327.
[11]	J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144 (1992), 163-188. doi: 10.1007/BF02099195.
[12]	R. T. Glassey, Asymptotic behavior of solutions to certain nonlinear schrödinger-hartree equations, Comm. Math. Phys., 53 (1977), 9-18. doi: 10.1007/BF01609164.
[13]	D. Hundertmark and Y.-R. Lee, Decay estimates and smoothness for solutions of the dispersion managed non-linear schrödinger equation, Commun. Math. Phys., 286 (2009), 851-873. doi: 10.1007/s00220-008-0612-4.
[14]	R. A. Hunt, An extension of the marcinkiewicz interpolation theorem to lorentz spaces, Bull. Amer. Math. Soc., 70 (1964), 803-807. doi: 10.1090/S0002-9904-1964-11242-8.
[15]	R. A. Hunt, On $L(p, q)$ spaces, Enseignement Math., 12 (1966), 249-276.
[16]	J. Kawakami, Scattering and blow up for nonlinear schrödinger equation with the averaged nonlinearity, J. Math. Anal. Appl., 543 (2025), 52. doi: 10.1016/j.jmaa.2024.128932.
[17]	M. Keel and T. Tao, Endpoint strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.
[18]	C. Kurtzke, Suppression of fiber nonlinearities by appropriate dispersion management, IEEE Phot. Tech. Lett., 5 (1993), 1250-1253. doi: 10.1109/68.248444.
[19]	P. M. Lushnikov, Dispersion-managed soliton in a strong dispersion map limit, Optics Lett., 26 (2001), 1535-1537. doi: 10.1364/OL.26.001535.
[20]	J. Murphy, Subcritical scattering for defocusing nonlinear schrödinger equations, https://pages.uoregon.edu/jamu/expository.pdf.
[21]	J. Murphy and T. Van Hoose, Well-posedness and blowup for the dispersion-managed nonlinear schrödinger equation, Proc. Amer. Math. Soc., 151 (2023), 248-2502. doi: 10.1090/proc/16243.
[22]	K. Nakanishi and T. Ozawa, Remarks on scattering for nonlinear schrödinger equations, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 45-68.
[23]	D. Pelinovsky and V. Zharnitsky, Averaging of dispersion managed solitons: Existence and stability, SIAM J. Appl. Math., 63 (2003), 745-776. doi: 10.1137/S0036139902400477.
[24]	W. A. Strauss, Nonlinear scattering theory, "Scattering Theory in Mathematical Physics", Reidel, Dordrecht, (1974), 53-78. doi: 10.1007/978-94-010-2147-0_3.
[25]	R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions to wave equations, Duke Math. J., 44 (1977), 705-714.
[26]	V. Zharnitsky, E. Grenier, K. R. T. C. Jones and S.K. Turitsyn, Stabilizing effects of dispersion management, Phys. D., 152 (2001), 794-817. doi: 10.1016/S0167-2789(01)00213-5.