# American Institute of Mathematical Sciences

January  2020, 40(1): 47-80. doi: 10.3934/dcds.2020003

## Unconditional uniqueness for the derivative nonlinear Schrödinger equation on the real line

 1 Maxwell Institute for Mathematical Sciences, School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD, UK 2 National Center for Theoretical Sciences, National Taiwan University, No. 1 Sec. 4 Roosevelt Rd., Taipei 10617, Taiwan

Received  October 2018 Published  October 2019

We prove the unconditional uniqueness of solutions to the derivative nonlinear Schrödinger equation (DNLS) in an almost end-point regularity. To this purpose, we employ the normal form method and we transform (a gauge-equivalent) DNLS into a new equation (the so-called normal form equation) for which nonlinear estimates can be easily established in $H^s({\mathbb{R}})$, $s>\frac12$, without appealing to an auxiliary function space. Also, we prove that low-regularity solutions of DNLS satisfy the normal form equation and this is done by means of estimates in the $H^{s-1}({\mathbb{R}})$-norm.

Citation: Razvan Mosincat, Haewon Yoon. Unconditional uniqueness for the derivative nonlinear Schrödinger equation on the real line. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 47-80. doi: 10.3934/dcds.2020003
##### References:
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Oh, On the uniqueness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces, preprint. Google Scholar [11] N. Fukaya, M. Hayashi and T. Inui, A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation, Anal. PDE, 10 (2017), 1149-1167.  doi: 10.2140/apde.2017.10.1149.  Google Scholar [12] G. Furioli, F. Planchon and E. Terraneo, Unconditional well-posedness for semi-linear Schrodinger equations in $H^s$, Harmonic Analysis at Mount Holyoke (South Hadley, MA, 2001), Contemporary Mathematics, 320 (2003), 147–156. doi: 10.1090/conm/320/05604.  Google Scholar [13] A. Grünrock and S. Herr, Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal., 39 (2008), 1890-1920.  doi: 10.1137/070689139.  Google Scholar [14] Z. Guo, S. Kwon and T. 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Kishimoto, Unconditional uniqueness for the periodic cubic derivative nonlinear Schrödinger equations, preprint. Google Scholar [25] S. Kwon and T. Oh, On unconditional well-posedness of modified KdV, Int. Math. Res. Not. IMRN, (2012), 3509–3534. doi: 10.1093/imrn/rnr156.  Google Scholar [26] S. Kwon, T. Oh and H. Yoon, Normal form approach to unconditional well-posedness of nonlinear dispersive PDEs on the real line, Ann. Fac. Sci. Toulouse Math., (to appear). Google Scholar [27] J. H. Lee, Global solvability of the derivative nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 314 (1989), 107-118.  doi: 10.2307/2001438.  Google Scholar [28] C. Miao, Y. Wu and G. Xu, Global well-posedness for Schrödinger equation with derivative in $H^{1/2}(\mathbb{R})$, J. Differential Equations, 251 (2011), 2164-2195.  doi: 10.1016/j.jde.2011.07.004.  Google Scholar [29] E. Mjølhus, On the modulational instability of hydromagnetic waves parallel to the magnetic field, J. Plasma Physics, 16 (1976), 321-334.   Google Scholar [30] R. O. Mosincat, Well-posedness of the One-dimensional Derivative Nonlinear Schrödinger Equation, PhD Thesis, University of Edinburgh, 2018. Google Scholar [31] T. Oh, A remark on norm inflation with general initial data for the cubic nonlinear Schrödinger equations in negative Sobolev spaces, Funkcial. Ekvac., 60 (2017), 259-277.   Google Scholar [32] T. Oh and Y. Wang, Global well-posedness of the periodic cubic fourth order NLS in negative Sobolev spaces, Forum Math. Sigma, 6 (2018), e5, 80 pp. doi: 10.1017/fms.2018.4.  Google Scholar [33] D. E. Pelinovsky and Y. Shimabukuro, Existence of global solutions to the derivative NLS equation with the inverse scattering transform method, Int. Math. Res. Not., 2018, 5663–5728. doi: 10.1093/imrn/rnx051.  Google Scholar [34] D. Pornnonpparath, Small data well-posedness for derivative nonlinear Schrödinger equations, J. Differential Equations, 265 (2018), 3792-3840.  doi: 10.1016/j.jde.2018.05.016.  Google Scholar [35] A. Rogister, Parallel propagation of nonlinear low-frequency waves in high-$\beta$ plasma, Phys. Fluids, 14 (1971), 2733-2739.   Google Scholar [36] H. Takaoka, Well-posedness for the one dimensional Schrödinger equation with the derivative nonlinearity, Adv. Diff. Eq., 4 (1999), 561-580.   Google Scholar [37] T. Tao, Multilinear weighted convolution of $L^2$ functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908.  doi: 10.1353/ajm.2001.0035.  Google Scholar [38] Y. Y. Su Win, Unconditional uniqueness of the derivative nonlinear Schrödinger equation in energy space, J. Math. Kyoto Univ., 48 (2008), 683-697.  doi: 10.1215/kjm/1250271390.  Google Scholar [39] Y. Y. Su Win and Y. Tsutsumi, Unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrödinger equation, Hokkaido Math. J., 37 (2008), 839-859.  doi: 10.14492/hokmj/1249046372.  Google Scholar [40] Y. Wu, Global well-posedness of the derivative nonlinear Schrödinger equations in energy space, Anal. PDE, 6 (2013), 1989-2002.  doi: 10.2140/apde.2013.6.1989.  Google Scholar [41] Y. Wu, Global well-posedness on the derivative nonlinear Schrödinger equation, Anal. PDE, 8 (2015), 1101-1112.  doi: 10.2140/apde.2015.8.1101.  Google Scholar [42] Y. Zhou, Uniqueness of weak solution of the KdV equation, Int. Math. Res. Not., 1997,271–283. doi: 10.1155/S1073792897000202.  Google Scholar

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##### References:
 [1] A. Babin, A. Ilyin and E. Titi, On the regularization mechanism for the periodic Korteweg-de Vries equation, Comm. Pure Appl. Math., 64 (2011), 591-648.  doi: 10.1002/cpa.20356.  Google Scholar [2] H. A. Biagioni and F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc., 353 (2001), 3649-3659.  doi: 10.1090/S0002-9947-01-02754-4.  Google Scholar [3] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations Ⅰ. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107–156. doi: 10.1007/BF01896020.  Google Scholar [4] M. Christ, Power series solution of a nonlinear Schrödinger equation, In: Mathematical Aspects of Nonlinear Dispersive Equations, Ann. of Math. Stud., Princeton, NJ: Princeton Univ. Press, 163 (2007), 131–155.  Google Scholar [5] M. Christ, Nonuniqueness of weak solutions of the nonlinear Schrödinger equation, arXiv: 0503366. Google Scholar [6] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669.  doi: 10.1137/S0036141001384387.  Google Scholar [7] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, A refined global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 34 (2002), 64-86.  doi: 10.1137/S0036141001394541.  Google Scholar [8] M. B. Erdoğan, T. B. Gürel and N. Tzirakis, The derivative nonlinear Schrödinger equation on the half line, Ann. I. H. Poincaré Anal. Non Linéaire, 35 (2018), 1947-1973.  doi: 10.1016/j.anihpc.2018.03.006.  Google Scholar [9] M. B. Erdoğan and N. Tzirakis, Global smoothing for the periodic KdV evolution, Int. Math. Res. Not., 20 (2013), 4589-4614.  doi: 10.1093/imrn/rns189.  Google Scholar [10] J. Forlano and T. Oh, On the uniqueness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces, preprint. Google Scholar [11] N. Fukaya, M. Hayashi and T. Inui, A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation, Anal. PDE, 10 (2017), 1149-1167.  doi: 10.2140/apde.2017.10.1149.  Google Scholar [12] G. Furioli, F. Planchon and E. Terraneo, Unconditional well-posedness for semi-linear Schrodinger equations in $H^s$, Harmonic Analysis at Mount Holyoke (South Hadley, MA, 2001), Contemporary Mathematics, 320 (2003), 147–156. doi: 10.1090/conm/320/05604.  Google Scholar [13] A. Grünrock and S. Herr, Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal., 39 (2008), 1890-1920.  doi: 10.1137/070689139.  Google Scholar [14] Z. Guo, S. Kwon and T. Oh, Poincaré-Dulac normal form reduction for unconditional well-posedness of the periodic cubic NLS, Comm. Math. Phys., 322 (2013), 19-48.  doi: 10.1007/s00220-013-1755-5.  Google Scholar [15] Z. Guo and Y. Wu, Global well-posedness for the derivative nonlinear Schrodinger equation in $H^{1/2}(\mathbb{R})$, Disc. Cont. Dyn. Sys., 37 (2017), 257-264.  doi: 10.3934/dcds.2017010.  Google Scholar [16] N. Hayashi, The initial value problem for the derivative nonlinear Schrödinger equation in the energy space, Nonlinear Anal., 20 (1993), 823-833.  doi: 10.1016/0362-546X(93)90071-Y.  Google Scholar [17] N. Hayashi and T. Ozawa, On the derivative nonlinear Schrödinger equation, Physica D., 55 (1992), 14-36.  doi: 10.1016/0167-2789(92)90185-P.  Google Scholar [18] N. Hayashi and T. Ozawa, Finite energy solutions of nonlinear Schrödinger equations of derivative type, SIAM J. Math. Anal., 25 (1994), 1488-1503.  doi: 10.1137/S0036141093246129.  Google Scholar [19] S. Herr and V. Sohinger, Unconditional uniqueness results for the nonlinear Schrödinger equation, 2018, arXiv: 1804.10631. doi: 10.1142/S021919971850058X.  Google Scholar [20] R. Jenkins, J. Liu, P. Perry and C. Sulem, Global existence for the derivative nonlinear Schrödinger equation with arbitrary spectral singularities, arXiv: 1804.01506v2. Google Scholar [21] T. Kato, On nonlinear Schrödinger equations. Ⅱ. $H^s$-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306.  doi: 10.1007/BF02787794.  Google Scholar [22] D. J. Kaup and A. C. Newell, An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys., 19 (1978), 798-801.  doi: 10.1063/1.523737.  Google Scholar [23] N. Kishimoto, Unconditional uniqueness of solutions for nonlinear dispersive equations, Proceedings of the 40th Sapporo Symposium on Partial Differential Equations, (2015), 78–82. Google Scholar [24] N. Kishimoto, Unconditional uniqueness for the periodic cubic derivative nonlinear Schrödinger equations, preprint. Google Scholar [25] S. Kwon and T. Oh, On unconditional well-posedness of modified KdV, Int. Math. Res. Not. IMRN, (2012), 3509–3534. doi: 10.1093/imrn/rnr156.  Google Scholar [26] S. Kwon, T. Oh and H. Yoon, Normal form approach to unconditional well-posedness of nonlinear dispersive PDEs on the real line, Ann. Fac. Sci. Toulouse Math., (to appear). Google Scholar [27] J. H. Lee, Global solvability of the derivative nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 314 (1989), 107-118.  doi: 10.2307/2001438.  Google Scholar [28] C. Miao, Y. Wu and G. Xu, Global well-posedness for Schrödinger equation with derivative in $H^{1/2}(\mathbb{R})$, J. Differential Equations, 251 (2011), 2164-2195.  doi: 10.1016/j.jde.2011.07.004.  Google Scholar [29] E. Mjølhus, On the modulational instability of hydromagnetic waves parallel to the magnetic field, J. Plasma Physics, 16 (1976), 321-334.   Google Scholar [30] R. O. Mosincat, Well-posedness of the One-dimensional Derivative Nonlinear Schrödinger Equation, PhD Thesis, University of Edinburgh, 2018. Google Scholar [31] T. Oh, A remark on norm inflation with general initial data for the cubic nonlinear Schrödinger equations in negative Sobolev spaces, Funkcial. Ekvac., 60 (2017), 259-277.   Google Scholar [32] T. Oh and Y. Wang, Global well-posedness of the periodic cubic fourth order NLS in negative Sobolev spaces, Forum Math. Sigma, 6 (2018), e5, 80 pp. doi: 10.1017/fms.2018.4.  Google Scholar [33] D. E. Pelinovsky and Y. Shimabukuro, Existence of global solutions to the derivative NLS equation with the inverse scattering transform method, Int. Math. Res. Not., 2018, 5663–5728. doi: 10.1093/imrn/rnx051.  Google Scholar [34] D. Pornnonpparath, Small data well-posedness for derivative nonlinear Schrödinger equations, J. Differential Equations, 265 (2018), 3792-3840.  doi: 10.1016/j.jde.2018.05.016.  Google Scholar [35] A. Rogister, Parallel propagation of nonlinear low-frequency waves in high-$\beta$ plasma, Phys. Fluids, 14 (1971), 2733-2739.   Google Scholar [36] H. Takaoka, Well-posedness for the one dimensional Schrödinger equation with the derivative nonlinearity, Adv. Diff. Eq., 4 (1999), 561-580.   Google Scholar [37] T. Tao, Multilinear weighted convolution of $L^2$ functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908.  doi: 10.1353/ajm.2001.0035.  Google Scholar [38] Y. Y. Su Win, Unconditional uniqueness of the derivative nonlinear Schrödinger equation in energy space, J. Math. Kyoto Univ., 48 (2008), 683-697.  doi: 10.1215/kjm/1250271390.  Google Scholar [39] Y. Y. Su Win and Y. Tsutsumi, Unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrödinger equation, Hokkaido Math. J., 37 (2008), 839-859.  doi: 10.14492/hokmj/1249046372.  Google Scholar [40] Y. Wu, Global well-posedness of the derivative nonlinear Schrödinger equations in energy space, Anal. PDE, 6 (2013), 1989-2002.  doi: 10.2140/apde.2013.6.1989.  Google Scholar [41] Y. Wu, Global well-posedness on the derivative nonlinear Schrödinger equation, Anal. PDE, 8 (2015), 1101-1112.  doi: 10.2140/apde.2015.8.1101.  Google Scholar [42] Y. Zhou, Uniqueness of weak solution of the KdV equation, Int. Math. Res. Not., 1997,271–283. doi: 10.1155/S1073792897000202.  Google Scholar
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