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 - A, 2020, 40 (1) : 47-80. doi: 10.3934/dcds.2020003
References:
[1]

A. BabinA. 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

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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

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M. Christ, Nonuniqueness of weak solutions of the nonlinear Schrödinger equation, arXiv: 0503366. Google Scholar

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J. CollianderM. KeelG. StaffilaniH. 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

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J. CollianderM. KeelG. StaffilaniH. 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

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M. B. ErdoğanT. 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. FukayaM. 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. GuoS. 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

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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

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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. MiaoY. 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

show all references

References:
[1]

A. BabinA. 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. CollianderM. KeelG. StaffilaniH. 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. CollianderM. KeelG. StaffilaniH. 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. ErdoğanT. 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. FukayaM. 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. GuoS. 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. MiaoY. 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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