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

Razvan Mosincat; Haewon Yoon

doi:10.3934/dcds.2020003

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

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

Razvan Mosincat ^1, and Haewon Yoon ^2,

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.

Keywords: Derivative nonlinear Schrödinger equation, unconditional well-posedness, normal form method.

Mathematics Subject Classification: 35Q55.

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:

[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
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[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

show all references

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

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