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On a Class of Solutions to the Generalized Derivative Schrödinger Equations

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Abstract

In this work we shall consider the initial value problem associated to the generalized derivative Schrödinger (gDNLS) equations

$$\partial_tu=\text{i}\partial_x^2u+\mu|u|^\alpha\partial_xu,\;\;x,t\in\mathbb{R},\;\;0<\alpha\leq1\;\;\text{and}\;\;|\mu|=1,$$

and

$$\partial_tu=\text{i}\partial_x^2u+\mu\partial_x(|u|^\alpha{u}),\;\;x,t\in\mathbb{R},\;\;0<\alpha\leq1\;\;\text{and}\;\;|\mu|=1.$$

Following the argument introduced by Cazenave and Naumkin we shall establish the local well-posedness for a class of small data in an appropriate weighted Sobolev space. The other main tools in the proof include the homogeneous and inhomogeneous versions of the Kato smoothing effect for the linear Schrödinger equation established by Kenig–Ponce–Vega.

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References

  1. Ambrose, D. M., Simpson, G.: Local existence theory for derivative nonlinear Schrödinger equations with non-integer power nonlinearities. SIAM J. Math. Anal., 47, 2241–2264 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Biagioni, H., Linares, F.: Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations. Trans. Amer. Math. Soc., 353, 3649–3659 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cazenave, T., Naumkin, I.: Local existence, global existence, and scattering for the nonlinear Schrödinger equation. Comm. Contemp. Math., 19, 1650038, 20 pp. (2017)

    Google Scholar 

  4. Colin, M., Ohta, M.: Stability of solitary waves for derivative nonlinear Schrödinger equation. Ann. Inst. H. Poincaré Anal. Non Linéaire, 23, 753–764 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Colliander, J., Keel, M., Staffilani, G., et al.: Global well-posedness for the Schrödinger equations with derivative. SIAM J. Math. Anal., 33, 649–669 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Colliander, J., Keel, M., Staffilani, G., et al.: A refined global well-posedness for the Schrödinger equations with derivative. SIAM J. Math. Anal., 34, 64–86 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Constantin, P., Saut, J. C.: Local smoothing properties of dispersive equations. J. Amer. Math. Soc., 1, 413–446 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fukaya, N., Hayashi, M., Inui, T.: A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation. Anal. PDE, 10, 1149–1167 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Grünrock, A., Herr, S.: Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data. SIAM J. Math. Anal., 39, 1890–1920 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guo, B., Wu, Y.: Orbital stability of solitary waves for the nonlinear derivative Schrödinger equation. J. Differential Equations, 123, 35–55 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guo, Z., Wu, Y.: Global well-posedness for the derivative nonlinear Schrödinger equation in H1/2ℝ. Discrete Cont. Dyn. Syst., 37, 257–264 (2017)

    Article  Google Scholar 

  12. Hao, C.: Well-posedness for one-dimensional derivative nonlinear Schrödinger equations. Comm. Pure Appl. Anal., 6, 997–1021 (2007)

    Article  MATH  Google Scholar 

  13. Hayashi, N.: The initial value problem for the derivative nonlinear Schrödinger equation in the energy space. Nonlinear Anal. TMA, 20, 823–833 (1993)

    Article  MATH  Google Scholar 

  14. Hayashi, M., Ozawa, T.: Well-posedness for a generalized derivative nonlinear Schrödinger equation. J. Differential Equations, 261, 5424–5445 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hayashi, N., Ozawa, T.: On the derivative nonlinear Schrödinger equation. Phys. D, 55, 14–36 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hayashi, N., Ozawa, T.: Finite energy solution of nonlinear Schrödinger equations of derivative type. SIAM J. Math. Anal., 25, 1488–1503 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. Herr, S.: On the Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition. Int. Math. Res. Not. IMRN, Art. ID 96763, 33 pp. (2006)

    Google Scholar 

  18. Jenkins, R., Liu, J., Perry, P., et al.: Global well-posedness and soliton resolution for the derivative nonlinear Schrödinger equation, arXiv: 1706.06252v1

    Google Scholar 

  19. Kato, T.: On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Advances in Mathematics Supplementary Studies, Studies in Applied Math., 8, 93–128 (1983)

    MathSciNet  Google Scholar 

  20. Kaup, D. J., Newell, A. C.: An exact solution for a derivative nonlinear Schrödinger equation. J. Math. Phys., 19, 789–801 (1978)

    Article  MATH  Google Scholar 

  21. Kenig, C. E., Ponce, G., Vega, L.: Small solutions to nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 10, 255–288 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kruzhkov, S. N., Faminskii, A. V.: Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation. Math. U.S.S.R. Sbornik, 48, 93–138 (1984)

    Article  Google Scholar 

  23. Kwon, S., Wu, Y.: Orbital stability of solitary waves for derivative nonlinear Schrödinger equation, arXiv:1603.03745

    MATH  Google Scholar 

  24. Le, S. Coz, Wu, Y.: Stability of multi-solitons for the derivative nonlinear Schrödinger equation, arXiv:1609.04589

    Google Scholar 

  25. Linares, F., Ponce, G.: Introduction to Nonlinear Dispersive Equations (Second Edition), Universitext, Springer, New York, 2015

    Book  MATH  Google Scholar 

  26. Liu, X., Simpson, G., Sulem, C.: Stability of solitary waves for a generalized derivative nonlinear Schrödinger equation. J. Nonlinear Sci., 23(4), 557–583 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Miao, C., Tang, X., Xu, G.: Stability of the traveling waves for the derivative Schrödinger equation in the energy space, arXiv:1702.07856

    MATH  Google Scholar 

  28. Mio, K., Ogino, T., Minami, K., et al.: Modified nonlinear Schrödinger equation for Alfvén Waves propagating along magnetic field in cold plasma. J. Phys. Soc., 41, 265–271 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mizohata, S.: On the Cauchy problem, Notes and Reports in Mathematics in Science and Engineering 3, Academic Press Inc., Orlando, FL, 1985

    Google Scholar 

  30. Mjolhus, E.: On the modulational instability of hydromagnetic waves parallel to the magnetic field. J. Plasma Phys., 16, 321–334 (1976)

    Article  Google Scholar 

  31. Moses, J., Malomed, B. A., Wise, F. W.: Self-steepening of ultrashort optical pulses without self-phasemodulation. Phys. Rev. A, 76, 1–4 (2007)

    Article  Google Scholar 

  32. Nahas, J., Ponce, G.: On the persistent properties of solutions to semi-linear Schrödinger equation. Comm. Partial Differential Equations, 34, 1208–1227 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ozawa, T.: On the nonlinear Schrödinger equations of derivative type. Indiana Univ. Math. J., 45, 137–163 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  34. Santos, G. N.: Existence and uniqueness of solutions for a generalized Nonlinear Derivative Schrödinger equation. J. Differential Equations, 259, 2030–2060 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sjölin, P.: Regularity of solutions to the Schrödinger equations. Duke Math. J., 55, 699–715 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  36. Takaoka, H.: Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity. Adv. Differential Equations, 4, 561–580 (1999)

    MathSciNet  MATH  Google Scholar 

  37. Takaoka, H.: Global well-posedness for the Schrödinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces. Electron. J. Differential Equations, 43, 1–23 (2001)

    MATH  Google Scholar 

  38. Tang, X., Xu, G.: Stability of the sum of two solitary waves for (gDNLS) in the energy space, arXiv:1702.07858

    MATH  Google Scholar 

  39. Tsutsumi, M., Fukuda, I.: On solutions of the derivatives nonlinear Schrödinger equation. Existence and uniqueness theorem. Funkcialaj Ekvacioj, 23, 259–277 (1980)

    MathSciNet  MATH  Google Scholar 

  40. Vega, L.: The Schrödinger equation: pointwise convergence to the initial data. Proc. Amer. Math. Soc., 102, 874–878 (1988)

    MathSciNet  MATH  Google Scholar 

  41. Wu, Y.: Global well-posedness of the derivative nonlinear Schrödinger equation. Analysis & PDE, 8, 1101–1112 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the anonymous referees whose comments improve the presentation of this work. Felipe Linares was partially supported by CNPq and FAPERJ/Brazil.

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Authors and Affiliations

  1. IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, RJ, Brazil

    Felipe Linares

  2. University of California, Santa Barbara, CA, 93106, USA

    Gustavo Ponce

  3. Universidade Federal do Piauí - UFPI, Campus Ministro Petrônio Portella, Teresina, 64049-550, PI, Brazil

    Gleison N. Santos

Authors
  1. Felipe Linares
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  2. Gustavo Ponce
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  3. Gleison N. Santos
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Correspondence to Felipe Linares.

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Dedicado a Nuestro Amigo Carlos E. Kenig

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Linares, F., Ponce, G. & Santos, G.N. On a Class of Solutions to the Generalized Derivative Schrödinger Equations. Acta. Math. Sin.-English Ser. 35, 1057–1073 (2019). https://doi.org/10.1007/s10114-019-7540-4

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  • DOI: https://doi.org/10.1007/s10114-019-7540-4

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