Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On the ill-posedness for a nonlinear Schrödinger-Airy equation

Author: Xavier Carvajal
Journal: Quart. Appl. Math. 71 (2013), 267-281
MSC (2010): Primary 35Q55, 35Q53
DOI: https://doi.org/10.1090/S0033-569X-2012-01297-1
Published electronically: October 18, 2012
MathSciNet review: 3087422
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Abstract | References | Similar Articles | Additional Information

Abstract: Using ideas of Kenig, Ponce and Vega and an explicit solution with two parameters, we prove that the solution map of the initial value problem for a particular nonlinear Schrödinger-Airy equation fails to be uniformly continuous.

Also, we will approximate the solution to the nonlinear Schrödinger-Airy equation by the solution to the cubic nonlinear Schrödinger equation and prove ill-posedness in a more general case than above. This method was originally introduced by Christ, Colliander and Tao for the modified Korteweg-de Vries equation.

Finally, we consider the general case and we prove ill-posedness for all values of the parameters in the equation.

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  • 1. M.J. Ablowitz, J. Hammack, D. Henderson, and C.M. Schober, Long-time dynamics of the modulational instability of deep water waves, Phys. D.(152-153) (2001), 416-433. MR 1837921 (2002h:76017)
  • 2. H. Biagioni and F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc. 353 (2001), 3649-3659. MR 1837253 (2002e:35215)
  • 3. T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29 (1967), 559-592.
  • 4. J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data, Selecta Math. New Ser. 3 (1997), 115-159. MR 1466164 (2000i:35173)
  • 5. X. Carvajal, Local well-posedness for a higher order nonlinear Schrödinger equation in Sobolev spaces of negative indices, Electron. J. Diff. Eqns., 13 (2004), 1-10. MR 2036197 (2004k:35345)
  • 6. X. Carvajal, Sharp global well-posedness for a higher order Schrödinger equation, J. Fourier Anal. Appl. 12 (2006), 53-70. MR 2215677 (2007d:35253)
  • 7. X. Carvajal and F. Linares, A higher order nonlinear Schrödinger equation with variable coefficients, Differential and Integral Equations, 16 (2003), 1111-1130. MR 1989544 (2004e:35207)
  • 8. X. Carvajal and M. Panthee, Unique continuation for a higher order nonlinear Schrödinger equation, J. Math. Anal. Appl., 303 (2005), 188-207. MR 2113876 (2006b:35298)
  • 9. X. Carvajal and M. Panthee, On uniqueness of solution for a nonlinear Schrödinger-Airy equation, Nonlinear Anal., (2006). MR 2183834 (2006f:35261)
  • 10. M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math. 125 (2003), 1235-1293. MR 2018661 (2005d:35223)
  • 11. P.A. Clarson and C.M. Cosgrove, Painlevé analysis of the nonlinear Schrödinger family of equations, Journal of Physics A: Math. and Gen. 20 (1987), 2003-2024. MR 893304 (89c:35136)
  • 12. J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, A refined global well-posedness result for Schrödinger equations with derivative, SIAM J. Math. Anal. 34 (2002), 64-86. MR 1950826 (2004c:35381)
  • 13. J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for periodic and nonperiodic KdV and mKdV, J. Amer. Math. Soc., 16 (2003), 705-749. MR 1969209 (2004c:35352)
  • 14. G. Fonseca, F. Linares and G. Ponce, Global well-posedness for the modified Korteweg-de Vries equation, Comm. Partial Differential Equations, 24 (1999), 683-705. MR 1683054 (2000a:35210)
  • 15. A. Hasegawa and Y. Kodama, Nonlinear pulse propagation in a monomode dielectric guide, IEEE Journal of Quantum Electronics, 23 (1987), 510-524.
  • 16. A. Ionescu and C. E. Kenig, Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, J. Amer. Math. Soc., 20, no. 3 (2007), 753-798. MR 2291918 (2008f:35350)
  • 17. C. E. Kenig, G. Ponce and L. Vega, Well-Posedness and Scattering Results for the Generalized Korteweg-de Vries Equation via the Contraction Principle, Comm. Pure and Applied Math., 46 (1993), 527-620. MR 1211741 (94h:35229)
  • 18. C. E. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Mathematical Journal, Vol 106, No. 3, (2001), 617-633. MR 1813239 (2002c:35265)
  • 19. C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9, No. 2, (1996), 573-603. MR 1329387 (96k:35159)
  • 20. Y. Kodama, Optical solitons in a monomode fiber, Journal of Statistical Phys., 39 (1985), 597-614. MR 807002 (86m:78021)
  • 21. C. Laurey, The Cauchy Problem for a Third Order Nonlinear Schrödinger Equation, Nonlinear Anal., 29 (1997), 121-158. MR 1446222 (98c:35154)
  • 22. L. Molinet, J. -C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM. J. Math. Anal. 4 (2001), 982-988. MR 1885293 (2002k:35281)
  • 23. H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091. MR 0398275 (53:2129)
  • 24. T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations, 11 (1998), 201-222. MR 1741843 (2000m:35167)
  • 25. K. Porsezian and K. Nakkeeran, Singularity Structure Analysis and Complete Integrability of the Higher Order Nonlinear Schrödinger equations, Chaos, Solitons and Fractals (1996), 377-382. MR 1381301 (97a:78032)
  • 26. K. Porsezian, P. Shanmugha, K. Sundaram and A. Mahalingam, Phys. Rev. 50E,1543 (1994). MR 1381874 (96k:35171)
  • 27. Soonsik Kwon, Well-posedness and ill-posedness of the fifth-order modified KdV equation, Electron. J. Diff. Eqns., 01 (2008), 1-15. MR 2368888 (2008j:35158)
  • 28. G. Staffilani, On the Generalized Korteweg-de Vries-Type Equations, Differential and Integral Equations 10 (1997), 777-796. MR 1741772 (2001a:35005)
  • 29. C. Sulem and P. L. Sulem, The nonlinear Schrödinger equation: self-focusing and wave collapse, Applied Mathematical Sciences, Springer-Verlag 139 (1999), 350 pages. MR 1696311 (2000f:35139)
  • 30. H. Takaoka, Well-posedness for the higher order nonlinear Schrödinger equation Adv. Math. Sci. Appl. 10 (2000), no. 1, 149-171. MR 1769176 (2001c:35224)
  • 31. T. Tao, Nonlinear dispersive equations, local and global analysis, CBMS Regional Conference Series in Mathematics, American Mathematical Society (2006). MR 2233925 (2008i:35211)
  • 32. H. Takaoka, Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity, Adv. Diff. Eq. 4 (1999), 561-680. MR 1693278 (2000e:35221)
  • 33. Y. Tsutsumi, $ L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), 115-125. MR 915266 (89c:35143)
  • 34. N. Tzvetkov, Remark on the local ill-posedness for KdV equation, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 1043-1047. MR 1735881 (2001f:35359)
  • 35. H. Wang, Global well-posedness of the Cauchy problem of a higher-order Schrödinger equation, Electron. J. Diff. Eqns., 04 (2007), 1-11. MR 2278418 (2007i:35209)

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Additional Information

Xavier Carvajal
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, C.P. 68530, C.E.P. 21944-970, Rio de Janeiro, R.J. Brazil
Email: carvajal@im.ufrj.br

DOI: https://doi.org/10.1090/S0033-569X-2012-01297-1
Keywords: Schrödinger equation, Korteweg-de Vries equation, ill-posedness
Received by editor(s): June 3, 2011
Published electronically: October 18, 2012
Additional Notes: The author was partially supported by FAPERJ, Brazil under grants E-26/111.564/2008 and E-26/ 110.560/2010 and by the National Council of Technological and Scientific Development (CNPq), Brazil, by the grant 303849/2008-8.
Article copyright: © Copyright 2012 Brown University

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