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
Full-text PDF Free Access
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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.
References
- 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. Advances in nonlinear mathematics and science. MR 1837921, DOI https://doi.org/10.1016/S0167-2789%2801%2900183-X
- H. A. Biagioni and F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc. 353 (2001), no. 9, 3649–3659. MR 1837253, DOI https://doi.org/10.1090/S0002-9947-01-02754-4
- T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29 (1967), 559-592.
- J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data, Selecta Math. (N.S.) 3 (1997), no. 2, 115–159. MR 1466164, DOI https://doi.org/10.1007/s000290050008
- Xavier Carvajal, Local well-posedness for a higher order nonlinear Schrödinger equation in Sobolev spaces of negative indices, Electron. J. Differential Equations (2004), No. 13, 10. MR 2036197
- Xavier Carvajal, Sharp global well-posedness for a higher order Schrödinger equation, J. Fourier Anal. Appl. 12 (2006), no. 1, 53–70. MR 2215677, DOI https://doi.org/10.1007/s00041-005-5028-3
- X. Carvajal and F. Linares, A higher-order nonlinear Schrödinger equation with variable coefficients, Differential Integral Equations 16 (2003), no. 9, 1111–1130. MR 1989544
- X. Carvajal and M. Panthee, Unique continuation property for a higher order nonlinear Schrödinger equation, J. Math. Anal. Appl. 303 (2005), no. 1, 188–207. MR 2113876, DOI https://doi.org/10.1016/j.jmaa.2004.08.030
- X. Carvajal and M. Panthee, On uniqueness of solution for a nonlinear Schrödinger-Airy equation, Nonlinear Anal. 64 (2006), no. 1, 146–158. MR 2183834, DOI https://doi.org/10.1016/j.na.2005.06.016
- Michael Christ, James Colliander, and Terrence Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math. 125 (2003), no. 6, 1235–1293. MR 2018661
- Peter A. Clarkson and Christopher M. Cosgrove, Painlevé analysis of the nonlinear Schrödinger family of equations, J. Phys. A 20 (1987), no. 8, 2003–2024. MR 893304
- 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), no. 1, 64–86. MR 1950826, DOI https://doi.org/10.1137/S0036141001394541
- J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\Bbb R$ and $\Bbb T$, J. Amer. Math. Soc. 16 (2003), no. 3, 705–749. MR 1969209, DOI https://doi.org/10.1090/S0894-0347-03-00421-1
- German Fonseca, Felipe Linares, and Gustavo Ponce, Global well-posedness for the modified Korteweg-de Vries equation, Comm. Partial Differential Equations 24 (1999), no. 3-4, 683–705. MR 1683054, DOI https://doi.org/10.1080/03605309908821438
- A. Hasegawa and Y. Kodama, Nonlinear pulse propagation in a monomode dielectric guide, IEEE Journal of Quantum Electronics, 23 (1987), 510-524.
- Alexandru D. Ionescu and Carlos E. Kenig, Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, J. Amer. Math. Soc. 20 (2007), no. 3, 753–798. MR 2291918, DOI https://doi.org/10.1090/S0894-0347-06-00551-0
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), no. 4, 527–620. MR 1211741, DOI https://doi.org/10.1002/cpa.3160460405
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 (2001), no. 3, 617–633. MR 1813239, DOI https://doi.org/10.1215/S0012-7094-01-10638-8
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), no. 2, 573–603. MR 1329387, DOI https://doi.org/10.1090/S0894-0347-96-00200-7
- Yuji Kodama, Optical solitons in a monomode fiber, J. Statist. Phys. 39 (1985), no. 5-6, 597–614. Transport and propagation in nonlinear systems (Los Alamos, N.M., 1984). MR 807002, DOI https://doi.org/10.1007/BF01008354
- Corinne Laurey, The Cauchy problem for a third order nonlinear Schrödinger equation, Nonlinear Anal. 29 (1997), no. 2, 121–158. MR 1446222, DOI https://doi.org/10.1016/S0362-546X%2896%2900081-8
- L. Molinet, J. C. Saut, and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal. 33 (2001), no. 4, 982–988. MR 1885293, DOI https://doi.org/10.1137/S0036141001385307
- Hiroaki Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39 (1975), no. 4, 1082–1091. MR 398275, DOI https://doi.org/10.1143/JPSJ.39.1082
- T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations 11 (1998), no. 2, 201–222. MR 1741843
- K. Porsezian and K. Nakkeeran, Singularity structure analysis and the complete integrability of the higher order nonlinear Schrödinger-Maxwell-Bloch equations, Chaos Solitons Fractals 7 (1996), no. 3, 377–382. MR 1381301, DOI https://doi.org/10.1016/0960-0779%2895%2900069-0
- K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, Coupled higher-order nonlinear Schrödinger equations in nonlinear optics: Painlevé analysis and integrability, Phys. Rev. E (3) 50 (1994), no. 2, 1543–1547. MR 1381874, DOI https://doi.org/10.1103/PhysRevE.50.1543
- Soonsik Kwon, Well-posedness and ill-posedness of the fifth-order modified KdV equation, Electron. J. Differential Equations (2008), No. 01, 15. MR 2368888
- Gigliola Staffilani, On the generalized Korteweg-de Vries-type equations, Differential Integral Equations 10 (1997), no. 4, 777–796. MR 1741772
- Catherine Sulem and Pierre-Louis Sulem, The nonlinear Schrödinger equation, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999. Self-focusing and wave collapse. MR 1696311
- Hideo Takaoka, Well-posedness for the higher order nonlinear Schrödinger equation, Adv. Math. Sci. Appl. 10 (2000), no. 1, 149–171. MR 1769176
- Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR 2233925
- Hideo Takaoka, Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity, Adv. Differential Equations 4 (1999), no. 4, 561–580. MR 1693278
- Yoshio Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), no. 1, 115–125. MR 915266
- Nickolay Tzvetkov, Remark on the local ill-posedness for KdV equation, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 12, 1043–1047 (English, with English and French summaries). MR 1735881, DOI https://doi.org/10.1016/S0764-4442%2800%2988471-2
- Hua Wang, Global well-posedness of the Cauchy problem of a higher-order Schrödinger equation, Electron. J. Differential Equations (2007), No. 04, 11. MR 2278418
References
- 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)
- 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)
- T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29 (1967), 559-592.
- J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data, Selecta Math. New Ser. 3 (1997), 115-159. MR 1466164 (2000i:35173)
- 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)
- X. Carvajal, Sharp global well-posedness for a higher order Schrödinger equation, J. Fourier Anal. Appl. 12 (2006), 53-70. MR 2215677 (2007d:35253)
- 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)
- 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)
- X. Carvajal and M. Panthee, On uniqueness of solution for a nonlinear Schrödinger-Airy equation, Nonlinear Anal., (2006). MR 2183834 (2006f:35261)
- 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)
- 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)
- 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)
- 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)
- 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)
- A. Hasegawa and Y. Kodama, Nonlinear pulse propagation in a monomode dielectric guide, IEEE Journal of Quantum Electronics, 23 (1987), 510-524.
- 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)
- 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)
- 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)
- 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)
- Y. Kodama, Optical solitons in a monomode fiber, Journal of Statistical Phys., 39 (1985), 597–614. MR 807002 (86m:78021)
- C. Laurey, The Cauchy Problem for a Third Order Nonlinear Schrödinger Equation, Nonlinear Anal., 29 (1997), 121-158. MR 1446222 (98c:35154)
- 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)
- H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091. MR 0398275 (53:2129)
- 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)
- 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)
- K. Porsezian, P. Shanmugha, K. Sundaram and A. Mahalingam, Phys. Rev. 50E,1543 (1994). MR 1381874 (96k:35171)
- 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)
- G. Staffilani, On the Generalized Korteweg-de Vries-Type Equations, Differential and Integral Equations 10 (1997), 777-796. MR 1741772 (2001a:35005)
- 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)
- 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)
- T. Tao, Nonlinear dispersive equations, local and global analysis, CBMS Regional Conference Series in Mathematics, American Mathematical Society (2006). MR 2233925 (2008i:35211)
- 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)
- Y. Tsutsumi, $L^2$–solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), 115-125. MR 915266 (89c:35143)
- 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)
- 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
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