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Real-valued non-analytic solutions for the generalized Korteweg-de Vries equation


Authors: A. Alexandrou Himonas and Gerson Petronilho
Journal: Proc. Amer. Math. Soc. 139 (2011), 2759-2766
MSC (2010): Primary 35Q53; Secondary 37K10
DOI: https://doi.org/10.1090/S0002-9939-2011-10983-3
Published electronically: February 22, 2011
MathSciNet review: 2801615
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Abstract: In both the periodic and non-periodic cases, non-analytic in time solutions to the Cauchy problem of the gKdV equation are constructed with real-valued analytic initial data when $ k$ is not a multiple of four. In the case that $ k=4\ell$, that is, the non-linearity is of the form $ u^{4\ell}\partial_xu$, where $ \ell$ is a positive integer, then non-analytic in time solutions are available only for complex-valued initial data.


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  • [BGK] J. Bona, Z. Grujić and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 22, No. 6 (2005), 783-797. MR 2172859 (2006e:35282)
  • [BS] J. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation. Phil. Trans. Roy. Soc. London A 278, No. 1287 (1975), 555-601. MR 0385355 (52:6219)
  • [BKPSV] B. Birnir, C. Kenig, G. Ponce, N. Svanstedt, and L. Vega, On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations. J. London Math. Soc. (2) 53, No. 3 (1996), 551-559. MR 1396718 (97d:35233)
  • [B1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, II, The KdV-equation. Geom. Funct. Anal. 3, No. 3 (1993), 209-262. MR 1215780 (95d:35160b)
  • [B2] J. Bourgain, On the Cauchy problem for periodic KdV-type equations. Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993), J. Fourier Anal. Appl. 1995, Special Issue, 17-86. MR 1364878 (96m:35270)
  • [B3] J. Bourgain, Periodic Korteweg-de Vries equation with measures as initial data. Selecta Math. (N.S.) 3, No. 2 (1997), 115-159. MR 1466164 (2000i:35173)
  • [CCT] M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Amer. J. Math. 125, No. 6 (2003), 1235-1293. MR 2018661 (2005d:35223)
  • [CKSTT1] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Sharp global well-posedness for KdV and modified KdV on $ \mathbb{R}$ and $ \mathbb{T}$. J. Amer. Math. Soc. 16, No. 3 (2003), 705-749. MR 1969209 (2004c:35352)
  • [CKSTT2] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Multilinear estimates for periodic KdV equations, and applications. J. Funct. Anal. 211 (2004), 173-218. MR 2054622 (2005a:35241)
  • [DHK] A. De Bouard, N. Hayashi, and K. Kato, Gevrey regularizing effect for the (generalized) Korteweg-de Vries equation and nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 12, No. 6 (1995), 673-725. MR 1360541 (96j:35213)
  • [GT] J. Ginibre and Y. Tsutsumi, Uniqueness of solutions for the generalized Korteweg-de Vries equation. SIAM J. Math. Anal. 20, No. 6 (1989), 1388-1425. MR 1019307 (90i:35240)
  • [GH1] J. Gorsky and A. Himonas, Construction of non-analytic solutions for the generalized KdV equation. J. Math. Anal. Appl. 303, No. 2 (2005), 522-529. MR 2122235 (2005m:35255)
  • [GH2] J. Gorsky and A. Himonas, On analyticity in space variable of solutions to the KdV equation. Geometric analysis of PDE and several complex variables, Contemp. Math. 368, Amer. Math. Soc., 2005, 233-247. MR 2126473 (2007g:35210)
  • [GK] Z. Grujić and H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions. Differential Integral Equations 15, No. 11 (2002), 1325-1334. MR 1920689 (2003h:35229)
  • [HHP] H. Hannah, A. Himonas and G. Petronilho, Gevrey regularity for the periodic gKdV equation. J. Differential Equations 250 (2011), 2581-2600.
  • [H] N. Hayashi, Analyticity of solutions of the Korteweg-de Vries equation. SIAM J. Math. Anal. 22, No. 6 (1991), 1738-1743. MR 1129407 (92h:35208)
  • [KT1] T. Kappeler and P. Topalov, Global well-posedness of KdV in $ H\sp {-1}(\mathbb{T},\mathbb{R})$. Duke Math. J. 135, No. 2 (2006), 327-360. MR 2267286 (2007i:35199)
  • [KT2] T. Kappeler and P. Topalov, Global well-posedness of mKdV in $ L\sp 2(\mathbb{T},\mathbb{R})$. Comm. Partial Differential Equations 30, No. 1-3 (2005), 435-449. MR 2131061 (2005m:35256)
  • [K] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Adv. Math. Suppl. Studies 8, Academic Press, 1983, 93-128. MR 759907 (86f:35160)
  • [KM] T. Kato and K. Masuda, Nonlinear evolution equations and analyticity. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 3, No. 6 (1986), 455-467. MR 870865 (88h:34041)
  • [KO] K. Kato and T. Ogawa, Analyticity and smoothing effect for the Korteweg-de Vries equation with a single point singularity. Math. Ann. 316, No. 3 (2000), 577-608. MR 1752786 (2001c:35217)
  • [KPV1] C. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math. 46 (1993), 527-620. MR 1211741 (94h:35229)
  • [KPV2] C. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc. 9 (1996), 573-603. MR 1329387 (96k:35159)
  • [KPV3] C. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations. Duke Math. J. 106, No. 3 (2001), 617-633. MR 1813239 (2002c:35265)
  • [KdV] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 5 39 (1895), 422-443.
  • [ST] J. Saut and R. Temam, Remarks on the Korteweg-de Vries equation. Israel J. Math. 24, No. 1 (1976), 78-87. MR 0454425 (56:12676)
  • [S] A. Sjöberg, On the Korteweg-de Vries equation: existence and uniqueness. J. Math. Anal. Appl. 29 (1970), 569-579. MR 0410135 (53:13885)
  • [T] T. Tao, Nonlinear dispersive equations. Local and global analysis. American Mathematical Society, Providence, RI, 2006. MR 2233925 (2008i:35211)
  • [Ta] S. Tarama, Analyticity of solutions of the Korteweg-de Vries equation. J. Math. Kyoto Univ. 44, No. 1 (2004), 1-32. MR 2062705 (2005e:35206)
  • [Tr] E. Trubowitz, The inverse problem for periodic potentials. Comm. Pure Appl. Math. 30 (1977), 321-337. MR 0430403 (55:3408)

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Additional Information

A. Alexandrou Himonas
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: himonas.1@nd.edu

Gerson Petronilho
Affiliation: Departamento de Matemática, Universidade Federal de São Carlos, São Carlos - SP 13565-905, Brazil
Email: gerson@dm.ufscar.br

DOI: https://doi.org/10.1090/S0002-9939-2011-10983-3
Keywords: Generalized Korteweg-de Vries equation, gKdV, Cauchy problem, periodic, non-periodic, analytic solutions
Received by editor(s): July 16, 2010
Published electronically: February 22, 2011
Additional Notes: The second author was partially supported by CNPq and Fapesp
Communicated by: Walter Craig
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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