Free access
Issue
ESAIM: M2AN
Volume 41, Number 1, January-February 2007
Page(s) 95 - 110
DOI http://dx.doi.org/10.1051/m2an:2007010
Published online 26 April 2007
  1. R. Beals, P. Deift and C. Tomei, Direct and inverse scattering on the line. Mathematical Surveys and Monographs 28, American Mathematical Society, Providence, RI (1988).
  2. J.L. Bona and Z. Grujić, Spatial analyticity for nonlinear waves. Math. Models Methods Appl. Sci. 13 (2003) 1–15. [CrossRef] [MathSciNet]
  3. J.L. 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 (2005) 783–797.
  4. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. GAFA 3 (1993) 107–156, 209–262. [CrossRef] [MathSciNet]
  5. J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17 (1872) 55–108.
  6. C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods in Fluid Dynamics. Springer, Berlin (1988).
  7. 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. [CrossRef] [MathSciNet]
  8. J.M. Cooley and J.W. Tukey, An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19 (1965) 297–301. [CrossRef] [MathSciNet]
  9. A. Doelman and E.S. Titi, Regularity of solutions and the convergence of the Galerkin method in the Ginzburg-Landau equation. Numer. Funct. Anal. Optim. 14 (1993) 299–321. [CrossRef] [MathSciNet]
  10. P.G. Drazin and R.S. Johnson, Solitons: an introduction, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (1989).
  11. A.B. Ferrari and E.S. Titi, Gevrey regularity for nonlinear analytic parabolic equations. Comm. Partial Differential Equations 23 (1998) 1–16. [MathSciNet]
  12. C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations. J. Functional Anal. 87 (1989) 359–369. [CrossRef]
  13. Z. Grujić and H. Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions. Diff. Integral Eq. 15 (2002) 1325–1334.
  14. N. Hayashi, Analyticity of solutions of the Korteweg-de Vries equation. SIAM J. Math. Anal. 22 (1991) 1738–1743. [CrossRef] [MathSciNet]
  15. N. Hayashi, Solutions of the (generalized) Korteweg-de Vries equation in the Bergman and Szegö spaces on a sector. Duke Math. J. 62 (1991) 575–591. [CrossRef] [MathSciNet]
  16. H. Kalisch, Rapid convergence of a Galerkin projection of the KdV equation. C. R. Math. 341 (2005) 457–460.
  17. T. Kappeler and P. Topalov, Global well-posedness of KdV in Formula . Duke Math. J. 7 135 (2006) 327–360.
  18. T. Kato and K. Masuda, Nonlinear evolution equations and analyticity I. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3 (1986) 455–467.
  19. C.E Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc. 9 (1996) 573–603. [CrossRef] [MathSciNet]
  20. D.J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular channel and on a new type of long stationary wave. Philos. Mag. 39 (1895) 422–443.
  21. H.-O. Kreiss and J. Oliger, Stability of the Fourier method. SIAM J. Numer. Anal. 16 (1979) 421–433. [CrossRef] [MathSciNet]
  22. C.D. Levermore and M. Oliver, Analyticity of solutions for a generalized Euler equation. J. Differential Equations 133 (1997) 321–339. [CrossRef] [MathSciNet]
  23. Y. Maday and A. Quarteroni, Error analysis for spectral approximation of the Korteweg-de Vries equation. RAIRO Modél. Math. Anal. Numér. 22 (1988) 499–529. [MathSciNet]
  24. J.E. Pasciak, Spectral and pseudospectral methods for advection equations. Math. Comput. 35 (1980) 1081–1092.
  25. E. Tadmor, The exponential accuracy of Fourier and Chebyshev differencing methods. SIAM J. Numer. Anal. 23 (1986) 1–10. [CrossRef] [MathSciNet]
  26. T. Taha and M. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations. III. Numerical, Korteweg-de Vries equation. J. Comput. Phys. 55 (1984) 231–253. [CrossRef] [MathSciNet]
  27. R. Temam, Sur un problème non linéaire. J. Math. Pures Appl. 48 (1969) 159–172. [MathSciNet]
  28. G.B. Whitham, Linear and Nonlinear Waves. Wiley, New York (1974).
  29. N.J. Zabusky and M.D. Kruskal, Interaction of solutions in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15 (1965) 240–243. [CrossRef]

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