On the long time behavior of KdV type equations

 Access to full text

 Full (PDF)

1. [1] M.J. Ablowitz and H. Segur. Solitons and the inverse scattering transform. SIAM, Philadelphia, 1981. Zbl0472.35002MR642018
2. [2] S. Alinhac. Blow-up for nonlinear hyperbolic equations. Progress in Nonlinear Differential Equations and their Applications. Birkhäuser, Boston, 1995. Zbl0820.35001MR1339762
3. [3] T. Benjamin. Internal waves of permanent form in fluids of great depth. J. Fluid Mech., 29:559–592, 1967. Zbl0147.46502
4. [4] T. Benjamin. The stability of solitary waves. Proc. London Math. Soc. (3), 328:153–183, 1972. MR338584
5. [5] T. Benjamin, J. Bona, and J. Mahony. Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Roy. Soc. London Ser. A, 272:47–78, 1972. Zbl0229.35013MR427868
6. [6] J. Bona. The stability of solitary waves. Proc. London Math. Soc. (3), 344:363–374, 1975. Zbl0328.76016MR386438
7. [7] J. Bona and R. Smith. The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. Roy. Soc. London Ser. A, 278:555–601, 1975. Zbl0306.35027MR385355
8. [8] J. Bona, P. Souganidis, and W. Strauss. Stability and instability of solitary waves of Korteweg-de Vries type. Proc. London Math. Soc. (3), 411:395–412, 1987. Zbl0648.76005MR897729
9. [9] J. Bona and F. Weissler. Similarity solutions of the generalized Korteweg-de Vries equation. Math. Proc. Cambridge Philos. Soc., 127:323–351, 1999. Zbl0939.35164MR1705463
10. [10] A. de Bouard and Y. Martel. Non existence of $L^2$-compact solutions of the Kadomtsev-Petviashvili II equation. Math. Ann., 328:525–544, 2004. Zbl1330.35389MR2036335
11. [11] J. Bourgain. Global solutions of nonlinear Schrödinger equations, volume 46 of AMS Colloquium Publications. American Mathematical Society, Providence, R.I., 1999. Zbl0933.35178MR1691575
12. [12] J. Bourgain. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations II. The KdV equation. Geom. Funct. Anal., 3:209–262, 1993. Zbl0787.35098MR1215780
13. [13] T. Cazenave. Semilinear Schrödinger equations, volume 10 of Courant Lecture Notes. 2003. Zbl1055.35003MR2002047
14. [14] T. Cazenave and P.-L. Lions. Orbital stability of standing waves for some nonlinear Schrödinger equation. Comm. Math. Phys., 85:549–561, 1982. Zbl0513.35007MR677997
15. [15] J.-Y. Chemin. Explosion géométrique pour certaines équations d’ondes non linéaires (d’après Serge Alinhac). In Sém. Bourbaki (1998/99), volume 266 of Astérisque, pages 7–20. Société Mathématique de France, 2000. Exp. 850. Zbl1049.35124MR1772668
16. [16] W. Eckhaus and P. Schuur. The emergence of solitons of the Korteweg-de Vries equation from arbitrary initial conditions. Math. Methods Appl. Sci., 5:97–116, 1983. Zbl0518.35074MR690898
17. [17] K. El Dika. Stabilité asymptotique des ondes solitaires de l’équation de Benjamin-Bona-Mahony. C. R. Acad. Sci. Paris Sér. I Math., 337:649–652, 2003. Zbl1032.35036MR2030105
18. [18] K. El Dika. Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation. Preprint, 2003. Zbl1032.35036MR2030105
19. [19] C. Fermanian, F. Merle, and H. Zaag. Stability of the blow-up profile of non-linear heat equations from a dynamical system point of view. Math. Ann., 317:347–387, 2000. Zbl0971.35038MR1764243
20. [20] S. Friedlander, W. Strauss, and M. Vishik. Nonlinear instability in an ideal fluid. Ann. Inst. H. Poincaré. Anal. Non Linéaire, 14:187–209, 1997. Zbl0874.76026MR1441392
21. [21] J. Ginibre and Y. Tsutsumi. Uniqueness of solutions for the generalized Korteweg-de Vries equation. SIAM J. Appl. Math., 20:1388–1425, 1989. Zbl0702.35224MR1019307
22. [22] L. Glangetas and F. Merle. A geometric approach of existence of blow-up solutions. Preprint, 1995. Zbl0808.35137
23. [23] M. Grillakis, J. Shatah, and W. Strauss. Stability theory of solitary waves in the presence of symmetry. J. Funct. Anal., 74:160–197, 1987. Zbl0656.35122MR901236
24. [24] L. Hörmander. The analysis of linear partial differential operators I. Springer-Verlag, 1983. Zbl0521.35001
25. [25] T. Kato. On the Cauchy problem for the (generalized) Korteweg-de Vries equation. volume 8 of Advances in Math. Suppl. Stud., pages 93–128. 1983. Zbl0549.34001MR759907
26. [26] C. Kenig and K. Koenig. On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations. Math. Res. Lett., 10:879–895, 2003. Zbl1044.35072MR2025062
27. [27] 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., 46:527–620, 1993. Zbl0808.35128MR1211741
28. [28] C. Kenig, G. Ponce, and L. Vega. On the concentration of blow-up solutions for the generalized KdV equation critical in $L^2$. volume 263 of Contemp. Math., pages 131–156. American Mathematical Society, 2000. Zbl0970.35125MR1777639
29. [29] G.L. Lamb Jr.Elements of soliton theory. John Wiley & Sons, New York, 1980. Zbl0445.35001MR591458
30. [30] C. Laurent and Y. Martel. Smoothness and exponential decay of $L^2$-compact solutions of the generalized KdV equation. Comm. Partial Differential Equations, 28:2093–2107, 2003. Zbl1060.35125MR2015414
31. [31] J. Maddocks and R. Sachs. On the stability of KdV multi-solitons. Comm. Pure Appl. Math., 46:867–901, 1993. Zbl0795.35107MR1220540
32. [32] Y. Martel. Multi-soliton-type solutions of the generalized KdV equations. Amer. J. Math. to appear. Zbl1047.35119MR2057725
33. [33] Y. Martel and F. Merle. Instability of solitons for the critical generalized Korteweg-de Vries equation. Geom. Funct. Anal., 11:74–123, 2001. Zbl0985.35071MR1829643
34. [34] Y. Martel and F. Merle. Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Rational Mech. Anal., 157:219–254, 2001. Zbl0981.35073MR1826966
35. [35] Y. Martel and F. Merle. A Liouville Theorem for the critical generalized Korteweg-de Vries equation. J. Math. Pures Appl., 79:339–425, 2000. Zbl0963.37058MR1753061
36. [36] Y. Martel and F. Merle. Stability of the blow-up profile and lower bounds on the blow-up rate for the critical generalized Korteweg-de Vries equation. Ann. of Math., 155:235–280, 2002. Zbl1005.35081MR1888800
37. [37] Y. Martel and F. Merle. Blow-up in finite time and dynamics of blow-up solutions for the $L^2$-critical generalized KdV equation. J. Amer. Math. Soc., 15:617–663, 2002. Zbl0996.35064MR1896235
38. [38] Y. Martel and F. Merle. Nonexistence of blow-up solution with minimal $L^2$-mass for the critical GKdV. Duke Math. J., 115:385–408, 2002. Zbl1033.35102MR1944576
39. [39] Y. Martel and F. Merle. Asymptotic stability of solitons for subcritical generalized KdV equations revisited. Preprint, 2004. Zbl0981.35073MR1826966
40. [40] Y. Martel, F. Merle, and Tai-Peng Tsai. Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations. Comm. Math. Phys., 231:347–373, 2002. Zbl1017.35098MR1946336
41. [41] F. Merle. Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math. J., 69:427–454, 1993. Zbl0808.35141MR1203233
42. [42] F. Merle. Asymptotics for $L^2$-minimal blow-up solutions of critical nonlinear Schrödinger equation. Ann. Inst. H. Poincaré. Anal. Non Linéaire, 13:553–565, 1996. Zbl0862.35013MR1409662
43. [43] F. Merle. Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations. In Proceeding of the International congress of Mathematicians (Berlin 1998), Doc. Math. Extra volume ICM (1998 III), pages 57–66. Deutsche Math. Vereinigung, 1998. Zbl0896.35123MR1648140
44. [44] F. Merle. Existence of blow-up solutions in the energy space for critical generalized KdV equation. J. Amer. Math. Soc., 14:555–578, 2001. Zbl0970.35128MR1824989
45. [45] F. Merle and P. Raphaël. Sharp upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Geom. Funct. Anal., 13:591–642, 2003. Zbl1061.35135MR1995801
46. [46] F. Merle and P. Raphaël. On universality of blow-up profile for $L^2$-critical nonlinear Schrödinger equation. Invent. Math., 156:565–672, 2004. Zbl1067.35110MR2061329
47. [47] F. Merle and P. Raphaël. Sharp lower bound on the blow-up rate for critical nonlinear Schrödinger equation. Preprint, 2004. Zbl1075.35077
48. [48] F. Merle and P. Raphaël. Blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann. of Math. to appear. Zbl1185.35263MR2150386
49. [49] F. Merle and P. Raphaël. Profiles and quantization of the blow-up mass for the critical nonlinear Schrödinger equation. Comm. Math. Phys. to appear. Zbl1062.35137MR2116733
50. [50] F. Merle and L. Vega. $L^2$-stability of solitons for KdV equation. Internat. Math. Res. Notices, pages 735–753, 2003. Zbl1022.35061MR1949297
51. [51] F. Merle and H. Zaag. A Liouville theorem for a vector valued nonlinear heat equation and applications. Math. Ann., 316:103–137, 2000. Zbl0939.35086MR1735081
52. [52] R. Miura. The Korteweg-de Vries equation : a survey of results. SIAM Rev., 18:412–459, 1976. Zbl0333.35021MR404890
53. [53] R. Pego and M. Weinstein. Eigenvalues, and instability of solitary waves. Philos. Trans. Roy. Soc. London Ser. A, 340:47–94, 1992. Zbl0776.35065MR1177566
54. [54] R. Pego and M. Weinstein. Asymptotic stability of solitary waves. Comm. Math. Phys., 164:305–349, 1994. Zbl0805.35117MR1289328
55. [55] P. Raphaël. Stability of the $loglog$ bound for blow-up solutions to the critical nonlinear Schrödinger equation. Math. Ann. to appear. Zbl1082.35143MR2122541
56. [56] J.-C. Saut. Sur quelques généralisations de l’équation de Korteweg-de Vries. J. Math. Pures Appl., 58:21–61, 1979. Zbl0449.35083MR533234
57. [57] J.-C. Saut. Remarks on generalized Kadomtsev-Petviashvili equations. Indiana Univ. Math. J., 42:1011–1026, 1993. Zbl0814.35119MR1254130
58. [58] P. Schuur. Asymptotic analysis of soliton problems, volume 1232 of Lect. Notes in Math. Springer-Verlag, Berlin, 1986. Zbl0643.35003MR874343
59. [59] S. Sulem and P.L. Sulem. The nonlinear Schrödinger equation. Self-focusing and wave collapse, volume 139 of Applied Mathematical Sciences. Springer-Verlag, New York, 1999. Zbl0928.35157MR1696311
60. [60] M. Weinstein. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys., 87:567–576, 1983. Zbl0527.35023MR691044
61. [61] M. Weinstein. Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Appl. Math., 16:472–491, 1985. Zbl0583.35028MR783974
62. [62] M. Weinstein. Lyapunov stability of ground states of nonlinear dispersive equations. Comm. Pure Appl. Math., 39:51–68, 1986. Zbl0594.35005MR820338
63. [63] M. Weinstein. On the structure and formation of singularities in solutions to nonlinear dispersive equations. Comm. Partial Differential Equations, 11:545–565, 1986. Zbl0596.35022MR829596

1. Nicolas Burq, Explosion pour l’équation de Schrödinger au régime du “log log”