$L^2$-stability of multi-solitons

 Access to full text

 Full (PDF)

1. M. Ablowitz, and P. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, 149. Cambridge University Press, Cambridge, 1991. Zbl0762.35001MR1149378
2. M. Ablowitz, D. Kaup, A. Newell, and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Studies in Appl. Math. 53 (1974), no. 4, 249–315. Zbl0408.35068MR450815
3. M. Ablowitz, and H. Segur, Solitons and the inverse scattering transform, SIAM Studies in Applied Mathematics, 4. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981. x+425 pp. Zbl0472.35002MR642018
4. J. Albert, J. Bona, and N. Nguyen, On the stability of KdV multisolitons, Differential Integral Equations 20 (2007), no. 8, 841–878. Zbl1212.35380MR2339841
5. Alejo, Miguel A., Muñoz C., and Vega, Luis, The Gardner equation and the $L^2$-stability of the $N$-soliton solution of the Korteweg–de Vries equation, to appear in Transactions of the AMS (arXiv:1012.5290). Zbl1278.35209
6. T.B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. London A 328, (1972) 153–183. MR338584
7. J.L. Bona, P. Souganidis and W. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type, Proc. Roy. Soc. London 411 (1987), 395–412. Zbl0648.76005MR897729
8. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV equation, Geom. Funct. Anal. 3 (1993), no. 3, 209-262. Zbl0787.35098MR1215780
9. K.W. Chow, R.H.J Grimshaw, and E. Ding, Interactions of breathers and solitons in the extended Korteweg-de Vries equation, Wave Motion 43 (2005) 158–166. Zbl1231.35196MR2186925
10. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T.Tao, Sharp global well-posedness for KdV and modified KdV on $ℝ$ and $𝕋$, J. Amer. Math. Soc. 16 (2003), no. 3, 705–749 (electronic). Zbl1025.35025MR1969209
11. C.S. Gardner, M.D. Kruskal, and R. Miura, Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion, J. Math. Phys. 9, no. 8 (1968), 1204–1209. Zbl0283.35019MR252826
12. F. Gesztesy, and B. Simon, Constructing solutions of the mKdV-equation, J. Funct. Anal. 89 (1990), no. 1, 53–60. Zbl0711.35121MR1040955
13. E. Fermi, J. Pasta and S. Ulam, Studies of nonlinear problems I, Los Alamos Report LA1940 (1955); reproduced in Nonlinear Wave Motion, A.C. Newell, ed., Am. Math. Soc., Providence, R. I., 1974, pp. 143–156. Zbl0353.70028
14. R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27 (1971), 1192–1194. Zbl1168.35423
15. C.E. Kenig, and Y. Martel, Global well-posedness in the energy space for a modified KP II equation via the Miura transform, Trans. Amer. Math. Soc. 358 no. 6, pp. 2447–2488. Zbl1106.35082MR2204040
16. 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 Appl. Math. 46, (1993) 527–620. Zbl0808.35128MR1211741
17. C.E. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 (2001), no. 3, 617–633. Zbl1034.35145MR1813239
18. 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 stationary waves, Philos. Mag. Ser. 5, 39 (1895), 422–443. Zbl26.0881.02
19. M.D. Kruskal and N.J. Zabusky, Interaction of “solitons” in a collisionless plasma and recurrence of initial states, Phys. Rev. Lett. 15 (1965), 240–243. Zbl1201.35174
20. P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21, (1968) 467–490. Zbl0162.41103MR235310
21. J.H. Maddocks, and R.L. Sachs, On the stability of KdV multi-solitons, Comm. Pure Appl. Math. 46, 867–901 (1993). Zbl0795.35107MR1220540
22. Y. Martel and F. Merle, Asymptotic stability of solitons of the subcritical gKdV equations revisited, Nonlinearity 18 (2005) 55–80. Zbl1064.35171MR2109467
23. Y. Martel and F. Merle, Description of two soliton collision for the quartic gKdV equation, preprint arXiv:0709.2672 (2007), to appear in Annals of Mathematics. Zbl1300.37045MR2831108
24. Y. Martel and F. Merle, Stability of two soliton collision for nonintegrable gKdV equations, Comm. Math. Phys. 286 (2009), 39–79. Zbl1179.35291MR2470923
25. Y. Martel and F. Merle, Inelastic interaction of nearly equal solitons for the quartic gKdV equation, Invent. Math. 183 (2011), no. 3, 563–648. Zbl1230.35121MR2772088
26. Y. Martel, and F. Merle, Asymptotic stability of solitons of the gKdV equations with general nonlinearity, Math. Ann. 341 (2008), no. 2, 391–427. Zbl1153.35068MR2385662
27. Y. Martel, F. Merle and T. P. Tsai, Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations, Comm. Math. Phys. 231 (2002) 347–373. Zbl1017.35098MR1946336
28. F. Merle; and L. Vega, $L^2$ stability of solitons for KdV equation, Int. Math. Res. Not. 2003, no. 13, 735–753. Zbl1022.35061MR1949297
29. R.M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Math. Phys. 9, no. 8 (1968), 1202–1204. Zbl0283.35018MR252825
30. R.M. Miura, The Korteweg–de Vries equation: a survey of results, SIAM Review 18, (1976) 412–459. Zbl0333.35021MR404890
31. T. Mizumachi, and D. Pelinovsky, Bäcklund transformation and $L^2$-stability of NLS solitons, preprint. Zbl1239.35148
32. T. Mizumachi, and N. Tzvetkov, Stability of the line soliton of the KP–II equation under periodic transverse perturbations, preprint. Zbl1233.35174MR2885592
33. C. Muñoz, On the inelastic 2-soliton collision for gKdV equations with general nonlinearity, Int. Math. Research Notices (2010) 2010 (9): 1624–1719. Zbl1198.35234MR2643578
34. C. Muñoz, The Gardner equation and the stability of multi-kink solutions of the mKdV equation, preprint arXiv:1106.0648. Zbl1336.35313
35. R.L. Pego, and M.I. Weinstein, Asymptotic stability of solitary waves, Commun. Math. Phys. 164, 305–349 (1994). Zbl0805.35117MR1289328
36. B. Thaller, The Dirac equation, Texts and Monographs in Physics. Springer-Verlag, Berlin, 1992. xviii+357 pp. Zbl0765.47023MR1219537
37. M.I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure. Appl. Math. 39, (1986) 51—68. Zbl0594.35005MR820338
38. M.V. Wickerhauser, Inverse scattering for the heat operator and evolutions in $2+1$ variables, Comm. Math. Phys. 108 (1987), 67–89. Zbl0633.35070MR872141
39. P. E. Zhidkov, Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Mathematics, vol. 1756, Springer-Verlag, Berlin, 2001. Zbl0987.35001MR1831831