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An extended discrete Hardy-Littlewood-Sobolev inequality
Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension
1. | Université Lille 1, U.F.R. de Mathématiques, 59 655 Villeneuve d'Ascq Cédex,, France |
References:
[1] |
T. Cazenave and F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Analysis, 14 (1990), 807-836.
doi: 10.1016/0362-546X(90)90023-A. |
[2] |
V. Combet, Multi-soliton solutions for the supercritical gKdV equations, Communications in Partial Differential Equations, 36 (2011), 380-419.
doi: 10.1080/03605302.2010.503770. |
[3] |
R. Côte, Y. Martel and F. Merle, Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations, Revista Matematica Iberoamericana, 27 (2011), 273-302.
doi: 10.4171/RMI/636. |
[4] |
T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geometric and Functional Analysis, 18 (2009), 1787-1840.
doi: 10.1007/s00039-009-0707-x. |
[5] |
T. Duyckaerts and S. Roudenko, Threshold solutions for the focusing 3d cubic Schrödinger equation, Revista Matematica Iberoamericana, 26 (2010), 1-56.
doi: 10.4171/RMI/592. |
[6] |
J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, Journal of Functional Analysis, 32 (1979), 1-32.
doi: 10.1016/0022-1236(79)90076-4. |
[7] |
M. Grillakis, Analysis of the linearization around a critical point of an infinite dimensional hamiltonian system, Communications on Pure and Applied Mathematics, 43 (1990), 299-333.
doi: 10.1002/cpa.3160430302. |
[8] |
M. Grillakis, J. Shatah and W. A. Strauss, Stability theory of solitary waves in the presence of symmetry. I, Journal of Functional Analysis, 74 (1987), 160-197.
doi: 10.1016/0022-1236(87)90044-9. |
[9] |
Y. Martel, Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, American Journal of Mathematics, 127 (2005), 1103-1140.
doi: 10.1353/ajm.2005.0033. |
[10] |
Y. Martel and F. Merle, Multi solitary waves for nonlinear Schrödinger equations, Annales de l'Institut Henri Poincaré/Analyse non linéaire, 23 (2006), 849-864.
doi: 10.1016/j.anihpc.2006.01.001. |
[11] |
Y. Martel, F. Merle and T.-P. Tsai, Stability in $H^1$ of the sum of $K$ solitary waves for some nonlinear Schrödinger equations, Duke Mathematical Journal, 133 (2006), 405-466.
doi: 10.1215/S0012-7094-06-13331-8. |
[12] |
F. Merle, Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearity, Communications in Mathematical Physics, 129 (1990), 223-240.
doi: 10.1007/BF02096981. |
[13] |
F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type $u_t = \Delta u + |u|^{p-1}u$, Duke Mathematical Journal, 86 (1997), 143-195.
doi: 10.1215/S0012-7094-97-08605-1. |
[14] |
G. Perelman, Some results on the scattering of weakly interacting solitons for nonlinear Schrödinger equations, Mathematical Topics, 14 (1997), 78-137. |
[15] |
G. Perelman, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations, Communications in Partial Differential Equations, 29 (2004), 1051-1095.
doi: 10.1081/PDE-200033754. |
[16] |
I. Rodnianski, W. Schlag and A. Soffer, Asymptotic Stability of N-soliton States of NLS, preprint, arXiv:math/0309114. |
[17] |
M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM Journal on Mathematical Analysis, 16 (1985), 472-491.
doi: 10.1137/0516034. |
show all references
References:
[1] |
T. Cazenave and F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Analysis, 14 (1990), 807-836.
doi: 10.1016/0362-546X(90)90023-A. |
[2] |
V. Combet, Multi-soliton solutions for the supercritical gKdV equations, Communications in Partial Differential Equations, 36 (2011), 380-419.
doi: 10.1080/03605302.2010.503770. |
[3] |
R. Côte, Y. Martel and F. Merle, Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations, Revista Matematica Iberoamericana, 27 (2011), 273-302.
doi: 10.4171/RMI/636. |
[4] |
T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geometric and Functional Analysis, 18 (2009), 1787-1840.
doi: 10.1007/s00039-009-0707-x. |
[5] |
T. Duyckaerts and S. Roudenko, Threshold solutions for the focusing 3d cubic Schrödinger equation, Revista Matematica Iberoamericana, 26 (2010), 1-56.
doi: 10.4171/RMI/592. |
[6] |
J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, Journal of Functional Analysis, 32 (1979), 1-32.
doi: 10.1016/0022-1236(79)90076-4. |
[7] |
M. Grillakis, Analysis of the linearization around a critical point of an infinite dimensional hamiltonian system, Communications on Pure and Applied Mathematics, 43 (1990), 299-333.
doi: 10.1002/cpa.3160430302. |
[8] |
M. Grillakis, J. Shatah and W. A. Strauss, Stability theory of solitary waves in the presence of symmetry. I, Journal of Functional Analysis, 74 (1987), 160-197.
doi: 10.1016/0022-1236(87)90044-9. |
[9] |
Y. Martel, Asymptotic N-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations, American Journal of Mathematics, 127 (2005), 1103-1140.
doi: 10.1353/ajm.2005.0033. |
[10] |
Y. Martel and F. Merle, Multi solitary waves for nonlinear Schrödinger equations, Annales de l'Institut Henri Poincaré/Analyse non linéaire, 23 (2006), 849-864.
doi: 10.1016/j.anihpc.2006.01.001. |
[11] |
Y. Martel, F. Merle and T.-P. Tsai, Stability in $H^1$ of the sum of $K$ solitary waves for some nonlinear Schrödinger equations, Duke Mathematical Journal, 133 (2006), 405-466.
doi: 10.1215/S0012-7094-06-13331-8. |
[12] |
F. Merle, Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearity, Communications in Mathematical Physics, 129 (1990), 223-240.
doi: 10.1007/BF02096981. |
[13] |
F. Merle and H. Zaag, Stability of the blow-up profile for equations of the type $u_t = \Delta u + |u|^{p-1}u$, Duke Mathematical Journal, 86 (1997), 143-195.
doi: 10.1215/S0012-7094-97-08605-1. |
[14] |
G. Perelman, Some results on the scattering of weakly interacting solitons for nonlinear Schrödinger equations, Mathematical Topics, 14 (1997), 78-137. |
[15] |
G. Perelman, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations, Communications in Partial Differential Equations, 29 (2004), 1051-1095.
doi: 10.1081/PDE-200033754. |
[16] |
I. Rodnianski, W. Schlag and A. Soffer, Asymptotic Stability of N-soliton States of NLS, preprint, arXiv:math/0309114. |
[17] |
M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM Journal on Mathematical Analysis, 16 (1985), 472-491.
doi: 10.1137/0516034. |
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