May  2014, 34(5): 1961-1993. doi: 10.3934/dcds.2014.34.1961

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

Received  September 2010 Revised  July 2013 Published  October 2013

For the $L^2$ supercritical generalized Korteweg-de Vries equation, we proved in [2] the existence and uniqueness of an $N$-parameter family of $N$-solitons. Recall that, for any $N$ given solitons, we call $N$-soliton a solution of the equation which behaves as the sum of these $N$ solitons asymptotically as $t \to +\infty$. In the present paper, we also construct an $N$-parameter family of $N$-solitons for the supercritical nonlinear Schrödinger equation in dimension $1$. Nevertheless, we do not obtain any classification result; but recall that, even in subcritical and critical cases, no general uniqueness result has been proved yet.
Citation: Vianney Combet. Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1961-1993. doi: 10.3934/dcds.2014.34.1961
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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