Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension

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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

Vianney Combet ^1,

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

Abstract
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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.

Keywords: supercritical, asymptotic behavior, multi-solitons, instability., NLS.

Mathematics Subject Classification: Primary: 35Q55, 35Q51; Secondary: 35B40, 37K4.

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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