Revista Matemática Iberoamericana

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Volume 27, Issue 1, 2011, pp. 273–302
DOI: 10.4171/RMI/636

Published online: 2011-04-30

Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations

Raphaël Côte[1], Yvan Martel[2] and Frank Merle[3]

(1) Ecole Polytechnique, Palaiseau, France
(2) École Polytechnique, Palaiseau, France
(3) Université de Cergy-Pontoise, France

Multi-soliton solutions, i.e. solutions behaving as the sum of $N$ given solitons as $t \to +\infty$, were constructed for the $L^2$ critical and subcritical (NLS) and (gKdV) equations in previous works (see [Merle, F.: Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearity. Comm. Math. Phys. 129 (1990), no. 2, 223-240], [Martel, Y.: Asymptotic $N$-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations. Amer. J. Math. 127 (2005), no. 5, 1103-1140] and [Martel, Y. and Merle, F.: Multi solitary waves for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), 849-864]). In this paper, we extend the construction of multi-soliton solutions to the $L^2$ supercritical case both for (gKdV) and (NLS) equations, using a topological argument to control the direction of instability.

Keywords: Multi-solitons, generalized Korteweg-de Vries equation, nonlinear Schrödinger equation, instability, supercritical problem.

Côte Raphaël, Martel Yvan, Merle Frank: Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations. Rev. Mat. Iberoam. 27 (2011), 273-302. doi: 10.4171/RMI/636