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

Abstract

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.