Multi-soliton solutions, i.e. solutions behaving as the sum of given solitons as , were constructed for the critical and subcritical (NLS) and (gKdV) equations in previous works (see [Merle, F.: Construction of solutions with exactly blow-up points for the Schrödinger equation with critical nonlinearity. Comm. Math. Phys. 129 (1990), no. 2, 223-240], [Martel, Y.: Asymptotic -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 supercritical case both for (gKdV) and (NLS) equations, using a topological argument to control the direction of instability.
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Raphaël Côte, Yvan Martel, Frank Merle, Construction of multi-soliton solutions for the -supercritical gKdV and NLS equations. Rev. Mat. Iberoam. 27 (2011), no. 1, pp. 273–302DOI 10.4171/RMI/636