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January, 2011 Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations
Raphaël Côte , Yvan Martel , Frank Merle
Rev. Mat. Iberoamericana 27(1): 273-302 (January, 2011).


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.


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Raphaël Côte . Yvan Martel . Frank Merle . "Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations." Rev. Mat. Iberoamericana 27 (1) 273 - 302, January, 2011.


Published: January, 2011
First available in Project Euclid: 4 February 2011

zbMATH: 1273.35234
MathSciNet: MR2815738

Primary: 35Q51 , 35Q53 , 35Q55

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

Rights: Copyright © 2011 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.27 • No. 1 • January, 2011
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