Conditional stability of multi-solitons for the 1D NLKG equation with double power nonlinearity

Yuan, Xu

doi:10.1016/j.anihpc.2020.11.008

Yuan, Xu ¹

¹ CMLS, École polytechnique, CNRS, Institut Polytechnique de Paris, F-91128 Palaiseau Cedex, France

Annales de l'I.H.P. Analyse non linéaire, septembre – octobre 2021, Tome 38 (2021) no. 5, pp. 1487-1524.

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Résumé

We consider the one-dimensional nonlinear Klein-Gordon equation with a double power focusing-defocusing nonlinearity

\partial_{t}^{2} u - \partial_{x}^{2} u + u - | u |^{p - 1} u + | u |^{q - 1} u = 0, on [0, \infty) \times R,

where

1 < q < p < \infty

. The main result states the stability in the energy space

H^{1} (R) \times L^{2} (R)

of the sums of decoupled solitary waves with different speeds, up to the natural instabilities. The proof is inspired by the techniques developed for the generalized Korteweg-de Vries equation and the nonlinear Schrödinger equation in a similar context by Martel, Merle and Tsai [14,15]. However, the adaptation of this strategy to a wave-type equation requires the introduction of a new energy functional adapted to the Lorentz transform.

MR Zbl

DOI : 10.1016/j.anihpc.2020.11.008

Classification : 35L71, 35B35
Mots-clés : 1D nonlinear Klein-Gordon equation, Multi-solitons, Conditional stability

Affiliations des auteurs :

Yuan, Xu ¹

¹ CMLS, École polytechnique, CNRS, Institut Polytechnique de Paris, F-91128 Palaiseau Cedex, France

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     author = {Yuan, Xu},
     title = {Conditional stability of multi-solitons for the {1D} {NLKG} equation with double power nonlinearity},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1487--1524},
     publisher = {Elsevier},
     volume = {38},
     number = {5},
     year = {2021},
     doi = {10.1016/j.anihpc.2020.11.008},
     mrnumber = {4300930},
     zbl = {1471.35034},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2020.11.008/}
}

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Yuan, Xu. Conditional stability of multi-solitons for the 1D NLKG equation with double power nonlinearity. Annales de l'I.H.P. Analyse non linéaire, septembre – octobre 2021, Tome 38 (2021) no. 5, pp. 1487-1524. doi : 10.1016/j.anihpc.2020.11.008. http://www.numdam.org/articles/10.1016/j.anihpc.2020.11.008/

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