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

Abstract

We consider the one-dimensional nonlinear Klein-Gordon equation with a double power focusing-defocusing nonlinearity∂t2u−∂x2u+u−|u|p−1u+|u|q−1u=0,on[0,∞)×R, where 1<q<p<∞. The main result states the stability in the energy space H1(R)×L2(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.