Multi-solitons for nonlinear Klein–Gordon equations

Abstract

AbstractIn this paper, we consider the existence of multi-soliton structures for the nonlinear Klein–Gordon (NLKG) equation in $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{R}^{1+d}$. We prove that, independently of the unstable character of NLKG solitons, it is possible to construct a $N$-soliton family of solutions to the NLKG equation, of dimension $2N$, globally well defined in the energy space $H^1\times L^2$ for all large positive times. The method of proof involves the generalization of previous works on supercritical Nonlinear Schrödinger (NLS) and generalized Korteweg–de Vries (gKdV) equations by Martel, Merle, and the first author [R. Côte, Y. Martel and F. Merle, Rev. Mat. Iberoam. 27 (1) (2011), 273–302] to the wave case, where we replace the unstable mode associated to the linear NLKG operator by two generalized directions that are controlled without appealing to modulation theory. As a byproduct, we generalize the linear theory described in Grillakis, Shatah, and Strauss [J. Funct. Anal. 74 (1) (1987), 160–197] and Duyckaerts and Merle [Int. Math. Res. Pap. IMRP (2008), Art ID rpn002] to the case of boosted solitons, and provide new solutions to be studied using the recent work of Nakanishi and Schlag [Zurich Lectures in Advanced Mathematics, vi+253 pp (European Mathematical Society (EMS), Zürich, 2011)] theory.