Multi-solitary waves for the nonlinear Klein-Gordon equation

Jacopo Bellazzini (1) , Marco Ghimenti (2) , Stefan Le Coz (3)

1 UNISS - Università degli Studi di Sassari = University of Sassari [Sassari]
2 DMA - Dipartimento di Matematica Applicata [Pisa]
3 IMT - Institut de Mathématiques de Toulouse UMR5219

We consider the nonlinear Klein-Gordon equation in $\R^d$. We call multi-solitary waves a solution behaving at large time as a sum of boosted standing waves. Our main result is the existence of such multi-solitary waves, provided the composing boosted standing waves are stable. It is obtained by solving the equation backward in time around a sequence of approximate multi-solitary waves and showing convergence to a solution with the desired property. The main ingredients of the proof are finite speed of propagation, variational characterizations of the profiles, modulation theory and energy estimates.

Mots clés: Klein-Gordon equation; multi-soliton; asymptotic behavior

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Jacopo Bellazzini, Marco Ghimenti, Stefan Le Coz. Multi-solitary waves for the nonlinear Klein-Gordon equation. Communications in Partial Differential Equations, 2014, 39 (8), pp.1479-1522. ⟨10.1080/03605302.2013.860988⟩. ⟨hal-00789123v2⟩