On multi-solitons for the energy-critical wave equation in dimension 5

Abstract

We construct K-solitons of the focusing energy-critical nonlinear wave equation in five-dimensional space, i.e. solutions u of the equation such thatwhere K ≥ 2 and for any k ∈ {1, . . . , K}, W k is the Lorentz transform of the explicit standing soliton W (x) = (1 + |x| 2 /15) −3/2 , with any speed ℓ k ∈ R 5 , |ℓ k | < 1 satisfying ℓ k ′ = ℓ k for k ′ = k, and an explicit smallness condition. The proof extends the refined method of construction of asymptotic multi-solitons from [11,12].

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

“…Such stability results are closely related the existence of asymptotic pure multisolitons for non-integrable dispersive and wave equations which have been established in several previous works, for both stable and unstable solitons, see [3,5,9,10,11,16] for (gKdV), (NLS), and the energy critical wave equation. For the nonlinear Klein-Gordon equation (1.2), the existence of asymptotic multi-solitons was established by Côte and Muñoz [6].…”

On existence and uniqueness of asymptotic $N$-soliton-like solutions of the nonlinear klein-gordon equation

“…By means of a technical lemma of analysis (we refer to Lemma 7.1 in Appendix), this monotonicity property allows us to see that any multi-soliton in the class with polynomial convergence to zero converges in fact exponentially (see subsection 4.1.4), provided one assumes suitable integrability conditions in the neighborhood of +∞ (and indeed > 3). Such a "weak" monotonicity property has a priori been used so far only for the construction of multi-solitons or multi-bound states of the energy-critical wave equation [20,29,30]. We also underline that, by Lemma 7.1, we directly obtain the adequate exponential convergence rate which allows us to identify 1 ; this is in contrast with [4].…”

Construction of excited multi-solitons for the 5D energy-critical wave equation

“…We define the continuous function χ N (t, x) = χ N (t, x 1 ) as follow (see [14,21] for a similar choice of cut-off function), for t > 0,…”

“…We refer to [3,14] for results on the existence of multi-solitons for the nonlinear Klein-Gordon (NLKG) and 5D energy critical wave equation that inspired the present work. See also [6,13,18,21] for other existence results.…”