Inelasticity of soliton collisions for the 5D energy critical wave equation

Abstract

For the focusing energy critical wave equation in 5D, we construct a solution showing the inelastic nature of the collision of two solitons for any choice of sign, speed, scaling and translation parameters, except the special case of two solitons of same scaling and opposite signs. Beyond its own interest as one of the first rigorous studies of the collision of solitons for a non-integrable model, the case of the quartic gKdV equation being partially treated in [32,33,34], this result can be seen as part of a wider program aiming at establishing the soliton resolution conjecture for the critical wave equation. This conjecture has already been established in the 3D radial case in [10] and in the general case in 3, 4 and 5D along a sequence of times in [13].Compared with the construction of an asymptotic two-soliton in [35], the study of the nature of the collision requires a more refined approximate solution of the two-soliton problem and a precise determination of its space asymptotics. To prove inelasticity, these asymptotics are combined with the method of channels of energy from [10,23].

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

“…, 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].…”

“…Outline of the proof. The strategy of the proof of Theorem 1 combines the use of a refined approximate solution for the K-soliton problem from [12], with uniform estimates and compactness, as in [11]. See also references in [11,12] for earlier works.…”

“…The smallness condition (1.4) is still probably technical and we conjecture that the result holds for any choice of speeds. However, by our method, (1.4) seems to be the best condition using the refined approximate solution from [12]. Moreover, constructing approximate solutions improving the one in [12] seems a difficult task.…”

“…In [11], the nature of the localization does not allow non-collinear multi-solitons, even under a smallness condition. The main improvements in the present work are the following: (1) the use of a refined approximate solution from [12]; (2) the introduction of a new localized energy functional to deal with non-collinear solitons. The smallness condition (1.4) is still probably technical and we conjecture that the result holds for any choice of speeds.…”

Construction of multi-bubble solutions for the energy-critical wave equation in dimension 5

“…In [29], a solution of (1.1) containing an arbitrary number K of bounded traveling solitons is constructed under some restrictions on the speeds ℓ k of the solitons. We also refer to [30] proving inelasticity of soliton interactions in the same context. Such works clearly relate questions (ii) and (iii) since the nonlinear interactions between the two solitons are responsible either for the blow up behavior or for the inelasticity property.…”