Refined asymptotics around solitons for gKdV equations

الأصل · EN

We consider the generalized Korteweg-de Vries equation ∂ₜ u + ∂ₓ (∂ₓ² u + f(u))=0, (t,x)∈ [0,T)× R with general C² nonlinearity f. Under an explicit condition on f and c>0, there exists a solution in the energy space H¹ of the type u(t,x)=Qc(x-x₀-ct), called soliton. Stability theory for Qc is well-known. In previous works, we have proved that for f(u)=uᵖ, p=2,3,4, the family of solitons is asymptotically stable in some local sense in H¹, i.e. if u(t) is close to Qc (for all t≥ 0), then u(t,.+ρ(t)) locally converges in the energy space to some Qc₊ as t→ +∞, for some c+ c. Then, the asymptotic stability result could be extended to the case of general assumptions on f and Qc. The objective of this paper is twofold. The main objective is to prove that in the case f(u)=uᵖ, p=2,3,4, ρ(t)-c+ t has limit as t→ +∞ under the additional assumption x+ u∈ L². The second objective of this paper is to provide large time stability and asymptotic stability results for two soliton solutions for the case of general nonlinearity f(u), when the ratio of the speeds of the solitons is small. The motivation is to accompany forthcoming works devoted to the collision of two solitons in the nonintegrable case. The arguments are refinements of previous works specialized to the case u(t) Qc₁+Qc₂, for 0< c₂ ≪ c₁.