Small multi solitons in a double power nonlinear Schrödinger equation

Abstract

We consider a nonlinear Schrödinger equation with double power nonlinearityiφt+Δφ+|φ|4dφ+|φ|p−1φ=0,(t,x)∈R×Rd,d≥2(NLS) for 1<p<1+4d. Let q=q(x) be the unique positive solution of the nonlinear elliptic equationΔu+|u|4du−u=0,u∈H1(Rd) and Qω be the unique positive solution of the nonlinear elliptic equationΔu+|u|4du+|u|p−1u−ωu=0,1<p<1+4d,u∈H1(Rd). Then we prove thatωq=sup{ω|‖Qω‖L2(Rd)<‖q‖L2(Rd)}>0. Furthermore for ω∈(0,ωq), the soliton eiωtQω(x) of (NLS) is orbitally stable. Moreover for K≥2 and k=1,2,⋅⋅⋅,K, taking ωk∈(0,ωq), γk∈R, xk∈Rd, vk∈Rd with vk≠vk′ to k≠k′ andRk(t,x)=Qωk(x−xk−vkt)ei(12vkx−14|vk|2t+ωkt+γk) with (t,x)∈R×Rd and ∑k=1K‖Rk(t)‖L2(Rd)<‖q‖L2(Rd), there exists a solution φ(t,x) of (NLS) such that limt→+∞⁡||φ(t,⋅)−∑k=1KRk(t,⋅)||H1(Rd)=0. This φ(t,x) is called the small multi soliton of (NLS).