Construction of the multi-soliton trains for a generalized derivative nonlinear Schr\''odinger equations by a fixed point method

We consider a derivative nonlinear Schr{\"o}dinger equation with general nonlinearlity: i$\partial$tu + $\partial$ 2 x u + i|u| 2$\sigma$ $\partial$xu = 0, In [12], the authors prove the stability of two solitary waves in energy space for $\sigma$ $\in$ (1, 2). As a consequence, there exists a two-soliton trains in energy space for $\sigma$ $\in$ (1, 2). Our goal in this paper is proving the existence of multi-soliton trains in energy space for $\sigma$ 5 2. Our proofs proceed by xed point arguments around the desired prole, using Strichartz estimates.