Orbital stability of solitary wave solutions of the derivative nonlinear Schrödinger equation in the endpoint case

In this paper, we consider the orbital stability of solitary waves for the derivative nonlinear Schrödinger equation $$i{q}_t+q_{xx}+i{\mathrm{\partial }}_x\left(\mathrm{\mid }q{\mathrm{\mid }}^{2}q\right)=0,(t,x)\in \mathbb{R}\times \mathbb{R}\mathrm{.}$$ For the endpoint solitary waves satisfying $c>0$ and $c^2=4\omega$, we prove the orbital stability in $H^1(\mathbb{R})$ without any additional assumptions. The proof uses conservation of mass and the second Hamiltonian ${\mathcal{H}}_2$ to keep the Bäcklund-transformed solution small in $H^1(\mathbb{R})$ uniformly in time. We establish an inverse Bäcklund compactness result to reconstruct the algebraic solitary wave as the spectral zero approaches the negative real axis.