Soliton, breather and rogue wave solutions of the nonlinear Schrödinger equation via Darboux transformation on a time–space scale.

Solving soliton equations on the time–space scale has always been a challenging issue. In this paper, we firstly generalize the Ablowitz–Kaup–Newel–Segur (AKNS) method to the time–space scale, concurrently obtain the nonlinear Schrödinger (NLS) equation on this scale, which unifies the continuous and the semi-discrete NLS equations. On this basis, the N -fold Darboux transformation is proposed for the NLS equation on a space scale. As applications, soliton, breather, and rogue wave solutions of NLS equation are derived from diverse seed solutions on a space scale. Specially, the rouge solution on a space scale is obtained for the first time. • The AKNS method is generalized to the time–space scale. • The NLS equation on a time–space scale is derived. • Continuous and the semi-discrete NLS equation are unified on a time–space scale. • Breather and rogue wave solutions of NLS equation are obtained on a space scale. [ABSTRACT FROM AUTHOR]