Soliton solutions for two kinds of fourth-order nonlinear nonlocal Schrödinger equations

Abstract

In this paper, we will take two symmetric reduction conditions to obtain two kinds of fourth-order nonlinear nonlocal Schrödinger (NLS) equations, namely the fourth-order nonlocal reverse space and reverse space–time NLS equations. For the former, we will construct one- and N-fold Darboux Transformations (DTs) to obtain the symmetry preserving and broken solutions. In addition, we will study the different combinations of collision scenarios of dark and anti-dark soliton solutions. This process will be expressed in the form of quasi-determinant. Finally, we will get the symmetry preserving and broken solutions can exist at the same time. For the latter, the one- and N-fold DTs will be constructed. Taking the seed solutions as zero and continue wave solutions, we will obtain different kinds of exact solutions under nonlocal constraints, such as soliton, breather and rogue wave solutions, which is different from the classical case.