Lagrangian nonlocal nonlinear Schrodinger equations

In 2013 Ablowitz and Musslimani (AM) obtained a nonlocal generalization of the NLS equation which is integrable, but does not have a standard Lagrangian structure. The present paper shows that there exist two new nonlocal NLS equations, similar to the AM model, which do possess Lagrangian structures. We show that these two models (LN1 and LN2) possess solitary wave solutions which remain trapped in the neighborhood of the origin (x = 0), and solitary waves which are able to escape from the origin. These two types of solutions are obtained by direct numerical solutions, and also by a variational method. In the case of LN2 model, the variational approach explains the existence of these two types of solutions. Collisions of breathers which obey the equation LN2 are numerically studied, and the results show that these breathers are robust solutions. (C) 2022 Elsevier Ltd. All rights reserved.