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We talk about the Ablowitz-Musslimani type nonlocal reductions. We show that nonlocal reductions of systems of integrable nonlinear partial differential equations are the special discrete symmetry transformations. We present some nonlocal reduced integrable equations obtained from nonlinear Schrödinger (NLS) and modified Korteweg-de Vries (mKdV) systems. We give some examples of soliton solutions of these nonlocal equations obtained by using the Hirota method. This is a joint work with Metin Gürses and Kostyantyn Zheltukhin.
References:
[1] Ablowitz MJ, Musslimani ZH. Integrable nonlocal nonlinear equations, Stud. Appl. Math. 2016; 139(1): 7.
[2] Gürses M, Pekcan A, Zheltukhin K. Discrete symmetries and nonlocal reductions, Phys. Lett. A, accepted.
[3] Gürses M, Pekcan A. Nonlocal nonlinear Schr\"{o}dinger equations and their soliton solutions, J. Math. Phys. 2018; 59: 051501.
[4] Gürses M, Pekcan A. Nonlocal nonlinear modified KdV equations and their soliton solutions, Commun. Nonlinear Sci. Numer. Simulat. 2019; 67: 427--448.