Bilinear forms and breather solutions for a variable-coefficient nonlocal nonlinear Schrödinger equation in an optical fiber

In this paper, we investigate a variable-coefficient nonlocal nonlinear Schrödinger equation in an optical fiber, which describes some optical pulses in a self-focusing medium. With the help of the Hirota bilinear method and symbolic computation, we obtain a set of the bilinear forms, the first-, second- and Nth-order breather solutions of the forementioned equation under a constraint. In particular, we graphically describe the effects of the variable coefficients on the first- and second-order breathers and we find that those variable coefficients change the spatial structures of the breathers. With the changes of the variable coefficients, the first-order breathers show periodicity, pulse train, split, breather wall and jagged shape and the second-order breathers show that one of the breathers becomes wider or the distance between the two breathers becomes closer. The results can provide a theoretical basis for the control of optical waves in nonlocal nonlinear media.