Controllable combined Peregrine soliton and Kuznetsov–Ma soliton in $${\varvec{\mathcal {PT}}}$$PT-symmetric nonlinear couplers with gain and loss

We investigate a (2+1)-dimensional-coupled variable coefficient nonlinear Schrödinger equation in parity time symmetric nonlinear couplers with gain and loss and analytically obtain a combined structure solution via the Darboux transformation method. When the imaginary part of the eigenvalue $$n$$n is smaller or bigger than 1, we can obtain the combined Peregrine soliton and Akhmediev breather, or Kuznetsov–Ma soliton, respectively. Moreover, we study the controllable behaviors of this combined Peregrine soliton and Kuznetsov–Ma soliton structure in a diffraction decreasing system with exponential profile. In this system, the effective propagation distance $$Z$$Z exists a maximal value $$Z_m$$Zm and the maximum amplitude of the KM soliton appears in the periodic locations $$Z_{i}$$Zi. By modulating the relation between values of $$Z_m$$Zm and $$Z_i$$Zi, we realize the control for the excitation of the combined Peregrine soliton and Kuznetsov–Ma soliton, such as the restraint, maintenance, and postpone.

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