The (2+1)-dimensional variable-coefficient coupled nonlocal Gross–Pitaevskii equation: Localized waves, nonlinear dynamics and controls

We investigate a (2+1)-dimensional variable-coefficient coupled nonlocal Gross–Pitaevskii equation, which includes a nonlocal spatial direction. This equation features variable diffraction and nonlocal nonlinearity, which can bring new phenomena of localized waves. By a similarity transformation, the soliton solutions and higher-order rogue wave solutions are derived. Then, we study the influence of the diffraction, nonlocal nonlinearity, and external potential in this system on localized waves. Results show that under the control of nonlocal nonlinearity, variable-coefficient diffraction and related parameters, more new nonlinear wave dynamics structures can be achieved, including long-time and periodic soliton interferences, dark soliton phenomena under attractive interactions, and higher-order rogue wave phenomena with non-eye-shaped structures and unusual quantities. Finally, we numerically reproduce the three types of localized wave solutions and analyze their dynamical behavior and excitations. The research results of this paper comprehensively explore for the first time the structures of localized waves under the simultaneous influence of nonlocal nonlinearity, diffraction, and external potential. The theoretical framework and the new localized wave structures can provide important references for quantum interferometry, interferometer technology, and the regulation of particle multiple distribution.