We study the (2 + 1)-dimensional coupled nonlinear Schrödinger equation with variable coefficients in a graded-index waveguide, and present a combined Akhmediev breather and Kuznetsov-Ma soliton solution with nonautonomous characteristics for certain functional relations. From this solution, by modulating the relation between the maximum effective propagation distance Z max and the periodic locations Z m based on the maximum amplitude of soliton solution, different types of controllable excitation behaviors such as limitation excitation, maintenance and postponement are demonstrated.

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By an improved projective equation approach and a linear variable separation approach, a new family of exact solutions of the (2+1)-dimensional Broek-Kaup system is derived. Based on the derived solitary wave solution and by selecting appropriate functions, some novel localized excitations such as instantaneous solitons and fractal solitons are investigated.

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