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The explicit split-operator algorithm is often used for solving the linearand nonlinear time-dependent Schr\"{o}dinger equations. However, when appliedto certain nonlinear time-dependent Schr\"{o}dinger equations, this algorithmloses time reversibility and second-order accuracy, which makes it veryinefficient. Here, we propose to overcome the limitations of the explicitsplit-operator algorithm by abandoning its explicit nature. We describe afamily of high-order implicit split-operator algorithms that arenorm-conserving, time-reversible, and very efficient. The geometric propertiesof the integrators are proven analytically and demonstrated numerically on thelocal control of a two-dimensional model of retinal. Although they are onlyapplicable to separable Hamiltonians, the implicit split-operator algorithmsare, in this setting, more efficient than the recently proposed integratorsbased on the implicit midpoint method.

An implicit split-operator algorithm for the nonlinear time-dependent Schr\"{o}dinger equation