We introduce and explore a Kerr soliton Ising machine with all-to-all connectivity and up to tripartite (cubic) interactions, unlocking efficient approaches for combinatorial optimization. The machine is an ensemble of solitons in a Kerr resonator, where a programmable optoelectronic feedback circuit generates amplitude mixing terms involving up to three arbitrary soliton spins in the system Hamiltonian. This capability offers a key advantage for solving canonical NP-complete problems like Boolean satisfiability (SAT) with three literal clauses, substantially reducing the spin overhead compared to quadratic-only Ising mappings. To explore our cubic Ising machine, we solve randomly generated 3-SAT instances of 250 variables and 1000 clauses, approaching the intrinsic satisfiability threshold. Our results highlight Kerr solitons as a flexible and energy-efficient analog computing substrate, with the potential to address challenges in combinatorial optimization, artificial intelligence, and machine learning.

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• Analog computation
• Nonlinear optics
• Probabilistic computing
• Coupled oscillators
• Solitons