His scientific interests lied mainly in the area to nonlinear optics. He was one of the founders of modern theoretical nonlinear optics. He obtained, in particular, a number of pioneering results on self-focusing and parametric interactions, which were later confirmed in experiments. In a series of works on self-focusing (since 1966) he has developed for the first time the aberration-free theory of wave self-action, found exact solutions for ray patterns accounting for nonlinear aberrations, and numerically investigated different regimes of spatial resonant self-focusing. Many of his theoretical predictions have been confirmed in experiments: aberrational structures due to wave self-focusing in nematics, limited lifetime of spatial solitons in media with relaxation-type nonlinearity, dynamics of thermal defocusing in the presence of forced and free convection, self-deflection of beams in nonlinear moving media, laser-induced transparency of a cloudy medium and an ozone layer, adaptive compensation of beam distortions in nonlinear media. Anatoly Sukhorukov has derived the general paraxial equation describing beam diffraction in anisotropic crystals, and originated the paraxial theory for three-wave interactions of beams and pulses under the presence of phase- and group-velocity mismatches, diffraction, and group velocity dispersion. He has developed a spatio-temporal analogy between optical beams and pulses, and predicted the effect of diffraction-dispersion incoherence for modulated waves, which limits the efficiency of frequency conversion and results in parametric mutual focusing in quadratic media. He has also predicted the phenomenon of anomalous diffraction in the parametric amplifier, and determined the optimal conditions for high-efficiency generation of the second and third harmonics in focused laser beams. He has first investigated the possibility of total energy conversion in the process of double phase-matched interaction between three frequency harmonics. He has developed the theory of up- and down parametric frequency conversion under the presence of group velocity mismatch, described the formation of three-frequency optical dissipative solitons, parametric pi – pulses, a giant parametric pulse (confirmed experimentally), calculated the modes of the parametrical amplifier with a pulsed pump, developed the theory of stimulated Raman scattering taking into account the relaxation time and group velocity mismatch, found the quasi-phase-matching conditions for slow and fast modes in 3D photonic crystals. He has predicted the existence of a new class of localized structures – parametric three-frequency solitons in media with quadratic nonlinearity (results published in 1974, confirmed experimentally in 1995), also discovered a new type of wave self-focusing, namely, mutual – focusing of just three beams at different frequencies in quadratic media. He has investigated the walk-off effect on quadratic solitons, analyzed dynamics of spatial soliton trapping and interactions in bulk media, in cavities (based on the full round-trip model), in gratings (model for counter-propagating waves); proposed parametric soliton model based on the analogy with coupling of three quasi-particles; considered the tunneling of slow parametric solitons through a bounded grating. He has analyzed the quasi-phase-matched hybrid interactions in periodically polled crystals (layer model), asymmetric modes of parametric solitons, properties of ultra-narrow quadratic solitons, propagation of ultra-short optical pulses inducing plasma generation, and parametric interactions of two non-axial vortices, methods of spatio-temporal vortices generation and recording, all-optical switching with parametric refraction and reflection. He developed the theory of dispersion managed interactions of few-cycle pulses in quadratically nonlinear layered media. He has investigated the nonlinear refraction, total internal reflection and scattering of optical beams and pulses in defocusing media with Kerr, cascaded quadratic, photorefractive, and thermal nonlinearities.
