Bernard is interested in applying mathematics to physical problems, especially nonlinear wave phenomena. In the past, he has studied problems related to Bose-Einstein condensates, fluid mechanics, plasma physics and lattice dynamics, using a variety of mathematical techniques from such different fields as integrable systems and solitons, dynamical systems, Hamiltonian dynamics, Riemann surfaces and algebraic geometry, Lie algebras, complex variables, asymptotics and perturbation theory.